Planar Ising Correlations

Progress in Mathematical Physics Volume 49 Editors-in-Chief Anne Boutet de Monvel, Université Paris VII Denis Diderot...

Author: John Palmer

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Progress in Mathematical Physics Volume 49

Editors-in-Chief Anne Boutet de Monvel, Université Paris VII Denis Diderot Gerald Kaiser, Center for Signals and Waves, Austin, TX

Editorial Board Sir M. Berry, University of Bristol C. Berenstein, University of Maryland, College Park P. Blanchard, Universität Bielefeld M. Eastwood, University of Adelaide A.S. Fokas, Imperial College of Science, Technology and Medicine C. Tracy, University of California, Davis

John Palmer

Planar Ising Correlations

Birkhäuser Boston • Basel • Berlin

John Palmer Department of Mathematics University of Arizona Tucson, AZ 85721 U.S.A. [email protected]

Mathematics Subject Classifications: 14Dxx, 81Qxx, 82Bxx, 82B20, 82B26, 82B44 Library of Congress Control Number: 2007927588 ISBN-13: 978-0-8176-4248-8

e-ISBN-13: 978-0-8176-4620-2

Printed on acid-free paper. c 2007 Birkhäuser Boston

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 www.birkhauser.com

(Lap/SB)

This book is dedicated to my family, the inimitable Racheli, Uri and Tal

Preface

This book is devoted to a study of the correlation functions for the two-dimensional Ising model on Z2 . This is a simplified model of a ferromagnetic nearest-neighbor interaction for spins on a two-dimensional lattice. The setting for the model is the equilibrium statistical mechanics of Gibbs, and the feature of the model that gives special interest to its study is the existence of a phase transition at finite temperature Tc . In Gibbs’s statistical mechanics one starts with a finite system and is given a function E that assigns to each configuration σ of an energy E (σ ). For the Ising model is a finite subset (think of a rectangle) of the integer lattice Z2 = {(m, n) : m, n = 0, ±1, ±2, . . .}, and a configuration σ is an assignment of +1 or −1 to each site in . Thus one can think of a configuration as a map, σ : → {+1, −1}. In one interpretation of the model +1 is “spin up” and −1 is “spin down.” The energy function of interest is Ji,j σi σj , E (σ ) = − i,j ∈

where the notation signifies that the sum is over all nearest-neighbor configurations i, j in . Two sites i, j are said to be nearest neighbors if i − j = (±1, 0) or i − j = (0, ±1). The real numbers Ji,j are called interaction constants and the interaction is said to be ferromagnetic if Ji,j ≥ 0. Other terms are natural to

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incorporate in the energy, such as a magnetic field term − i bσi , which for b > 0 tends to force spin-up configurations. However, the mathematical technique we employ does not work for b = 0 and is effective mostly for the translation-invariant situation in which there is one horizontal interaction constant J(0,0),(1,0) = J1 , and one vertical interaction constant J(0,0),(0,1) = J2 , so we will right away confine our remarks to this case. For ferromagnetic interactions the energy is minimized by the configurations that are either all +1 or all −1. Gibbs’s model for a system in equilibrium at temperature T is probabilistic. Each configuration σ occurs with a probability proportional to the Boltzmann weight, w(σ ) = e−E (σ )/kB T , where kB is Boltzmann’s constant, a conversion factor relating thermal and kinetic energy. More specifically, the probability that σ occurs is the normalized measure (the Gibbs distribution), w(σ ) , σ ∈C() w(σ )

Pr (σ ) =

where C() is the set of all configurations of . Note that for ferromagnetic interactions the Boltzmann weight favors configurations in which there are many nearest neighbors that are aligned. The behavior of the Boltzmann weights as a function of temperature is also of interest. For small values of T the differences between the weights assigned to different configurations are accentuated; for large values of T the difference between weights assigned to different configurations is washed out. The tension between the propensity of spins to line up at low temperature and the “randomizing” effect of high temperatures is resolved in a surprising way for this model. Observe that the value of a configuration σ at a site i is a natural ±1-valued random variable and all the probabilistic information in the model is contained in the correlation functions σi1 · · · σin = σi1 · · · σin Pr (σ ). σ ∈C()

Thermodynamic regularities are expected to arise only in the limit of very large systems, and one is led to consider the thermodynamic limit, lim σi1 · · · σin .

↑Z2

For the two-dimensional Ising model this limit is quite subtle. To understand the subtlety we return to a point that we glossed over in our description of the energy

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ix

function above. We wrote the energy function as a sum over nearest neighbors in , but to understand what happens in the thermodynamic limit it is better also to allow nearest neighbors i, j in which one of the sites, say i, is in and the other site j is not. To give the energy function meaning in this case, we need to assign spin values to sites on the “boundary” of (sites that are not in but that are nearest neighbors of sites in ). The surprising fact about the twodimensional Ising model is that for T less than a certain critical temperature Tc , the thermodynamic correlations depend on the boundary conditions, and for T greater than the critical temperature they do not depend on the boundary conditions. More specifically, if we let ·+ denote the expectation with plus boundary conditions (all the sites on the boundary of are assigned spin values +1), then writing σ = σ0 for the spin at site 0 = (0, 0), one has lim σ + >0

for T < Tc

lim σ + =0

for T > Tc .

↑Z2

and ↑Z2

Thus below the critical temperature, order can triumph through the boundary conditions, but above Tc thermal disorder triumphs. In fact, disorder triumphs above Tc not only for plus boundary conditions but for all choices [79], [6]. The critical value Tc is perhaps the simplest manifestation of a phase transition. It is also an example of what is called spontaneous symmetry breaking. To a mathematician schooled in group representations a “broken symmetry” sounds like no symmetry at all, but for the Ising model it is easy to explain what is going on. The energy function E (σ ) has a spin flip symmetry—the energy is unchanged by the substitution σ → −σ . Naively, one might anticipate that the expected spin σ would reflect this symmetry in the infinite-volume limit. However, to be invariant under σ → −σ , the expected value σ must be 0. The fact that there are thermodynamic equilibrium states for the Ising model in which σ = 0 is referred to as broken symmetry. In physics, broken symmetries sometimes seem to play a role that is more important than actual symmetries (see the contribution of Witten in [46]). For what appear to be accidental reasons, the correlations of the two-dimensional Ising model can be studied in microscopic mathematical detail in a neighborhood of the critical point. There is no other local model that incorporates a phase transition that can be analyzed at anything like the resolution that is possible for the Ising model, and for this reason the model has attracted enormous interest from physicists trying to understand critical phenomena. More than a thousand papers have been written on the subject, and this thwarts any attempt to summarize what is known and even makes it hard to give a fair account of the history. To get some feeling for the early history of the subject there are a number of review articles [100], [23], [102]. The book The Two-Dimensional Ising Model [89] by McCoy and Wu gives an account of what was known about the model before 1973. It details the calculation of various important quantities for the model in what is

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referred to as the “Pfaffian” approach. We refer the reader to this book for an excellent account of these calculations and also for a more complete discussion of the features of the model that are important for physics. Not too long after McCoy and Wu published their book, Wu, McCoy, Tracy, and Barouch made a discovery that illuminated a new facet of the “solvability” of the Ising model. They found [159] that the asymptotics of the two-point correlation functions near the critical point could be neatly expressed in terms of Painlevé functions of the third kind (see (6.78) and (6.79) below). This discovery inspired Sato, Miwa, and Jimbo to formulate their theory of holonomic quantum fields [137], a class of field theories that include the scaling limit of the Ising model and for which the expression of correlation functions in terms of solutions to holonomic differential equations is a general feature. A mathematical account of the Sato, Miwa, and Jimbo theory as it applies to the Ising model is the central concern of this book. In writing this book, I adopt a somewhat different attitude toward the subject from what is typical in the physics literature. The mathematical calculations that lead to exact results for quantities like the spontaneous magnetization (i.e., σ + above) have a complexity to them that many physicists writing on the subject feel obscures the essential physics. Emphasis has usually been on either simplifying derivations or bypassing them altogether with heuristics that give the right results. Finding these heuristics has been the source of much progress in the theory of critical phenomena [158], [68]. The reader should keep in mind that the scientific impact of the study of Ising models depends much less on the details of solvability than that the model provides a test bed for interesting conjectures. Not being a physicist, I take license in this book to celebrate the variety of mathematics that is brought to bear in giving an account of the Sato, Miwa, and Jimbo theory from its humble beginnings on a finite lattice to its incarnation as a Euclidean field theory on R 2 . I believe it an attractive feature of this model that algebra, analysis, and geometry all have some part to play in describing their theory. In algebra there is the combinatorics of duality, the theory of Clifford algebras, Pfaffians, and the Grassmann calculus for spin representations. Elementary functional analysis is used to control the infinite volume and scaling limits and to formulate the appropriate setting for the infinite spin representations of the orthogonal group. It is also used to prove existence results for the Green functions of the twisted Dirac operators that eventually figure in our understanding of the deformation analysis of the scaling functions. The function-theoretic geometry of elliptic curves plays a role in obtaining a more complete understanding of the one- and two-point correlations. The geometry of the det ∗ line bundle over the restricted Grassmannian is the setting for the introduction of τ functions. The connection between Green functions and projections associated with the localization of Dirac operators is important in formulating the “transfer matrix” account of the twisted Dirac operators. Vector bundles play a role in giving an alternative account of the significance of “monodromy-preserving deformations.” There is more that could be said, but in short, the two-dimensional Ising model is a playground

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of mathematical ideas. My goal is to provide an accessible account of the Sato, Miwa, and Jimbo theory to students of mathematics and physics, which, however, is slanted toward overcoming the difficulties that the mathematicians might have with a subject that has mostly evolved in the different culture of theoretical physics. It is my feeling that such students can benefit from seeing Onsager and Kaufmann’s original analysis of the transfer matrix of the Ising model play out in beautiful and unexpected mathematical developments. That this is a useful program for a practicing physicist is a proposition that I am not prepared to defend too strongly. For a physicist schooled in local operator product expansions, Kadanoff and Kamoto’s account [70] of the SMJ theory has much to recommend it. In their paper, all the “magic” is condensed into a few local operator expansions. If one is ready to accept these expansions then this is a very concise account. The original SMJ papers also employ local operator product expansions to the same end [136] but are not so concise because they expend considerable effort to justify these expansions using the Grassmann calculus developed in their first paper [133]. It is possible to develop the SMJ theory from a mathematical perspective that focuses on the lattice analogues of the local operator product expansions [118]. This is interesting and leads to the McCoy, Wu, and Perk [92], [90], [120], finite difference analogues of the SMJ deformation equations. However, controlling the continuum limit in this formalism is clumsy, and the nature of the tau function is quite obscure. The det ∗ formalism developed in this book has the advantage that the tau function is defined in a conceptually transparent manner, and the link between the tau function and its deformation-theoretic analysis is not magical but straightforward. This analysis was inspired by the Segal–Wilson work on KdV [142]. It could be of interest to a physicist that the central object in our analysis is a twisted Dirac operator; this is an object that also arises in path integral versions of the subject [37]. In addition to its mathematical interest our account can also be read as a primer on the sort of results one would like to have for other more complicated models. The techniques we use to prove results for the Ising model are too special to shed light on the analysis of more complicated models, but there is the expectation that aspects of what is true for the Ising model will also be true for more-realistic models. That the desired results are theorems for the Ising model makes these expectations more accessible to the mathematician. In the rest of this preface I present an account of what is to be found in the book with a brief recounting of history that is intended to help orient the reader. The story of the Ising model begins in 1925 [61] when Ising, a student of Lenz, determined that the one-dimensional model does not have a phase transition. In 1936 Peierls gave an argument to show that the two-dimensional model does have a phase transition. In 1941 Kramers and Wannier determined the critical temperature for this model based on a duality property [76], [77]. A breakthrough in making exact calculations in the model came in 1944 when Onsager determined the free energy per site in the thermodynamic limit [105] and showed that it had a logarithmic singularity at a critical temperature Tc . This was the first calculation of a thermodynamic quantity in a local model that exhibited a singularity, and it provided additional hard evidence that the infinite-volume limit was the appropriate

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place to study phase transitions. Onsager’s analysis was simplified by Kaufmann in [72]. She noted that the transfer matrix for periodic boundary conditions, which is the central player in Onsager’s analysis of the model, can best be understood as an element in a spin representation of the orthogonal group. Her work is the progenitor of the first chapter of our book. It turns out that not only is the transfer matrix an element of the spin representation of the orthogonal group but so is the spin operator (note that the “spin” in “spin operator” has nothing to do with the “spin” in “spin representation”). What’s more, as long as T = Tc , this remains true in the thermodynamic limit—both the transfer matrix and the spin operator are elements of an infinite-dimensional spin representation of the orthogonal group. From this point of view, understanding the infinite spin representation is the key to understanding the Ising correlations. In [133] Sato, Miwa, and Jimbo present a symbol calculus for the finite-dimensional spin representations of the orthogonal group that they use to suggest series expansions for the infinite-volume Ising correlations by analogy. We present their theory inAppendixAof this book together with an infinite-dimensional generalization that is appropriate for studying both the infinite-volume limits of the Ising correlations and later the scaling functions that describe the large-scale asymptotics of the correlations near the critial temperature. The Grassmann calculus that is developed there is, I believe, interesting in its own right and is intimately connected with the algebraic construct known as a Pfaffian, an algebraic square root of the determinant of a skew-symmetric matrix. A deeper explanation of this is given by Segal and Pressley’s Borel–Weil construction of the spin representation of an orthogonal group as the space of holomorphic sections of a Pfaffian bundle over a Grassmannian (a manifold that can be realized as a collection of subspaces of a vector space) [126]. We don’t consider this Borel–Weil construction, but Segal and Wilson’s related determinant bundle construction [142] is at the heart of the tau function analysis in Chapter 4. Interestingly enough, the Pfaffians that arise here are not the Pfaffians that arise in the so-called Pfaffian approach to the Ising model. The connection is understood [111] but has not been systematically developed. Chapter 1 applies elementary functional analysis to show that the infinite-volume Ising correlations for plus boundary conditions below Tc are Fock expectations of spin operators in an infinite-spin representation of the orthogonal group. The ingredients that define this representation are a Hilbert space W , a spin operator s, and a transfer matrix T (V ) that act on W , and a polarization (splitting) of W , W = W+ ⊕ W− . The matrix elements of s relative to this splitting are written A B s= , C D and the Grassmann calculus requires the inversion of the matrix element D to find explicit formulas for the spin correlations. The Wiener–Hopf technique is explained and used to obtain formulas for D −1 . Kramers–Wannier duality in an

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xiii

incarnation due to Kadanoff and Ceva [69] is used to give a representation of the spin correlations above Tc as correlations of disorder operators in the Fock space at the dual temperature below Tc . The chapter concludes with a look at some Clifford algebra formulas for even-spin correlations that are specialized to a Toeplitz determinant representation for the two-point correlations along a horizontal line due to Potts and Ward [125]. More might be done with these representations, which, unlike our other representations, do specialize to the critical temperature, but we pursue this later only to give the Montroll, Potts, and Ward derivation of the spontaneous magnetization. Although some proofs have been improved, this chapter is largely based on Palmer and Tracy [117]. The spectral analysis of the plus state transfer matrix on a finite lattice is adapted from Abraham and Martin Löf [5]. After the free energy per site, the next quantity to be explicitly computed for the Ising model was the spontaneous magnetization, M. There is more than one way to think of this quantity, but for us it is defined as the limit of σ + as tends to Z2 . In 1949, Onsager [106] announced the formula, 1

M = (1 − k 2 )− 8 , where k −1 = sh

2J1 kB T

sh

2J2 kB T

and T < Tc . The reader can find a graph of this function of T in Figure 2.2 below; M falls precipitously from approximately 1 well below Tc to 0 at and above Tc . Yang published the first derivation of the formula for M in [161]. In this paper Yang mentions that the elliptic substitutions that are central to his calculations were communicated to him in conversations with Onsager and Kaufmann. Onsager’s comfort level with special functions is attributed in a story to his having worked all the problems in Whittaker and Watson’s Modern Analysis. In honor of this story we will use Modern Analysis as our reference for the Jacobi theory of elliptic functions, which figures prominently in Chapter 2. Chapter 2 begins with a discussion of the uniformization of an elliptic curve that leads to the elliptic substitutions used by Yang. The elliptic curve in question is a complexification of the energy–momentum relation for the induced rotation of the transfer matrix—we refer to it more simply as the spectral curve, M. Jacobi’s version of elliptic function theory is ideally suited to the uniformization of the spectral curve given in Theorem 2.2.1 below. The uniformization of the complex spectral curve plays a surprising role in understanding the matrix elements of the spin operator relative to the polarization that defines the representation of the orthogonal group relevant to the plus state correlations of the Ising model below Tc . There is a geometric model for the polarization of the Hilbert space W = W+ ⊕ W− in which W is realized as the space of L2 sections of a line bundle E over a pair of cycles (images of circles) M± on the spectral curve. The subspace W+ (W− ) is just the space of L2 sections of E that are supported on M+ (M− ). Relative to an appropriate trivialization of the bundle E the matrix elements A, B, C, and D of s are all seen to be convolution operators

xiv

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with respect to the uniformization parameter (Theorem 2.4.1). These convolution operators are diagonalized by appropriate Fourier series. In particular the formula M 2 = det |D|, which was proved in Chapter 1, can now be explicitly computed, and we obtain the Onsager–Yang formula for the spontaneous magnetization (Theorem 2.4.2). The proof that Yang gave requires an inspired identification of the spontaneous magnetization. However, the diagonalization of the integral operator that follows this identification in his paper suggested the treatment in the first part of Chapter 2. The second part of Chapter 2 is devoted to some explicit expansions of the infinite-volume two-point Ising correlations that were first found in Palmer and Tracy [117] and Yamada [160]. The Fock space technique that we use to obtain these formulas is the same as can be found in Abraham [2]. The formulas we write down are essentially Fredholm expansions for infinite Pfaffians. A crucial simplification of these expansions arises from the explicit evaluation of certain finite-dimensional Pfaffians as products. The proofs of these formulas are exercises in elliptic function technology—matching poles and zeros. Chapter 2 closes with a discussion of the elegant Montroll, Potts, and Ward calculation of the spontaneous magnetization using a Szegö-type theorem on the asymptotics of Toeplitz determinants. Szegö apparently became interested in refinements in the study of Toeplitz determinants through conversations with Onsager (both were at Yale). The extensive development of the theory of Toeplitz determinants [27] that has emerged since then thus has its roots in Ising model calculations. Chapter 3 begins the analysis of the scaling function for the Ising correlations. We move to a pre-elliptic, “hyperbolic” version of the spectral representation that was the focus of Chapter 2. This allows us to concentrate on the behavior of a single auxiliary function γ in a neighborhood of 0 in momentum space for all the estimates needed to control the scaling limits. There are separate horizontal and vertical scales for the correlations, and with appropriate choices we see that the scaling functions no longer depend on the relative sizes of the horizontal and vertical interaction strengths. We see later that the scaling functions are actually rotationally invariant (Theorem 6.5.5). We find infinite Pfaffian formulas for the subcritical and supercritical scaling functions (Theorems 3.4.1 and 3.4.2). Next we take up the probabilistic interpretation for the scaling functions. Minlos’s theorem [146] is used to prove that the scaling functions are associated with cylinder set measures on the space of tempered distributions S ′ (R 2 ). This construction is also the place to look for the connection with central-limit-theorem-type behavior that guides physicists thinking about some aspects of critical phenomena. The chapter concludes with “hyperbolic” representations for the two-point functions and the corresponding expansions for the two-point scaling functions. Using the Fredholm determinant representation of these results, the two-point scaling functions have been analyzed in detail; in particular Tracy, and Tracy and Widom, have done the short-distance analysis [153], [154].

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Chapter 4 begins with a discussion of a finite difference equation that is intimately connected with the fermions that arise in the description of the Ising correlations. We show that a mixed fermion–spin correlation is the Green function for a finite difference operator that scales to a Dirac operator. Our goal is a characterization of the subcritical and supercritical scaling functions in terms of this Dirac operator. Missing from our treatment is a characterization of the lattice correlations in terms of the associated finite difference operator. This has been done for the Pfaffian formalism in [111]; the correlations are shown to be Pfaffians of finite difference operators acting on Z2 . However, the relation between these finite difference operators and the finite difference operators that appear in the transfer formalism, while straightforward, is a bit of a story in its own right. Because we don’t know how to tell this story in a way that makes the proof of the connection between the scaling functions and the tau functions of SMJ any clearer or shorter, we leave the contemplation of this matter to the interested reader. The Dirac operator of interest acts on “functions” that are branched at the spin sites with a transition map given by multiplication by −1 along the branch cut. The rest of the chapter is devoted to an analysis of the Green function for the Dirac operator acting on the sections of a bundle whose smooth sections have the appropriate branching behavior at a single site a ∈ C. The lattice Green function is seen to have a “real symmetry” that singles out a specific choice for this Dirac Green function. This Dirac operator is relevant for subcritical scaling. There is no confusion with the supercritical case since the boundary conditions relevant for supercritical scaling produce a Dirac operator that fails to be invertible when there is an odd number of spin sites (or branch cuts). Thus there is no “one-point” supercritical Green function. It is simple to construct the rotational eigenfunction expansions for this Green function, but some work is required to translate this to the contour integral representations for the Green function that are used to make the connection with the transfer formalism that emerged naturally on the lattice. We obtain contour integral representations for the rotational eigenfunctions for the Dirac equation, and the freedom to deform these contours plays a very important role in the calculations that take one from “radial quantization” to “transfer matrix quantization,” which is the subtext for this chapter. The “transfer formalism” for the Dirac operator localizes the operator in the complement of the horizontal strip containing the spin site a (the branch point). The boundary condition that defines the Dirac operator is 1 given by a subspace of H 2 (∂S), the Sobolev space of order 21 for ∂S, the boundary of the strip S. A natural projection on this subspace is defined in terms of the Green function for the Dirac operator. The formula for this projection is the glue that connects the scaling limit of the Ising correlations with the transfer formalism for solutions of the appropriately twisted Dirac operator. The next chapter is devoted to an account of the tau functions associated with the Dirac operator for n branch points. The subspaces that define the localization of the Dirac operator to the complement of the n strips that contain the branch points 1 are elements of a restricted Grassmannian of subspaces in H 2 (∂S), where ∂S is

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the boundary of the union of the strips S. Over this Grassmannian there is a determinant line bundle, first defined by Segal and Wilson [142]. Following Segal and Wilson’s definition for tau functions associated with the KdV equation we introduce tau functions for the Dirac operators defined for subcritical and supercritical boundary conditions, by introducing appropriate trivializations over restrictions of the determinant line bundle. The principal results are Theorems 5.1.1 and 5.2.1, which show that the squares of the subcritical and supercritical scaling functions are related to tau functions. The squares arise because the det ∗ formalism is not exactly the correct one for this problem. The appropriate formalism involves Pfaffian bundles instead of determinant bundles. However, because we use the tau function formalism to analyze the logarithmic derivative of the scaling functions and this is just altered by a factor of two, we did not take the trouble here to introduce the slightly more complicated Pfaffian formulation (which the reader can find explained in [111]). The proof of the relation between the scaling functions and the tau functions comes down to the observation that a natural “transfer matrix representation” of the tau function has the same structure as the representation of the scaling functions based on the SMJ product deformation formalism described in Appendix A. The identification of the matrix elements of the one-point subspace projection of Theorem 4.6.1 with the Grassmann calculus ingredients for the description of the spin operator detailed in Theorem 5.1.1 completes the analysis. Some complications arise from the fact that the transfer formalism natural for the Dirac operator does not precisely match up with the SMJ product deformation formalism (as interpreted in Theorem A.7.4). The reason for the difference is obvious but still manages to be irritating. The payoff for the connection between scaling functions and tau functions is that we can prove that the SMJ monodromy-preserving deformation analysis applies to the tau functions. Chapter 6 is the heart of the book and is devoted to an account of this analysis. We first prove some existence results for the subcritical Green functions that were used in the preceding chapter. Then we show that the derivative of this Green function with respect to the branch points is a finite-rank operator (Theorem 6.3.3). From our perspective this is the crucial result that allows one to make a deformation-theoretic analysis of the logarithmic derivative of the tau function. We then give a trace formula for the logarithmic derivative of the subcritical tau function and use the derivative formula for the Green function to rewrite the trace in terms of local expansion coefficients of Dirac wave functions Wj , j = 1, . . . , n (Theorem 6.4.3). From this point on, the analysis follows that in the third paper in the SMJ series on holonomic fields quite closely. We show that there is a canonical basis {wj (x)} for the L2 solutions to the Dirac equation with prescribed monodromy −1 at branch points a1 , . . . , an ∈ C. The infinitesimal rotational symmetries of the Dirac equation extend the equations satisfied by w = (w1 , . . . , wn ) in the x variable so that it becomes a horizontal section for a flat connection with regular singularities at a1 , . . . , an (Theorem 6.5.1). In fact, w(x) also depends smoothly on the ai variables and is a horizontal section for a flat connection in the (x, a) variables (Theorem 6.5.2). Following SMJ we show (Theorem 6.5.3) that the coefficients of this connection satisfy nonlinear

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deformation equations, which in turn guarantee that the connection constructed from them is flat. The wave functions Wj can be expressed in terms of the canonical basis wj , and this allows us to find the local expansion coefficients for Wj in deformation-theoretic terms. Hence we also see that the logarithmic derivative of the tau function can be expressed in deformation-theoretic terms (Theorem 6.4.3). This is the Sato, Miwa, and Jimbo generalization of the Wu, McCoy, Tracy, and Barouch result for the two-point scaling functions. We also use the deformation analysis to prove that the scaling functions for the Ising model have unique rotationally invariant extensions off the limited domain on which we are able to prove convergence using the transfer matrix method (Theorem 6.5.5). Finally, we indicate how the Dirac operator technology has been used to prove the Luther– Peschel formula (and a generalization to the subcritical odd correlations) for the short-distance behavior of the n-point scaling functions. We conclude this introduction with a brief account of some of the recent and not so recent developments for the two-dimensional Ising model that are not discussed in this book. Periodic boundary conditions make it possible to discuss the Ising model on discrete cylinders and tori [30], [34]. Continuum limits of these models give rise to Euclidean field theories on the cylinder and the torus that are distinct from the field theory on R 2 that we study [81]. It is possible to formulate continuum versions of the Ising model for other two-dimensional symmetric spaces. This has been studied for the Poincaré disk [116], [50] with some results for the Riemann sphere as well. It is possible to solve the Ising model on a half-space so that boundary effects can be studied [16]. The Ising model has been solved on lattices other than the square lattice [42]. There is a development of discrete function theory in which the duality for the Ising model has a suggestive place [95]. The diagonal transfer matrices of the Ising model commute for common values of the elliptic modulus k; Onsager’s observation of this fact led Baxter to formulate a far-reaching generalization of solvability for lattice models that one can read about in his book [20]. This in turn led to the development of “quantum inverse scattering” [75] and quantum affine algebras [38]. The corner transfer matrix formalism of Baxter specializes to the Ising model in a fashion that produces a discrete model with properties that mimic conformal field theory [54]. There is a “Z invariant” formulation of the Ising model [21] that allows “solvable” deformations away from translation invariance (see [127], [9]). Analytic properties of the susceptibility have recently been investigated in some detail [13], [162]. The symmetries of the conformal field theory associated with the Ising model have been investigated as a first step in understanding the special nature of such symmetries for twodimensional local field theories [83]. The deformation theory for the Ising model also turns out to be relevant for the discussion of the moduli space of ground states for N = 2 supersymmetric field theories [37]. The behavior of the model with random interaction strengths is of interest and has been studied [89]. This incomplete list makes it clear that in this book we study only a narrowly circumscribed part of the story of the Ising model. Even so, to give a mathematical account of the Ising correlations on Z2 we must harvest the fruit of several generations of work by mathematical physicists.

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Being able to contemplate this model from so many different perspectives gives its study some of the wonder in the first lines of Blake’s “Auguries of Innocence,” To see a World in a grain of sand And Heaven in a wild flower, To hold Infinity in the palm of your hand And Eternity in an hour. If this book engages the reader’s curiosity in the pleasurable mysteries of the Ising model, a subject with facets that are illuminated by a surprising spectrum of mathematical ideas, it will have served its purpose.

Acknowledgements I give special thanks to Craig Tracy, Doug Pickrell andAlan Carey. Craig first taught me about the lsing model. He is a good friend and it is a delight to collaborate with him. Over the years Doug patiently instructed me in the advantages of a geometric point of view. He did his best to turn me into a mathematician. Alan is a pleasure to work with and I thank him for the opportunity to visit him in Australia.

Contents

Preface 1 The Thermodynamic Limit 1.1 The Transfer Matrix in a Pure Phase . . . . . . . 1.2 The Semi-infinite-Volume Limit . . . . . . . . . 1.2.1 Induced Rotations . . . . . . . . . . . . 1.2.2 Fock Representations . . . . . . . . . . . 1.2.3 The Spectrum of TM . . . . . . . . . . . 1.3 The Thermodynamic Limit for T < Tc . . . . . . 1.4 Duality and the Thermodynamic Limit Above Tc 1.5 Even Correlations at All Temperatures . . . . . .

vii . . . . . . . .

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1 1 8 9 13 21 30 45 58

2 The Spontaneous Magnetization and Two-Point Spin Correlation 2.1 The Spectral Curve for the Transfer Matrix . . . . . . . . . . . 2.2 The Uniformization of M . . . . . . . . . . . . . . . . . . . . 2.3 Trivializing E over M± . . . . . . . . . . . . . . . . . . . . . . 2.4 The Matrix Elements of s for the Q Polarization . . . . . . . . . 2.5 The Two-Point Function Below Tc . . . . . . . . . . . . . . . . 2.6 The Two-Point Function Above Tc . . . . . . . . . . . . . . . . 2.7 The Correlation Length and Two-Point Asymptotics . . . . . . . 2.8 The Montroll, Potts, and Ward Calculation of σ . . . . . . . .

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63 64 68 72 74 83 89 96 101

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3 Scaling Limits 105 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Hyperbolic Formulas for the Kernels of D −1 , BD −1 , and D −1 C . 110

xx

Contents

3.3 Convergence Estimates . . . . . . . . . . . . . . . 3.4 Pfaffian Formulas for the Scaling Functions . . . . 3.5 A Probabilistic Interpretation for the Scaling Limits 3.6 Expansions for the Two-Point Scaling Functions . . 4 The One-Point Green Function 4.1 Introduction . . . . . . . . . . . . . . . . 4.2 Free Fermions . . . . . . . . . . . . . . . 4.3 Contour Integral Representations . . . . . 4.4 Local Expansions . . . . . . . . . . . . . 4.5 The Green Function for One Branch Point 4.6 Localization and Projections . . . . . . .

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113 124 131 140

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147 147 149 160 168 175 181

5 Scaling Functions as Tau Functions 197 5.1 The Subcritical Tau Function . . . . . . . . . . . . . . . . . . . . 198 5.2 The Supercritical Tau Function . . . . . . . . . . . . . . . . . . . 211 6 Deformation Analysis of Tau Functions 6.1 Boundary Conditions for m − D on E . . . . . . . . . . . . . . 6.2 Existence Theory for the Subcritical Green Function . . . . . . 6.3 Existence for the Basis {Wj } and the Derivative of G(x, y; a) . 6.4 The Derivative of log τ . . . . . . . . . . . . . . . . . . . . . . 6.5 Holonomic Systems and the Deformation Equations . . . . . . . 6.5.1 A Deformation-Theoretic Expression for d log τ . . . . . 6.5.2 Rotational Invariance for τ [±] . . . . . . . . . . . . . . 6.5.3 The MTWB Result for the Two-Point Scaling Functions 6.6 Short-Distance Behavior of the Scaling Functions . . . . . . . .

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223 224 236 240 247 252 261 264 267 271

A Spin Representations of the Orthogonal Group A.1 Grassmann Algebra . . . . . . . . . . . . . . A.2 Fock Representations of the Clifford Algebra A.3 The Grassmann Calculus . . . . . . . . . . . A.4 Pfaffians . . . . . . . . . . . . . . . . . . . . A.5 Vacuum Expectations for Elements of G . . . A.6 The SMJ Product Deformation Formalism . . A.7 Infinite Dimensions . . . . . . . . . . . . . .

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273 273 277 279 293 304 310 319

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B Relations Between Transforms 337 B.1 Boltzmann Weights . . . . . . . . . . . . . . . . . . . . . . . . . 337 B.2 Representations of the Complex Orthogonal Space W . . . . . . . 338 References

345

List of Notation

357

Index

361

1 The Thermodynamic Limit

1.1 The Transfer Matrix in a Pure Phase The two-dimensional Ising model that we study is a model for the equilibrium distribution of spins (or magnetic moments) that sit on a two-dimensional lattice and interact ferromagnetically with their nearest neighbors. More specifically, let Z2 denote the lattice of pairs (i1 , i2 ) of integers ij ∈ Z. We need to start by restricting our attention to a finite part of the lattice Z2 , and so for M and N positive integers we introduce M,N = {(i1 , i2 ) ∈ Z2 : |i1 | ≤ M, |i2 | ≤ N }, which we also refer to by for brevity. A configuration σ of spins on the set is an assignment of +1 (spin up) or −1 (spin down) to each site in . In other words, such a spin configuration is a map σ : → {+1, −1}. The interaction between spins in a configuration is described by assigning an energy E (σ ) to each configuration σ . In the model we wish to look at, the configuration energy is given by E (σ ) = − Jij σi σj . (1.1) i,j ⊂

Here Jij is a real-valued function of the pair of sites (i, j ), and the notation i, j ⊂ means that the sum on the right is restricted to pairs of sites (i, j ) that are nearest neighbors; i is one lattice unit away from j in either the horizontal or

2

1. The Thermodynamic Limit

vertical direction with both sites i and j in . Many other interesting terms might be admitted in the interaction energy such as h σi , which reflects an external magnetic field h. However, the special mathematical techniques we employ do not generalize to this case, and so we confine our attention to energy functions that are specializations of (1.1). We now consider the statistical ensemble of all possible configurations σ on , each configuration occurring with probability proportional to the Boltzmann weight E (σ ) exp − , kB T

where kB is the Boltzmann constant and T is temperature. It is an observation of Gibbs that the statistics of this ensemble ought to reproduce the equilibrium thermodynamics of a classical system with energy function E at temperature T (why this is so is a deep question, but the reader can find an interesting mathematical discussion of the matter in chapter 5 of Haag [59]). The total weight, E (σ ) Z = exp − , kB T σ ∈C

with C the set of all configurations on is called the partition function. Now suppose that A is a subset of . We define σA = σi , i∈A

the product of the spin variables at the sites in A. The expected value of σA is given by E (σ ) 1 σA = , (1.2) σA exp − Z σ ∈C kB T

and is referred to as a spin correlation function. Almost everything we do in this book is concerned with understanding these correlations in the limit → Z2 for a very special choice of interaction constants: the translation-invariant choice Jij = J1 > 0 if i and j are horizontally separated and Jij = J2 > 0 if i and j are vertically separated. A first reason for the interest in the limit → Z2 is that real systems in the bulk have a huge number of constituent atoms; there are approximately 6 × 1023 atoms (Avogadro’s number) in a 4-gram sample of helium (the atomic weight for helium is roughly 4). Consequently, one might expect that statistical regularities in the behavior of large systems (i.e., thermodynamics) could show up in the infinitevolume limit → Z2 , which is also referred to as the thermodynamic limit for this reason. Furthermore, if one examines the formula for the correlation function σA , it is evidently an analytic function of T for T = 0 (and near the real axis where the denominator is not 0). However, the phenomena associated with what are called phase transitions clearly involve thermodynamic quantities that do not

1.1 The Transfer Matrix in a Pure Phase

3

behave analytically as a function of temperature. An example is provided by the magnetization of a bar magnet as a function of temperature. If you heat an iron bar magnet, its magnetic field steadily decreases with temperature until it reaches 0 at a temperature that is called the Curie point (for iron this is about 768 degrees centigrade). For all temperatures above the Curie point the magnetic field of the iron bar is 0. Nontrivial analytic functions cannot vanish on an interval, so it is impossible to see this ideal behavior in our model for the one-point function σi (the average magnetization) on a finite lattice . Nonetheless, it is a remarkable fact that the Ising model does exhibit such nonanalytic behavior in the limit → Z2 . Onsager’s calculation of the infinitevolume limit of the free energy per site in this model in 1944 provided the first example of a such a nonanalytic thermodynamic function. It also fueled interest in the model as a test bed for the notion that the infinite-volume limit provides a suitable framework for understanding phase transitions. The motivation for much of the considerable work that has been done on the model since Onsager’s original paper is a desire for a better understanding of the phase transition that appears in the thermodynamic limit. Before we turn to the mathematical reformulation of the correlations (1.2) that makes possible an analysis of the thermodynamic limit, it might be helpful to make some remarks about what happens in this limit in intuitive terms. Interactions with Jij ≥ 0 are said to be ferromagnetic. For ferromagnetic interactions, each nonzero term −Jij σi σj in the energy is negative when σi and σj are aligned (both +1 or both −1) and positive when σi and σj are unaligned (one is +1 the other −1). In the Boltzmann distribution given above this has the effect of giving more weight to spin configurations in which there are many nearest neighbors that are aligned compared to configurations in which there are many nearest neighbors that are unaligned. One could say that the equilibrium distribution prefers order to disorder. The effect of temperature in the probability distribution is to accentuate the difference in weight between ordered and disordered configurations at low temperature and to wash out the difference in weight between order and disorder at high temperatures; in the limit T → ∞ all configurations are given equal weight. We will see that in the model we are considering there is a critical temperature Tc below which order prevails in the infinite-volume limit and above which disorder prevails. One way to give a precise expression to this phenomenon is to consider the infinite-volume limit of the correlations subject to the restriction that the sum over configurations include only those configurations in which all the spins on the boundary of are +1. These “plus” boundary conditions result in an infinite-volume limit for the one-point function σi that is strictly positive for T < Tc and zero for T > Tc . In fact, for T < Tc the infinite-volume limit of correlation functions is sensitive to the boundary conditions. If instead of plus boundary conditions one admits only configurations that are −1 on the boundary of , the limiting value of the onepoint function is strictly negative for T < Tc ; on the other hand, periodic boundary conditions lead to a limiting value of the one-point function that is 0. For T > Tc it is known [79] that the infinite-volume limit of the correlations does not depend on the boundary conditions.

4


There is an elegant thermodynamic formalism that incorporates a theory of these infinite-volume “Gibbs states” (see Ruelle [130]). In the context of this theory it is known that for T < Tc , every Gibbs state for the model is a convex combination of the “plus” and “minus” states. Above Tc there is a unique Gibbs state. We do not develop this theory here, but we take away from this discussion the idea that the infinite-volume state obtained from plus boundary conditions is interesting. We will see later that this state has a cluster property that is the hallmark of a pure phase. We now introduce the constructions that allow us to analyze the thermodynamic limit for plus boundary conditions below Tc (this critical temperature is identified in the course of the analysis). We use the transfer matrix formalism as adapted to deal with plus boundary conditions by Abraham and Martin-Löf [5]. Let C+ denote the maps from to {−1, 1} that are +1 on the boundary of (so, for example, if σ ∈ C+ , then σ (M, j ) = +1 for all j ∈ {−N, . . . , N}). We now define E (σ ) + (1.3) = Z exp − kB T + σ ∈C

and σA + =

1 E (σ ) , σ exp − A + kB T Z +

(1.4)

σ ∈C

and we suppose henceforth that we are looking at the translation-invariant case, J(i,i±e1 ) = J1 > 0, J(i,i±e2 ) = J2 > 0, where e1 = (1, 0) and e2 = (0, 1). The essence of the transfer matrix formalism is to split the sums over all configurations C+ into a multiple sum over the configurations of the rows. Let C (row) denote the unrestricted space of configurations of a row, that is, the set of maps from {−M, . . . , M} into {−1, 1} (no plus boundary condition). We now introduce “matrices” that allow us to reexpress the sums in (1.3) and (1.4) as multiple sums over the row configurations. Suppose that b ∈ R and ρ, σ ∈ C (row). We define   M−1 V1 (σ ) := exp  (1.5) K1 σj σj +1  , j =−M



where

V2 (ρ, σ ) := e−2b exp  Kα :=

Jα kB T

M

j =−M



K2 (j )ρj σj  ,

for α = 1, 2,

(1.6)


and K2 (j ) :=

b K2

5

for |j | = M, for |j | = M.

Note that a configuration σ in C determines a collection σ i ∈ C (row) for i = −N, . . . , N as follows: σji := σ (i, j ),

the ith row configuration is evaluated at the j th column. Distinguishing the row configurations with superscripts is a device that has some temporary utility. With this understanding it is easy to see that V1 (σ i ) is the Boltzmann weight associated with the horizontal bonds in the ith row for the configuration σ , and apart from the parameter b, V2 (σ i+1 , σ i ) represents the Boltzmann weight associated with the vertical bonds connecting the ith and (i + 1)th rows. Let + denote the element in C (row) whose values are all +1 and define Z (b) = V2 (+, σ N −1 )V1 (σ N −1 ) σ i ∈C (row) (1.7) i=−N+1,...,N−1

× · · · × V1 (σ −N +1 )V2 (σ −N +1 , +), where we used the shorthand notation := σ i ∈C (row) i=−N+1,...,N−1

σ −N+1 σ −N +2

···

,

σ N−2 σ N −1

in which each sum in the multiple sum on the right ranges over C (row). Then we have + Z = lim Z (b). b→+∞

To see this, observe that the terms in Z (b) that involve b can be rewritten as i+1 i i+1 i exp{b(σ−M σ−M − 1)} exp{b(σM σM − 1)}.

Thus the only terms that survive the limit b → +∞ are those for which i+1 i i+1 i σ−M σ−M = σM σM = +1,

for all i = −N, . . . , N − 1. However, since σ N = + and σ −N = + in the sum defining Z (b), it follows that only configurations in C+ survive the limit b → +∞ (Abraham and Martin-Löf [5]). Now write Ai for the intersection of A with the ith row and σAi for the product of the σji with (i, j ) ∈ Ai and σ∅ := 1. Then we have 1 σA + V2 (+, σ N −1 )V1 (σ N−1 )σAN = lim b→+∞ Z (b) i σ ∈C (row) i=−N +1,...,N −1

× · · · × σA−N +1 V1 (σ −N +1 )V2 (σ −N +1 , +). (1.8)

6


The argument that the only configurations that survive the limit b → +∞ are those in C+ is the same as it was for the partition function Z (b). It is natural in (1.7) and (1.8) to regard the multiple sums as arising from the multiplication of matrices associated with V2 (ρ, σ ) and V1 (σ ). To do this it is useful to introduce a vector space that has a basis that is indexed by C (row). The number of elements in C (row) is 22M+1 , so any vector space with this dimension would work. However, it is convenient to introduce the tensor product M

j =−M

C2j := C2−M ⊗ · · · ⊗ C2M

for this purpose. Here C2j is just a copy of C2 . The map that identifies the elements 2 of C (row) with a basis for M j =−M Cj is C (row) ∋ σ → eσ :=

M

j =−M

1+σj 2 1−σj 2

.

So, for example, the configuration + gets mapped into the vector that is the tensor product of 2M + 1 copies of 10 . This is convenient because for σ ∈ C (row), multiplication by σj on eσ is given by the simple tensor product 1 0 · · ⊗ 1 ⊗ σj eσ = 1 ⊗ · ⊗ 1 ⊗ · · · ⊗ 1eσ . (1.9) 0 −1 M+j

If X is a 2 × 2 complex matrix, it is convenient to write

Xj := 1 ⊗ · · · ⊗ 1 ⊗X ⊗ 1 ⊗ · · · ⊗ 1,

(1.10)

M+j

so that (1.9) translates to

σj =

1 0 0 −1

(1.11)

j

in the tensor product space. The simplicity of the action of the “spin operator” σj 2 is the reason for introducing the vector space M j =−M Cj . Next we wish to compute the action of the operators associated with the matrices V1 and V2 . The action of V1 is diagonal in the basis {eσ }, and is given by   M−1 eσ → V1 (σ )eσ = exp K1 σj σj +1  eσ , j =−M

which implies that

V1 = exp

M−1

j =−M

K1 σj σj +1 ,

(1.12)


7

with the understanding that the action of the spin operators σj and σj +1 in this formula is given by (1.11). Now we compute the action of V2 in the tensor product. The standard relation between linear transformations and matrices relative to a basis implies V2 eσ =

ρ

V2 (ρ, σ )eρ = e−2b

= e−2b =e where each sum

−2b

M ρ

j =−M

M

j =−M

M ρ

j =−M

exp K2 (j )ρj σj

exp K2 (j )ρj σj eρ

1+ρj 2 1−ρj 2

exp K2 (j )σj , exp −K2 (j )σj

is over all ρ ∈ C (row). The vectors on the ends have limits 1+σ±M exp(bσ±M ) −b lim e = 1−σ2±M , exp(−bσ±M ) b→+∞

ρ

2

and for j = −M + 1, . . . , M − 1 one has K e 2 exp(K2 σj ) = −K2 exp(−K2 σj ) e

e−K2 e K2

1+σj 2 1−σj 2

.

One need check this only for the two values σj = ±1. Thus in the limit b → +∞ the action of V2 is given by K K e 2 e−K2 e 2 e−K2 lim V2 eσ = 1 ⊗ −K2 ⊗ · · · ⊗ −K2 ⊗ 1eσ . (1.13) e e K2 e e K2 b→+∞ We are interested in the action of V2 only in the limit b → +∞, so from now on we suppose that the action of V2 is given by (1.13). Next observe that we can write K 1 0 1 e 2 e−K2 2 exp K ∗ = sh 2K (1.14) (2 ) 2 2 1 0 , e−K2 eK2 where K2∗ is defined by the relation, sh(2K2 ) sh(2K2∗ ) = 1

(1.15)

and we use the shorthand notation, sh A := sinh A

and

ch A = cosh A.

Then combining (1.13) and (1.14), we see that in the notation of (1.10) we have 1

V2 = (2 sh 2K2 )M− 2 exp

M−1

j =−M+1

K2∗

0 1

1 . 0 j

(1.16)

8


The results (1.11), (1.12), and (1.16) make it possible to translate the expression for the correlations (1.8) into the tensor product representation. Before we do this it is useful to symmetrize the transfer matrix. Based on (1.12) we introduce 1

V12 := exp

M−1 K1 σj σj +1 , 2 j =−M

with σj given by (1.11). Now define a symmetrized transfer matrix VM by 1

1

VM = V12 V2 V12 .

(1.17)

From (1.13) we see that V2 is a positive self-adjoint transformation with respect to the inner product for which {eσ } is an orthonormal basis. The eigenvectors for V2 are just tensor products of eigenvectors for K e 2 e−K2 , e−K2 eK2 with eigenvalues that are products of the positive eigenvalues 2 ch K2 and 2 sh K2 . 1

Since V12 is also clearly self-adjoint, it follows that VM is a positive self-adjoint transformation. We summarize these developments in the following proposition. Proposition 1.1.1 Let A denote a finite collection of sites in . Then the expected value of σA with plus boundary conditions, (1.4), is given by, σA + =

+|VM σAN−1 VM σAN −2 · · · σA−N+1 VM |+ +|VM2N |+

,

(1.18)

where +|X|+ is the (+, +) matrix element of the linear transformation X with respect to the basis {eσ }. 1

1

1

Proof. Each occurrence of V1 in (1.8) can be factored V1 = V12 V12 , and since V12 commutes with the spin operators in each row, it is clear that all “internal” products V2 V1 can be replaced by VM . To deal with the products on the ends observe that the 1

configuration + is an eigenvector for V12 with eigenvalue eMK1 . Note in addition that the matrix elements in (1.18) are given by inner products ({eσ } is orthormal and + is an element of this basis). An immediate consequence is that the extra 1

factors of V12 that appear at the ends in (1.18) compared to (1.8) simply produce factors e2MK1 in both the numerator and denominator that cancel.

1.2 The Semi-infinite-Volume Limit We are interested, of course, in the infinite-volume limit for (1.18). The transfer matrix method we are using makes it natural to take this limit in two stages.

1.2 The Semi-infinite-Volume Limit

9

First we examine the semi-infinite-volume limit N → ∞ with M fixed, and then we determine the limit as M → ∞. Note that there are convergence results for “plus” boundary conditions [79] that show that the correlations obtained in this two-step process are the same as those obtained by having the sides of a square box tend simultaneously to infinity. The N → ∞ limit is intimately connected with the spectral analysis of the transfer matrix VM . The set A of spin sites in the correlation (1.18) is fixed as N → ∞. This implies that by rewriting (1.18) in an obvious way, all the N-dependence in the formula can be lodged in the appearance of vectors VMN + that occur in both the numerator and the denominator. Expanding the vector + in eigenvectors for VM , one sees that if the eigenvectors associated with the largest eigenvalue for VM make a contribution to the expansion of +, then only those terms involving these eigenvectors survive the limit N → ∞. Our goal in what follows is to determine the eigenvectors associated with the largest eigenvalues for VM and to show that these eigenvectors are not orthogonal to the state +. This allows us to obtain a formula for the semi-infinite correlations as a vacuum expectation in a Fock representation, Theorem 1.2.1; a formula that makes possible the calculation the M → ∞ limit. It is simplest for us to proceed somewhat indirectly. We analyze the transfer matrix VM and the spin operators σj that appear in (1.18) by realizing both these operators as elements in a finite-dimensional spin representation of the orthogonal group. There are many equivalent Fock realizations of this spin representation, and we find one in which the vacuum vector in the Fock realization can be identified with an eigenvector associated with the largest eigenvalue for VM . This method allows us to proceed without actually figuring out the largest eigenvalue for VM . Knowing the largest eigenvalue is important for the infinite-volume free-energy calculation. The reader can find this calculation in the paper of Kaufman [72].

1.2.1 Induced Rotations Following Kaufman, and Abraham and Martin-Löf, we begin our analysis of the semi-infinite-volume limit by introducing a finite-dimensional spin representation of the orthogonal group realized as a group of automorphisms of the Brauer– Weyl [28] representation of the Clifford relations. To simplify some later developments we index the elements of the Clifford algebra by half-integers rather than integers. Let IM := {−M, −M +1, . . . , M}. For k ∈ IM − 12 (the set of half-integer translates of the elements in IM ), we define pk :=

=

0 1

1 0 ⊗ ··· ⊗ 0 1 M+k+ 12

k− 12

j =−M

0 1

1 1 0 ⊗ ⊗ 1 ⊗ ··· ⊗ 1 0 0 −1

1 1 0 . 0 j 0 −1 k+ 1 2

(1.19)

10


For k ∈ IM + qk :=

=

1 2

define

0 1

1 0 ⊗ ··· ⊗ 0 1 M+k− 12

k− 23

j =−M

0 1

1 0 −i ⊗ ⊗ 1 ⊗ ··· ⊗ 1 0 i 0

1 0 −i , 0 j i 0 k− 1

(1.20)

2

where the empty product in (1.19) for k = −M − 21 is interpreted as the identity, as is the empty product in (1.20) for k = −M + 21 . It is straightforward to check that pk pℓ + pℓ pk = 2δkℓ , qk qℓ + qℓ qk = 2δkℓ , pk qℓ + qℓ pk = 0,

(1.21)

for all the relevant indices, so that we have a representation of the Clifford relations. In order to discuss the complex spin representation associated with this represen′ tation of the Clifford relations we introduce the vector space WM of complex linear ′ combinations of the “vectors” pk and qk . For v ∈ WM we write qM+ 1 p−M− 1 v = y0 (v) √ 2 + x0 (v) √ 2 + 2 2

M− 12

pk qk xk (v) √ + yk (v) √ . 2 2 1

(1.22)

k=−M+ 2

√ The reason for the 2 factors is explained momentarily, and we have singled out the coordinates of p−M− 1 and qM+ 1 , since these vectors are mostly “specta2 2 tors” in our analysis. It is also useful to introduce the complementary vector space WM , which we define as the complex linear span of the vectors pk and qk for ′ as a finite-dimensional k = −M + 21 , . . . , M − 21 . We regard the vector space WM Hilbert space with Hermitian symmetric inner product defined by M− 21

v, w := x0 (v)x0 (w) + y0 (v)y0 (w) +

k=−M+ 12

xk (v)xk (w) + yk (v)yk (w),

and we define the nondegenerate bilinear form (v, w) := v, ¯ w, obtained from the conjugation p−M− 1 qM+ 1 v¯ := y0 (v) √ 2 + x0 (v) √ 2 + 2 2

M− 21

qk pk xk (v) √ + yk (v) √ . 2 2 1

k=−M+ 2


11

The map

′ WM

x0 (v) ⊕ ∋v→ y0 (v)

M− 21

qk pk xk (v) √ + yk (v) √ 2 2 1

(1.23)

k=−M+ 2

′ identifies WM with the orthogonal direct sum C2 ⊕ WM . The expressions (1.19) and (1.20)determine an irreducible representation of M ′ 2 the Clifford algebra Cliff(WM ) on j =−M Cj [28]. As noted in the appendix (see (A.4)) and (A.5)), it is convenient for us to think of the generator relations for Cliff(W ) as

vw + wv = (v, w)e, for v, w ∈ W, where (v, w) is a distinguished bilinear form on the complex vector space W . √ The normalization by 2 in (1.22) is included so that an extra factor of 2 is not incorporated into the Clifford relations (even though this is possibly more standard usage). The calculations that follow in this section do not require any results from the appendix but use the explicit anticommutators (1.21) for pℓ and qℓ ; we refer to these relations as the Clifford relations. One should just keep in mind that it qℓ pℓ ′ √ √ is and not {qℓ , pℓ }. In subsequent , 2 that is the orthonormal basis for WM 2 sections, after we introduce the Fock representations of the Clifford relations, it is important to keep the proper normalization in mind. 2 Suppose that V is a linear transformation on M j =−M Cj . We say that V is an ′ → element of the Clifford group if there exists a linear transformation T (V ) : WM ′ ′ WM such that for all v ∈ WM we have V vV −1 = T (V )v, 2 each side being regarded as a linear transformation on M j =−M Cj . The map T (V ) is called the induced rotation associated with V ; it is a complex orthogonal map and it determines V up to a scalar multiple. A complex orthogonal map is one that preserves the symmetric bilinear form (·, ·). As noted above, both the spin operator σj and the transfer matrix VM are in the Clifford group. We now compute the induced rotations associated with these operators. The spin operators are simplest to deal with, and without difficulty the reader may check that σj pk σj−1 = sgn(j − k)pk for k ∈ IM − 21 ,

(1.24)

σj qk σj−1 = sgn(j − k)qk for k ∈ IM + 21 . The simplicity of this result is one reason the elements pk and qk are indexed by half-integers.

12


Next we compute the induced rotation for V1 and V2 . Using the elementary identities 0 1 = ipj − 1 qj + 1 , 1 0 j 2 2 (1.25) 1 0 1 0 = iqj + 1 pj + 1 , 0 −1 j 0 −1 j +1 2 2 the formulas (1.12) and (1.16) translate into M− 21

V1 = exp iK1

qk pk ,

k=−M+ 21 M− 12

V2 = (2 sh(2K2 ))

(1.26)

M− 23

exp iK2∗

pk qk+1 .

k=−M+ 12

Exponentials of quadratic elements in the Clifford algebra are well known to be elements of the Clifford group. We can compute T (V1 ) and T (V2 ) from (1.26) using the Taylor series expansion (for λ ∈ C), exp(λX)v exp(−λX) =

∞ λn n=0

n!

adn (X)v,

(1.27)

where ad0 (X) is the identity, ad(X) is the commutator, ad(X)v = [X, v] = Xv − vX,

n

and ad (X) is the n-fold iterated commutator,

adn (X)v = [X, · · · , [X, v], · · · ]. n

Neither p−M− 1 nor qM+ 1 appears in the exponents that determine V1 and V2 , and 2 2 as a consequence we have the obvious relations Vi p−M− 1 Vi−1 = p−M− 1 , 2

2

Vi qM+ 1 Vi−1 = qM+ 1 , 2

(1.28)

2

for i = 1, 2. The induced rotations T (V1 ) and T (V2 ) are thus the identity on p−M− 1 and qM+ 1 , and this is behind the earlier remark that they are “spectators.” 2

2

Using the Clifford relations, one finds that for ℓ = −M + 21 , . . . , M − 12 ,   M− 21   qk pk , qℓ  = −2iK1 pℓ , iK1 k=−M+ 21



 iK1

M− 12

k=−M+ 12



 qk pk , pℓ  = 2iK1 qℓ .

(1.29)


13

From (1.27) and (1.29) it follows that V1 qℓ V1−1 = ch(2K1 )qℓ − ish(2K1 )pℓ , V1 pℓ V1−1 = ish(2K1 )qℓ + ch(2K1 )pℓ .

(1.30)

The quadratic element of the Clifford algebra that appears in the exponential determining V2 does not contain either q−M+ 1 or pM− 1 . Thus, 2

V2 q−M+ 1 V2−1 2 V2 pM− 1 V2−1 2

2

= q−M+ 1 , 2

= pM− 1 .

(1.31)

2

Again using the Cifford relations one finds that for ℓ = −M + 21 , . . . , M − 32 ,   M− 23  ∗  pk qk+1 , qℓ+1  = 2iK2∗ pℓ , iK2 k=−M+ 12



 ∗ iK2

M− 23

k=−M+ 21



 pk qk+1 , pℓ  = −2iK2∗ qℓ+1 ,

which in concert with (1.27) implies that

V2 qℓ+1 V2−1 = ish(2K2∗ )pℓ + ch(2K2∗ )qℓ+1 , V2 pℓ V2−1 = ch(2K2∗ )pℓ − ish(2K2∗ )qℓ+1 .

(1.32)

In (1.28), (1.31), and (1.32) we have found the action of the induced rotations, T (V1 ) and T (V2 ). From these results it is straightforward to calculate T (VM ), and this determines VM up to a scalar multiple. Because VM enters into (1.18) the same number of times in the numerator as it does in the denominator, (1.18) remains valid if VM is replaced by any multiple of VM . We take advantage of this when we transcribe (1.18) into a purely representation-theoretic setting, obtaining a model for the action of some multiple of VM by making use of T (VM ).

1.2.2 Fock Representations It is useful at this point to recall some facts about the Fock representations of the Clifford relations, which are explained in more detail in Appendix A. Suppose that W is an even-dimensional complex Hilbert space with a Hermitian symmetric inner product x, y that is conjugate linear in the first slot. Let x → x¯ denote a conjugation on W (a conjugate linear map whose square is the identity). Then (x, y) = x, ¯ y defines a nondegenerate bilinear form on W . Recall that an isotropic subspace of W is one on which the bilinear form (·, ·) vanishes identically. An isotropic splitting of W , also called a polarization, is a direct sum decomposition W = W+ + W− ,

14


where both W+ and W− are isotropic subspaces of W . If W+ and W− are orthogonal with respect to the Hermitian inner product, we write W = W+ ⊕ W− and refer to the splitting as a Hermitian polarization. The application to the Ising model involves only Hermitian polarizations, and except in the appendix all polarizations are Hermitian polarizations. If W = W+ ⊕ W− is a polarization we write w = w+ + w− for the components of w relative to this splitting. We parametrize a polarization W+ ⊕ W− by the map Qw = w+ − w− , which we also refer to as a polarization. The Clifford algebra over the orthogonal space W is the associative algebra with unit e that is generated by the elements of W subject to the Clifford relations vw + wv = (v, w)e. Associated with each polarization W+ ⊕ W− of W there is a Fock representation of the Clifford algebra. This representation lives on the alternating tensor algebra over W+ , Alt(W+ ), which is defined by Alt(W+ ) = C ⊕ Alt1 (W+ ) ⊕ Alt2 (W+ ) ⊕ · · · ⊕ Altn (W+ ),

where Altj (W+ ) is the space of alternating j tensors over W+ and n = dim(W+ ). The Fock representation of the Clifford relations for W associated with the polarization W+ ⊕ W− is given by W ∋ w → FQ (w) := a ∗ (w+ ) + a (w − ) ,

(1.33)

where a ∗ (w+ ) is the creation operator associated with w+ , a ∗ (w+ )v = w+ ∧ v,

and a (w − ) is the annihilation operator associated with w− , a(w) = a ∗ (w)∗ for w ∈ W+ .

Note that for any w− ∈ W− we have w − ∈ W+ , and a ∗ (w)∗ is the adjoint of a ∗ (w) with respect to the Hermitian inner product on Alt(W+ ) (see the appendix (A.75)). The vacuum vector |0 := 1 ⊕ 0 ⊕ 0 ⊕ · · · ⊕ 0 is characterized as the unique vector that is killed by all the elements of W− in this representation. In fact, any irreducible ∗ representation of the Clifford relations with a vector |0 that is killed by all the elements in W− is unitarily isomorphic in a natural way to this Fock representation. A ∗ representation of the Clifford algebra Cliff(W ) is one in which the conjugate x¯ is mapped to x ∗ , the adjoint of the image of x (this is clearly true for (1.33)).

Remark 1.2.1. The reader who is more familiar with the development of these matters in the physics literature might find it helpful at this point to recognize that since the subspaces W+ and W− correspond to creation and annihilation operators, the condition that the subspaces are isotropic is just a translation of the fact that creation operators anticommute among themselves, and the same is true for annihilation operators (for antisymmetric quantization).


15

The Brauer–Weyl representation above with the inner product for which the basis {eσ } is orthonormal is unitarily equivalent to a Fock representation. All Fock representations are equivalent in finite dimensions, so this can be seen in many different ways. One way, which is simple to check, is to note that the vector 1 1 1 ⊗ ⊗ ··· ⊗ (1.34) 1 1 1 is a vacuum vector for the isotropic splitting ′ = Spank∈IM − 1 {pk + iqk+1 } ⊕ Spank∈IM − 1 {pk − iqk+1 }, WM 2

2

that is, this vector is annihilated by pk − iqk+1 for k ∈ IM − 12 . Also, both pk and qk are self-adjoint with respect to the inner product for which {eσ } is an orthonormal basis. This means that the representation given by (1.19) and (1.20) is a ∗ representation. We now make use of our freedom to choose a Fock representation to pick one in which the vacuum vector becomes the eigenvector associated with the largest eigenvalue of the transfer matrix. Actually, we find that the mulitiplicity of the largest eigenvalue for VM is 2. This multiplicity is resolved in the course of dealing with the spectator coordinates (x0 , y0 ). Recall that ′ = C2 ⊕ WM , WM

(1.35)

where C2 is spanned by p−M− 1 and qM+ 1 . Then because T (V1 ) and T (V2 ) are 2 2 both the identity on this two-dimensional subspace and leave WM invariant, we can write T (VM ) = 1 ⊕ TM ,

relative to the splitting (1.35), where TM is just the restriction of T (VM ) to WM . ′ We now give an isotropic splitting of WM by introducing isotropic splittings for both C2 and WM . Let C± denote complex linear span, $ # C p−M− 1 ± iqM+ 1 , 2

2

regarded as a subspace of C2 (recall (1.23)). Then C2 = C+ ⊕ C− is an isotropic splitting. Next we describe an isotropic splitting of WM whose associated vacuum + vector is an eigenvector for VM associated with its largest eigenvalue. Let WM denote the span of all eigenvectors of TM associated with eigenvalues less than 1 and − let WM denote the span of all the eigenvectors of TM associated with eigenvalues greater than 1. The splitting of WM we are interested in is + − WM = WM ⊕ WM .

The representations (1.12) and (1.13) show that the symmetrized transfer matrix VM has an induced rotation that is a positive self-adjoint operator. Following Abraham and Martin-Löf, in the next subsection we will make a spectral analysis of TM that shows that it does not have eigenvalue 1.

16


If 1 is not in the spectrum of TM , the spectral theorem for Hermitian symmetric + − operators immediately implies that WM ⊕ WM is an orthogonal direct sum decom+ position. There is a simple argument to show that WM is isotropic. Suppose that + u, v ∈ WM are eigenvectors for TM with TM u = αu

and TM v = βv.

Then since TM is a complex orthogonal mapping, (u, v) = (TM u, TM v) = αβ(u, v), but since 0 ≤ α < 1 and 0 ≤ β < 1, it follows that (u, v) = 0, and hence that + − WM is isotropic. The same argument works to show that WM is isotropic. Next we indicate how to obtain a model for the action of a multiple of VM in the Fock representation associated with the isotropic splitting + − ′ WM = C+ ⊕ WM ⊕ C− ⊕ WM (1.36) ′ of WM . It is convenient for us to realize this Fock representation on the space + Alt(C+ ) ⊗ Alt(WM )

with the representation of the Clifford generators given by 0 1 ⊗ 1, p−M− 1 → 1 0 2 0 i qM+ 1 → ⊗ 1, −i 0 2 1 0 WM ∋ v → ⊗ (c (v+ ) + a (v¯− )) , 0 −1 + acting on Alt(C+ ) ⊗ Alt(WM ). The reader can check that the appropriate vacuum vector is given in this representation by 1 ⊗ |0M , 0 + where |0M = 1 ⊕ 0 ⊕ · · · ⊕ 0 is the standard vacuum in Alt(WM ). Next we introduce the model for VM that we work with. Let TM+ denote the + restriction of TM to WM and define the linear transformation Ŵ(TM+ ) acting on + Alt(WM ) by (1.37) Ŵ(TM+ ) = 1 ⊕ TM+ ⊕ TM+ ⊗ TM+ ⊕ · · · ⊕ TM+ ⊗ · · · ⊗ TM+ . 2M

Then the reader can easily verify that T Ŵ(TM+ ) = TM . Thus T 1 ⊗ Ŵ(TM+ ) = 1 ⊕ TM , and it follows that for some constant α ∈ R, the representation of VM in + the Fock representation Alt(C+ ) ⊗ Alt(WM ) is given by VM = α1 ⊗ Ŵ(TM+ ).

(1.38)


17

The representation (1.37) and the fact that TM+ is a strict contraction (recall that 1 is not in the spectrum of TM ) show that the largest eigenvalue of Ŵ(TM+ ) is 1 + with the unique eigenvector given by the vacuum vector |0M in Alt(WM ). The representation (1.38) then shows that α can be identified with the largest eigenvalue of VM . Also note that the eigenvectors for VM associated with the largest eigenvalue for VM span the two-dimensional subspace C2 ⊗ |0M . We want to make use of the representation (1.38) to pass to the N → ∞ limit in (1.18). Before we can do this we need to translate the spin operators σj and the vector + into the Fock representation we are now working in. Observe that σj is determined up to a constant multiple by its induced rotation (1.24). For j = −M, . . . , M define a linear transformation sj on WM by sj pk := −sgn(k − j )pk sj qk := −sgn(k − j )qk ,

(1.39)

for k = −M + 21 , . . . , M − 21 . Then (1.24) translates into T (σj ) =

1 0

0 ⊕ sj . −1

+ ) with the property that Now write σ¯ j for a linear transformation on Alt(WM

T (σ¯ j ) = sj , σ¯ j2 = 1.

(1.40)

Such a linear transformation exists, since each sj is a complex orthogonal map (this is shown in Theorem A.3.3 of the appendix). Note that there are only two possible choices for σj , differing from one another by a sign. A little calculation shows that 0 1 1 0 T ⊗ σj = ⊕ sj . 1 0 0 −1 Thus σj = ±

0 1

1 ⊗ σ¯ j , 0

since both transformations have the same induced rotations and both sides square to 1. We now fix the choice of sign in σ¯ j by requiring that 0 1 ⊗ σ¯ j . (1.41) σj = 1 0 Observe also that + is characterized up to a multiple as the unique common eigenvector for σ¯ j (j = −M, . . . , M) associated with the eigenvalue 1.

18


For the purposes of a later argument it is useful to introduce the subspace   M

 (1.42) C2j  , j =−M

plus

M

of vectors in j =−M C2j with “plus boundary conditions” and to characterize this + subspace in the representation space Alt(C+ ) ⊗ Alt(WM ). The subspace we have in mind consists of all the vectors of the form 1 1 ⊗v⊗ , 0 0 2 for v ∈ jM−1 =−M+1 Cj . This subspace is characterized as the +1 eigenspace of both σ¯ −M and σ¯ M . We see from (1.19) that σ¯ −M = p−M− 1 and from (1.25) that 2

M− 21

σ¯ −M σ¯ M =

k=−M+ 12

iqk pk := ,

(1.43)

where is a “volume element.” + Since p−M− 1 = 01 10 ⊗ 1 in Alt(C+ ) ⊗ Alt(WM ), the +1 eigenspace for σ−M 2 consists of vectors of the form 1 ⊗w 1 + ). The subspace (1.42) thus consists of the vectors of this form for w ∈ Alt(WM that are also eigenvectors for σ¯ −M σ¯ M with eigenvalue +1. We show that the +1 eigenspace of the on the right-hand side of (1.43) is the subspace of even + elements in Alt(WM ). To see this, first note that 2 = 1 and pℓ = −pℓ and

qℓ = −qℓ , so that is an element of the Clifford group with induced rotation + −1 on WM . Let N denote the number operator in Alt(WM ), + N |Altj (WM ) = j. + Then (−1)N F (w)(−1)N = −F (w) for the Fock representation on Alt(WM ). N Since F ( ) and (−1) have the same induced rotation in the (irreducible) Fock representation, it follows that F ( ) = α(−1)N for some constant α ∈ C. However, both and (−1)N square to the identity, so α = ±1. Next we check that α = 1. Observe that if we define

ak∗ : = 21 (pk + iqk ), ak : = 12 (pk − iqk ),


19

then WM = span{ak∗ } ⊕ span{ak }

is a polarization. The number operator in the Fock representation associated with this polarization is k ak∗ ak , and we compute ∗ ∗ (−1) k ak ak = eiπ ak ak = 1 − 2ak∗ ak = iqk pk = , k

k

k

where we used the fact that (ak∗ ak )2 = ak∗ ak . However, the vacuum for the operators + + {ak } in Alt(WM ) is an even element of Alt(WM ) (this is true for any vacuum vector arising from a polarization of W ), and it follows that (−1)N , and F ( ) agree on + ). this vector and hence on all vectors in Alt(WM Thus we obtain   % & M

1 even + 2  ≃ Cj ⊗ w : w ∈ Alt (WM ) . (1.44) 1 j =−M

plus

The vector + is in the subspace (1.42), so we find that there is an even tensor (+) + in Alt(WM ) with 1 += ⊗ (+). (1.45) 1 We turn to the N → ∞ limit in (1.18). Substituting (1.38), (1.41), and (1.45) into (1.18), the inner products in the “multiplicity” space Alt(C+ ) drop out of the formula to give σ + =

(+), Ŵ(TM+ )N −j σ¯ Aj · · · σ¯ A−j Ŵ(TM+ )N −j (+) (+), Ŵ(TM+ )2N (+)

,

(1.46)

where j is chosen such that all the sites in A lie between the −j th and the j th rows (j is fixed as N → ∞). At this point the original spin operators σj will no longer make an appearance. Henceforth, we drop the “bar” in σ¯ j , writing σj = σ¯ j instead. Since TM+ is a strict contraction, (1.37) implies lim Ŵ(TM+ )N (+) = 0|(+) · |0.

N →∞

(1.47)

The following lemma makes it possible for us to employ (1.47) in the calculation of the N → ∞ limit of (1.46). Lemma 1.2.1 The inner product 0|(+) is nonzero.

Proof. We adapt an argument from Abraham and Martin-Löf [5]. Let α denote the largest eigenvalue for the transfer matrix VM . Then (1.47) implies that +, VMN + = |0|(+)|2 . N →∞ αN lim

20


Write V for the restriction of VM to the subspace   M

 C2j  . j =−M

plus

We see from (1.44) that V has a simple largest eigenvalue that coincides with that of VM , and hence that +, VMN + (+), V N (+) (+), V N (+) . = lim = lim N →∞ N →∞ N →∞ αN αN Tr V N lim

Next we estimate

(+), V N (+) Tr V N

by returning to the partition-function interpretation for numerator and denominator. The numerator is, of course, the partition function for plus boundary conditions around the rectangle [−M, M] × [−N, N ]. Because of the restriction to the subspace (1.42), the denominator is the partition function for plus boundary conditions on the vertical sides and periodic boundary conditions on the top and bottom. This periodic partition function is the sum of 22M−1 terms each of which is a partition function associated with identical top and bottom configurations each given by σ ∈ C + (row). Each term in the partition function summed over spin configurations on associated with the boundary configuration σ on the top and bottom 2M−1 2K 2M−1 rows has a weight that is at most a factor e2K2 bigger than the · e 2 weight associated with the same spin configuration on except changed to plus boundary conditions on the top and bottom rows. Only the vertical bonds connecting the top and bottom configurations to the rest of need be taken into account, since the horizontal weights are maximized for σ = +. Thus we have the lower bound (+), V N (+) ≥ 2−2M+1 e−2K2 (4M−2) , Tr V N

independent of N. Note that we worked with V rather than VM precisely to eliminate any N-dependence in this last estimate (there is no change in the boundary conditions on the vertical sides). This finishes the proof of the lemma. Making use of (1.47) in (1.46) together with lemma just proved, we obtain the following theorem.

Theorem 1.2.1 Suppose that the sites in A all have second coordinates between −j and j . Then for = M,N we have + + lim σA + = 0|σAj Ŵ(TM )σAj −1 · · · Ŵ(TM )σA−j |0,

N →∞

(1.48)

+ ), Ŵ(TM+ ) is given by (1.37) and σj where |0 is the normalized vacuum in Alt(WM is characterized by (1.40) and (1.41). By convention, σ∅ = identity.


21

Remark 1.2.2. It is not hard to see that the spin operators σj all leave the subspace + Alteven (WM ) invariant, as does the transfer matrix Ŵ(TM+ ). Thus (1.48) could be formulated on this smaller subspace, and given the connection with plus boundary conditions it is even natural to do so. However, we are going to use (1.48) to analyze the thermodynamic limit M → ∞ only for T < Tc . Above the critical temperature we use a variant of Kramers–Wannier duality to give an alternative representation of the correlation functions. In this alternative representation the + associated spin operators no longer preserve the even subspace of Alt(WM ), and + it is convenient to work on all of Alt(WM ). We turn now to the missing ingredient in the proof of (1.48).

1.2.3 The Spectrum of TM In this section we follow the analysis in Abraham and Martin-Löf [5] to find the 1

1

spectrum of TM . Since TM = T (V12 )T (V2 )T (V12 ) restricted to WM , it is similar to T (V1 )T (V2 ) restricted to WM (both T (V1 ) and T (V2 ) leave WM invariant). We do not need the eigenvectors for TM , so it suffices to find the spectrum of the restriction of T (V1 )T (V2 ) to WM . The nonzero vector v ∈ WM is an eigenvector for T (V1 )T (V2 ) with eigenvalue λ if and only if T (V2 )v = λT (V1 )−1 v.

(1.49)

We now introduce coordinates

M− 12

v=

k=−M+ 21

xk qk + yk pk

relative to the ordered basis {q−M+ 1 , p−M+ 1 , . . . , qM− 1 , pM− 1 }. 2

2

2

2

Norms of vectors do not play a role in this section, so we can drop the √12 needed to convert {qk , pk } to an orthonormal basis. We have already calculated the matrices for T (V1 ) and T (V2 ) in this basis in (1.30)–(1.32). Introducing the notation ci = ch 2Ki and ci∗ = ch 2Ki∗ , si = sh 2Ki and si∗ = sh 2Ki∗ ,

for i = 1, 2, we obtain



     T (V2 ) =     



1 c2∗ −is2∗

is2∗ c2∗ ..

. c2∗ −is2∗

is2∗ c2∗

          1

22


and

−1

T (V1 )



c1 is1   =  



−is1 c1 ..

   .  −is1  c1

. c1 is1

The eigenvector–eigenvalue equation (1.49) translates into the end conditions $ # x−M+ 1 = λ c1 x−M+ 1 − is1 y−M+ 1 , 2 2 $2 # (1.50) yM− 1 = λ is1 xM− 1 + c1 yM− 1 , 2

2

2

and the interior conditions ∗ c2 is1 xk + c1 yk yk is2∗ , = λ c1 xk+1 − is1 yk+1 −is2∗ c2∗ xk+1

(1.51)

for k = −M + 12 , . . . , M − 32 . The strategy for solving this eigenvalue problem is to treat (1.51) as a translation-invariant finite difference relation that can be solved for all k. For the general solution to have a nonzero specialization that satisfies (1.50), the eigenvalue λ must be restricted. To begin, we look for solutions to (1.51) given by 1 xk azk− 2 := , (1.52) 1 yk bzk+ 2 with a, b, and z complex numbers. After a small algebraic simplification, (1.51) becomes ∗ c2 is2∗ b c1 is1 z−1 b = λ , −is2∗ c2∗ a −is1 z c1 a which translates into the eigenvector equation c1 c2∗ − s1 s2∗ z−1 i(c1 s2∗ − s1 c2∗ z−1 ) b b = λ . −i(c1 s2∗ − s1 c2∗ z) c1 c2∗ − s1 s2∗ z a a

The characteristic polynomial for the matrix on the left-hand side is z + z−1 λ2 − 2 c1 c2∗ − s1 s2∗ λ + 1. 2 Now suppose that λ(z) is one of the roots of (1.54). One finds that b q(z) = a 1 is an eigenvector for (1.53) with eigenvalue λ(z) for i c1 s2∗ − s1 c2∗ z−1 c1 c2∗ − s1 s2∗ z − λ(z) =− ∗ q(z) = . c1 c2 − s1 s2∗ z−1 − λ(z) i c1 s2∗ − s1 c2∗ z

(1.53)

(1.54)

(1.55)

(1.56)


Thus

23

1 zk− 2 xk := 1 yk q(z)zk+ 2

is a solution to (1.56) when λ =λ(z) is a root of (1.54). The characteristic polynomial (1.54) is symmetric under the map z → z−1 . Transform the ansatz (1.52) by this symmetry to get −k+ 1 2 xk az = . −k− 12 yk bz Calculation shows that this vector satisfies (1.51) with the same eigenvalue λ = λ(z), provided 1 −q(z)z−k+ 2 xk = , 1 yk z−k− 2 where q(z) is again given by (1.56). Thus for λ = λ(z) a root of (1.54), 1 1 xk −q(z)z−k+ 2 zk− 2 +β := α 1 1 yk q(z)zk+ 2 z−k− 2

(1.57)

is a solution to (1.51) for any choice of α, β, and z. In what follows we suppose that xk and yk are defined by (1.57) for all k ∈ Z + 12 . Next we satisfy the endpoint conditions (1.50) by choosing α, β, and z appropriately. First transform the endpoint conditions (1.50) by making use of the difference relations # $ λ c1 x−M+ 1 − is1 y−M+ 1 = −is2∗ y−M− 1 + c2∗ x−M+ 1 , 2 2 2 2 $ # ∗ ∗ λ is1 xM− 1 + c1 yM− 1 = c2 yM− 1 + is2 xM+ 1 , 2

2

2

2

which follow from (1.51). With this relation and the elementary identities c2∗ − 1 = 2 sh2 K2∗ and s2∗ = 2 ch K2∗ sh K2∗ , the endpoint conditions (1.50) become sh K2∗ x−M+ 1 − i ch K2∗ y−M− 1 = 0, 2

i

ch K2∗ xM+ 1 2

2

+

sh K2∗ yM− 1 2

= 0.

Substitute (1.57) in this relation and multiply by i to obtain (ch K2∗ q + i sh K2∗ )z−M (−i sh K2∗ q + ch K2∗ )zM α 0 = . (1.58) (i sh K2∗ q − ch K2∗ )zM (ch K2∗ q + i sh K2∗ )z−M β 0 To obtain a nonzero solution βα to this equation, the determinant of the matrix on the left-hand side must vanish. This is equivalent to ch K2∗ q + i sh K2∗ 2 z4M = − . (1.59) −i sh K2∗ q + ch K2∗

24


This relation is an “eigenvalue” equation for z that restricts the possible values of λ = λ(z). Define ch K2∗ q + i sh K2∗ u := . −i sh K2∗ q + ch K2∗ In order to analyze the eigenvalue equation (1.59) we provide an alternative characterization for λ and u. The eigenvalue–eigenvector equation that preceeds (1.53) can be written ∗ c1 −is1 z−1 c2 is2∗ q q =λ . 1 −is2∗ c2∗ 1 is1 z c1 Since

it follows that

c2∗ −is2∗

ch K2∗ −i sh K2∗

ch K2∗ is2∗ ∗ = −i sh K2∗ c2 i sh K2∗ ch K2∗

i sh K2∗ ch K2∗

2

,

q ch K2∗ q + i sh K2∗ , = 1 −i sh K2∗ q + ch K2∗

and hence also [ u1 ] is an eigenvector for the “symmetrized” matrix ch K2∗ i sh K2∗ c1 −is1 z−1 ch K2∗ i sh K2∗ −i sh K2∗ ch K2∗ is1 z c1 −i sh K2∗ ch K2∗ with eigenvalue λ. Multiply these matrices to obtain the product A(z) B(z) , C(z) A(z)

(1.60)

where z + z−1 A(z) : = c2∗ c1 − s2∗ s1 , 2 −1 z − z−1 ∗ ∗ z+z B(z) : = i s2 c1 − c2 s1 , + is1 2 2 z − z−1 z + z−1 . + is1 C(z) : = −i s2∗ c1 − c2∗ s1 2 2 Since this matrix is similar to the matrix on the left-hand side of (1.53), the characteristic equation for (1.60) is again given by (1.54). The eigenvector equation for [ u1 ] is easily seen to imply u(z)2 =

B(z) . C(z)

(1.61)

In order to analyze the eigenvalue equation (1.59), we first examine the function B(z). Multiply B(z) by 2z and then factor the resulting quadratic polynomial; as the reader can verify, 2zB(z) = i(1 − c2∗ )s1 (z − α1 ) (z − α2 ) ,

(1.62)


where α1 := and α2 :=

25

thK1 2(K2 −K1∗ ) = (c1∗ − s1∗ )(c2 + s2 ) ∗ =e thK2

1 ∗ = e2(K2 +K1 ) = (c1∗ + s1∗ )(c2 + s2 ), thK1 thK2∗

where thK is the hyperbolic tangent of K. Since C(z) = −B(z−1 ), we find that (1.61) and (1.62) imply u(z)2 = −

z − α1 z − α2 B(z) =− . B(z−1 ) 1 − α1 z 1 − α 2 z

(1.63)

Thus the eigenvalue equation (1.59) becomes z4M =

z − α1 z − α2 . 1 − α 1 z 1 − α2 z

(1.64)

It is useful to observe that the eigenvalues for TM are always real, since it is selfadjoint. Thus the only eigenvalues of (1.60) that concern us are real eigenvalues. This means that A(z) (which is half the sum of the eigenvalues), and hence z + z−1 must be real to be of interest. But z + z−1 is real only for z on the unit circle or z real. Suppose to start that |z| = 1. The matrix (1.60) then has the form A B , B¯ A with A real and A2 − |B|2 = 1. Hence there exists a positive real γ such that A = ch γ and |B| = sh γ . It makes sense to define a real-valued function γ (z) ≥ 0 and an S 1 -valued function v(z) such that −1

ch γ (z) := c2∗#c1 − s2∗ s1 z+z2 , $ −1 −1 v(z) sh γ (z) := i s2∗ c1 − c2∗ s1 z+z2 + is1 z−z2 .

(1.65)

The eigenvalue problem for (1.60) then has solutions ±v ch γ v sh γ ±v = e±γ . 1 v¯ sh γ ch γ 1

Thus we can identify λ with either eγ or e−γ and u with either v or (respectively) −v (this is the alternative characterization of λ and u mentioned above). At this point in the analysis we encounter the first manifestation of a critical point in the Ising model. For 0 < α < 1 the function S 1 ∋ z →

z−α 1 − αz

26


1.5

1 K2 0.5

0 0.5

K1

1

1.5

Figure 1.1: The critical curve sh 2K1 sh 2K2 = 1. The points closer to the x-and y-axes have coordinates that correspond to temperatures above Tc . has winding number 1 about the origin. For 1 < α the same function has winding number −1 (the number of zeros minus the number of poles inside the unit disk). Since α2 > 1 for Kj > 0 (j = 1, 2), it follows that the right-hand side of (1.64) has winding number −2 when α1 > 1 and winding number 0 when α1 < 1. When α1 = 1 the winding number for the right-hand side of (1.64) is −1. Later we see that α1 = 1 is the condition that singles out the critical point in the Ising model determined by the vanishing of the spontaneous magnetization. For the present we simply recast this condition in terms of the temperature parameter. Recall that J Kj = kBjT and K2∗ is defined by sh 2K2∗ sh 2K2 = 1. If we regard Jj as fixed (j = 1, 2), then the condition α1 = 1 or K1 = K2∗ becomes a condition on the temperature T that fixes T at its critical value Tc . The critical temperature is determined by the equation

2J1 sh kB Tc

2J2 sh kB Tc

= 1.

(1.66)

Since K1 is a decreasing function of T , and K2∗ is an increasing function of T , it is clear that the condition α1 > 1 or K1 > K2∗ is the same as T < Tc , and in a similar fashion α1 < 1 or K1 < K2∗ is equivalent to T > Tc . The critical curve in (K1 , K2 ) determined by sh(2K1 ) sh(2K2 ) = 1 is depicted in Figure 1.1. We now look at the eigenvalue equation (1.64) in the three different cases α1 > 1, α1 < 1, and α1 = 1 (and at first restrict our attention to |z| = 1). Suppose that α1 > 1 and introduce the principal value for the argument, z = |z|ei arg z , with − π < arg z ≤ π.


27

1 3 0.8 2 0.6

1 -3

-2

-1

0 -1

1

2

0.4

3

0.2 -2 -3

-3

-2

-1

0

1

2

3

Figure 1.2: Left: Graphical location for the four nontrivial positive roots of 2Mω − nπ = ϕ(ω) with M = 2, below Tc . Right: The associated points (ωn , e−γn ). Write ω = arg z and ϕi (ω) := arg

eiω − αi , 1 − αi eiω

so that the eigenvalue equation (1.64) can be written 2Mω − πn = ϕ(ω) :=

ϕ1 (ω) + ϕ2 (ω) , 2

(1.67)

where n is an integer. For i = 1, 2, 1 − αi2 ϕi′ (ω) = ' ' , '1 − αi eiω '2

from which it follows that each of the functions ϕi (ω) is a monotonically decreasing function from ϕi (−π) = π to ϕi (π) = −π . The same can be said for the average value ϕ(ω) that appears on the right-hand side of (1.67). For each of the integers n = −2M − 1, −2M, . . . , 2M, 2M + 1, the line intesects the curve

ω → 2Mω − πn ω → ϕ(ω)

over a point ωn with −π ≤ ωn ≤ π (see Figure 1.2). Included among these solutions are the special roots ω−2M−1 = −π,

ω0 = 0,

ω2M+1 = π.

However, none of these roots lead to a nonzero eigenvector for TM . The reader can check that although nonzero (α, β) exist for such ω, they give 0 when plugged

28


into the eigenvector (1.57). This check is not necessary, however, since we will shortly identify all the eigenvalues for TM . The remaining 4M solutions ωn with n = ±1, ±2, . . . , ±2M have associated eigenvalues # $ e

±γ eiωn

.

These are not all distinct, since the z → z−1 symmetry in the expression for ch γ (z) implies that γ eiω = γ e−iω . Theorem 1.2.2 Suppose that α1 = exp 2(K2 − K1∗ ) > 1, so that we are below Tc . Then the spectrum of TM of the 4M distinct values e±γn for iωconsists n = 1, 2, . . . , 2M, where γn = γ e n and ωn is the positive root of 2Mω−nπ = ϕ(ω) lying between 0 and π . We also have ch γ eiω ≥ ch(2K1 − 2K2∗ ) > 1. Proof. Since ch γ eiω = c1 c2∗ − s1 s2∗ cos ω, it follows that ω → ch γ eiω is a strictly increasing function for ω ∈ (0, π). In particular, ch γ eiω ≥ ch γ (1) = c1 c2∗ − s1 s2∗ = ch(2K1 − 2K2∗ ). (1.68) It also follows that ω → γ eiω is a strictly increasing function for ω ∈ (0, π) and hence that 0 < γ1 < γ2 < · · · < γ2M

(recall that we defined γ to be nonnegative). Thus the 4M values e±γn for n = 1, 2, . . . , 2M are all distinct. If we can show that each of these is associated with a nonzero eigenvector for TM , then because TM acts on a 4M-dimensional space, we have found all of the eigenvalues for TM . The issue for us is thus whether the solutions eiωn to (1.64) lead to nontrivial eigenvectors (1.57) for TM . Suppose that (α, β) is a nonzero solution to (1.58) associated with zn = eiωn . For the eigenvector (1.57) to be trivial we must have in particular k− 12

αzn

−k+ 21

− βq(zn )zn

= 0,

for k = −M + 12 , . . . , M + 12 . Since (1.56) implies that |q(zn )| = 1, it follows that both α = 0 and β = 0 and α = q(zn )zn−2k+1 . β The special cases k = 21 and k = − 21 for this last equation imply that zn2 = 1. It follows that any root zn = eiωn of (1.64) with 0 < ωn < π must lead to a nontrivial eigenvector of TM , and this finishes the proof. Next we turn to the case α1 < 1. In this situation the number of roots of 2Mω − nπ = ϕ(ω) for 0 < ω < π changes from 2M when α1 is close to 1 (near


29

1.5

-3

-2

1

0.4

0.5

0.2

-1 0 -0.5

1

2

3

-3

-2

-1

0

1

2

3

-0.2 -1 -0.4 -1.5

Figure 1.3: Left: Graphical location of the four nontrivial positive roots for 2Mω − nπ = ϕ(ω), with M = 2, just above Tc . Right: The three roots well above Tc . the critical temperature) to 2M − 1 when α1 is far from 1 (high above the critical temperature). The reader can see examples of this for K1 = 0.43 and K2∗ = 0.45 pictured on the left-hand side of Figure 1.3 and for K1 = 0.2 and K2∗ = 0.7 on the right-hand side. We can describe the transition in the following manner. As T increases from the critical temperature Tc , the smallest positive root ω1 decreases until it becomes 0 at some point above the critical temperature. Beyond this temperature the eigenvalue equation (1.67) has only 2M − 1 positive roots. To find the missing eigenvalue for TM we return to the “eigenvalue equation” z4M =

z − α1 z − α2 1 − α1 z 1 − α2 z

and seek a nontrivial real solution for z (which is close to z = 1 near the point at which ω1 “disappears”). We won’t bother to analyze this rigorously since our treatment of the thermodynamic limit for T > Tc proceeds via Kramers–Wannier duality, but for the reader’s benefit we sketch what happens. Of principal interest for us is what happens for large M. With α1 < 1 and α2 fixed it is easy to see that as soon as M is large enough so that the slope 2M exceeds ϕ ′ (0), the situation always resembles the right-hand side of Figure 1.3. For z real and between 0 and 1 the graph of the left-hand side of the eigenvalue equation z → z4M approaches the segment [0,1) on the x-axis as M → ∞. The graph of the right-hand side of the eigenvalue equation for z crosses the x-axis at z = α1 ∈ (0, 1), since we suppose T > Tc . Near z = α1 the right-hand side is approximately linear, with slope (1 − α12 )−1

α1 − α 2 = 0. 1 − α 1 α2

Thus there is a real root z1 of the eigenvalue equation that approaches α1 as M → ∞. The value α1 for z has special interest, since, as the reader can check, z = α1 (or α1−1 ) in the eigenvalue equation (1.54) leads to the eigenvalue λ = 1. At issue for dealing with the semi-infinite volume limit in Theorem 1.2.1 is the question whether TM can have 1 as an eigenvalue above Tc . This does not happen.

30


1 1.5 0.8

1

0.6

0.5 -3

-2

-1 0 -0.5

1

2

0.4

3

0.2 -1 -1.5

-3

-2

-1

0

1

2

3

Figure 1.4: Left: Graphical location for the four nontrivial positive roots of 2Mω − nπ = ϕ(ω) with M = 2 at Tc . Right: The associated points (ωn , e−γn ) at Tc . However, the spectrum of TM has an asymptotically degenerate largest eigenvalue as M → ∞, and this substantially complicates the analysis of the thermodynamic limit. We do not attempt to analyze this limit for plus boundary conditions and T > Tc . Finally, we consider the critical case α1 = 1. It is not difficult to see that the eigenvalue equation 2Mω − nπ = ϕ(ω) has 2M positive roots for n = 1, 2, . . . , 2M and that these roots are associated with 4M independent eigenvectors for TM . For fixed M this allows us to take the semi-infinite volume limit in the same way that works for T < Tc . However, as the reader can see in Figure 1.4, the spectral curve ω → exp(−γ (eiω )) rises to 1 at ω = 0. For this reason the gap in the spectrum of TM about 1 is not uniform in M, and consequently the proof we give for the convergence of the full infinite-volume limit does not go through for T = Tc as it does for T < Tc . Indeed, the representation-theoretic formulas we obtain for the infinite-volume correlations below Tc do not even make sense at T = Tc . We say more about this later.

1.3 The Thermodynamic Limit for T < Tc We now examine the M → ∞ limit for the correlations (1.48) below the critical temperature, that is, for T < Tc . The ingredients that go into the representation (1.48) are (1) the induced rotation for the transfer matrix, (2) the associated isotropic splitting, and (3) the spin operators σi . We use the Pfaffian formulas developed in the appendix in order to analytically control the M → ∞ limit, but it is straightforward (and helpful) to first identify the infinite-volume counterparts for (1), (2), and (3). We start by identifying the induced rotation of the transfer matrix in the infinitevolume limit. Introduce the standard Hilbert space of sequences W := ℓ2 (Z + 12 , C2 ),

1.3 The Thermodynamic Limit for T < Tc

31

where the inner product of two sequences (xk , yk ) and (xk′ , yk′ ) is x¯k xk′ + y¯k yk′ . k∈Z+ 12

It is helpful to think of this as the space of linear combinations qk pk xk √ + yk √ 2 2 k pk for an orthonormal basis √qk2 , √ . 1 1 2 Recall that TM = T V12 T (V2 ) T V12 and that away from the boundary,

the action of T (V2 ) is given by (1.32),

T (V2 ) qk+1 = c2∗ qk+1 + is2∗ pk , T (V2 ) pk = −is2∗ qk+1 + c2∗ pk . This is suitable as a natural infinite-volume limit for the action of T (V2 ) on ℓ2 (Z + 1 , C2 ). In a similar fashion we find that away from the boundary, the action of 2 1 T V12 is given by (see (1.30)) 1 T V12 qk = ch(K1 )qk − i sh(K1 )pk , 1 T V12 pk = i sh(K1 )qk + ch(K1 )pk , which we adopt as the infinite-volume action for T

1 V12 on ℓ2 (Z + 21 , C2 ). Now

introduce the Fourier transform 1 1 x x(z) := √ zk− 2 k , y(z) yk 2π 1

(1.69)

k∈Z+ 2

where z = eiθ ∈ S 1 . Calculation shows that the action on Fourier transforms is given by 1 x(z) ch(K1 ) i sh(K1 ) x(z) 2 T V1 = , y(z) −i sh(K1 ) ch(K1 ) y(z) ∗ x(z) c2 −is2∗ z x(z) T (V2 ) . = ∗ −1 y(z) is2 z c2∗ y(z) Finally, we define

1 1 2 T (V ) := T V1 T (V2 ) T V12

32


as the infinite-volume counterpart to the induced rotation for the transfer matrix. By a calculation that is completely analogous to that used to compute (1.60), one finds that x(z) x(z) = Tz (V ) , (1.70) T (V ) y(z) y(z) where Tz (V ) is the 2 × 2 matrix with entries

z + z−1 Tz (V )ii = a(z) := c1 c2∗ − s1 s2∗ , where i = 1, 2, 2 z + z−1 z − z−1 Tz (V )12 = b(z) := i s1 c2∗ − c1 s2∗ , − is2∗ 2 2 and Tz (V )21 = b(z). The characteristic equation for the eigenvalues λ of Tz (V ) is again given by −1 2 ∗ ∗z + z λ − 2 c1 c2 − s1 s2 λ + 1 = 0. (1.71) 2 We find as before (for the same function γ = γ (z) > 0) that % & 0 w ch γ w sh γ Tz (V ) = = exp γ , w sh γ ch γ w 0

(1.72)

where w = w(z) takes values on the unit circle, so that ww ¯ = 1. The vector w 1 is an eigenvector for this last matrix with eigenvalue eγ . Hence w(z)2 = b(z)/b(z). Factoring 2zb(z), we obtain

so that

2zb(z) = i(1 − c1 )s2∗ 1 − α1−1 z (1 − α2 z), 1 − α1−1 z (1 − α2 z) α1 − z α2 − z−1 w(z)2 = − =− · . −1 α2 − z α1 − z−1 z − α1 (z − α2 )

The factorization on the right-hand side is chosen for the following reason. Since T < Tc , we know 1 < α1 < α2 and hence that z → (αj − z)±1 is analytic in the unit disk without zeros there. For use in later calculations we introduce ( (1.73) Aj (z) := αj − z,


33

defined and analytic for αj > 1 and for z in a neighborhood of the unit disk and normalized so that Aj (1) > 0. We define w+ (z) =

A1 (z) , A2 (z)

(1.74)

which is analytic for z in a neighborhood of the unit disk (|z| < α1 ) and normalized so that w+ (1) > 0. We also define w− (z) =

A2 (z−1 ) , A1 (z−1 )

(1.75)

which is analytic in a neighborhood of the exterior of the unit disk (|z| > α1−1 ) (including the point at ∞) and normalized so that w− (1) > 0. The value of b(z) = w(z) sh γ (z) at z = 1 is a negative multiple of i for α1 > 1. Thus for |z| = 1, w(z) = −iw+ (z)w− (z). (1.76)

In addition to being a useful formula for w(z), this factorization plays an important role in the Wiener–Hopf technique that is used in the analysis of the thermodynamic limit. + − The infinite-volume analogue of the polarization WM is the polarization ⊕ WM W+ ⊕ W− of W , where W+ is the spectral subspace associated with T (V ) for the interval (0, 1) and W− is the spectral subspace associated with T (V ) for the interval (1, ∞). Consulting (1.72), we see that W± can be identified with the ±1 eigenspace of the polarization 0 w Q := − . w¯ 0 Note the − sign in the definition of Q. The orthogonal projection Q± on W± is thus given by Q± = 12 (1 ± Q) . The bilinear form k xk xk′ + yk yk′ becomes ) dz x(z)x ′ (z−1 ) + y(z)y ′ (z−1 ) (1.77) iz S1

after Fourier transform. The easily checked identity w(z) = −w(z−1 )−1 = −w(z ¯ −1 ) implies that W+ and W− are isotropic subspaces with respect to the bilinear form (1.77). The third ingredient we need to translate into the infinite-volume setting is the spin operator σi . Certainly, at the level of induced rotations there is no problem in declaring that si pk = −sgn(k − i)pk , (1.78) si qk = −sgn(k − i)qk ,

34


for all i ∈ Z and k ∈ Z + 12 . Eack si is a complex orthogonal unitary operator on W . Our strategy is to understand the action of σi in terms of si , but because W is infinite-dimensional we need to know that si is actually covered by a transformation σi acting in the spin representation associated with the splitting W+ ⊕ W− . To understand the condition that si must satisfy for the existence of such a σi , we write A(si ) B(si ) Q+ si Q+ Q+ si Q− := si = C(si ) D(si ) Q− si Q+ Q− si Q+

for the matrix representation of si relative to the splitting W+ ⊕ W− . The condition for the existence of σi (acting on a dense linear domain D that contains the vacuum vector) is simply that si be a bounded complex orthogonal transformation with B(si ) and C(si ) Schmidt class operators on the Hilbert space W [148]. We confirm that B and C satisfy this condition in the course of finding representations for A, B, C, and D that are suitable for controlling the infinite-volume and scaling limits. The formulas we use to control these limits require knowlege of D(si )−1 . We explain how to invert D(si ) using the Wiener–Hopf technique [78]. The infinitevolume limit has translation-invariance. The operator si is given by ti sti−1 , where ti is the translation defined by ti pk = pk+i and ti qk = qk+i and we have written s := s0

for brevity. Since ti acts as multiplication by zi on Fourier transforms and this clearly commutes with Q, it follows that D(si ) = ti D(s)ti−1 . To invert D(si ) it is thus enough to invert D(s). The spectrum of s is {+1, −1}, and it is useful to identify the projections on the spectral subspaces for s. Let pk for k > 0, s+ pk := 0 for k < 0, qk for k > 0, s+ qk := 0 for k < 0, and s− := 1 − s+ . Then s = −s+ + s− and after Fourier transform, s+ can be identified with the orthogonal projection on those functions in L2 (S 1 , C2 ) that have holomorphic continuations into the unit disk {z : |z| < 1}. The complementary projection s− is the orthogonal projection on the functions in L2 (S 1 , C2 ) that have holomorphic continuations into the exterior of the unit disk {z : |z| > 1} that vanish at z = ∞. Of course, s+ is also the projection onto the −1 eigenspace for s, and s− is the projection on the +1 eigenspace for s. This notational inversion is dictated by the fact that the Hardy space interpretation of the range spaces s± L2 (S 1 ) is more important for us and s+ L2 (S 1 ) consists of functions whose Fourier coefficients are


35

nonzero only for positive values of the index k, and s− L2 (S 1 ) consists of functions whose Fourier coefficients are nonzero only for negative values of k. It is convenient to introduce the unitary map (thought of as a matrix-valued multiplication operator) 1 1 −w U=√ , (1.79) 1 2 w¯ which is defined such that 1 0 ∗ . U QU = 0 −1 Thus U identifies each of the subspaces W± with L2 (S 1 ). Transforming s by the similarity U , one finds that 1 s + ws w¯ sw − ws A(s) B(s) ∗ U sU = = , (1.80) ¯ − s w¯ s + wsw ¯ C(s) D(s) 2 ws where we have identified the operators A(s), B(s), C(s), and D(s) with the unitarily equivalent operators obtained by similarity transformation with U . The invertibility of D(s) is a central issue for us as is a formula for the inverse. To see how the Wiener–Hopf technique can be used to invert D(s), we write 2D(s) = s + wsw ¯ = w(ws ¯ + sw)

= w((s ¯ − + s+ )w(s− − s+ ) + (s− − s+ )w(s− + s+ )) = 2w(s ¯ − ws− − s+ ws+ ).

The following calculation illustrates the Wiener–Hopf technique and also shows −1 −1 that the inverse of s+ ws+ (on s+ L2 (S 1 ), of course) is given by iw+ s+ w− : −1 −1 −1 −1 s+ ws+ iw+ s+ w− = s+ wiw+ s+ w− −1 = s+ w− s+ w−

−1 = s+ w− (s− + s+ )w− = s+ . −1 In the first line, the removal of s+ is justified since w+ s+ has range s+ L2 (S 1 ); this −1 is a consequence of the fact that w+ (z) is holomorphic in the open unit disk and continuous on the closed unit disk. In the third line the insertion of s− is justified since s+ w− s− = 0. The analogous calculation for s− ws− shows that the inverse −1 −1 of this operator on s− L2 (S 1 ) is given by iw− s− w+ . Thus we find that −1 −1 −1 −1 −1 −1 D(s)−1 = i(w− s− w+ − w+ s+ w− )w = w− s− w− − w+ s+ w+ .

(1.81)

In the course of proving the main convergence result of this section it is of use to know that A(s) is also invertible. In the representation we are working with, A(s) = 21 (s + ws w) ¯ = wD(s)w, ¯ so the invertibility of A(s) is clear and we have −1 −1 A(s)−1 = w+ s− w+ − w− s+ w− .

(1.82)

36


It is straighforward to check that B(s) and C(s) are in the Schmidt class. First observe that the Cauchy integral formula implies ) ) 1 f (ζ ) f (ζ ) 1 s+ f (z) = lim dζ := dζ, z′ →z 2πi S 1 ζ − z′ 2π i S 1 ζ − zint where the integral is oriented counterclockwise and the limit z′ → z is the nontangential limit as z′ approaches z ∈ S 1 from the interior of the unit disk. One also finds that ) ) f (ζ ) 1 f (ζ ) 1 dζ, dζ := − s− f (z) = − lim z′ →z 2πi S 1 ζ − z′ 2π i S 1 ζ − zext where zext ∈ S 1 is the nontangential limit of z′ from the exterior of the unit disk. These representations imply that B = 12 (ws − sw), or ) 1 w(ζ ) − w(z) B(s)f (z) = − f (ζ ) dζ. 2πi S 1 ζ −z is not Limits are not required to define the integral since the kernel w(ζζ)−w(z) −z singular; in fact, the kernel is obviously square integrable since w(z) is smooth on the circle for T < Tc . Therefore B(s) is a Schmidt class operator [148]. Completely analogous considerations show that C(s) is in the Schmidt class. We have proved the following lemma. Lemma 1.3.1 The operators A(sj ) and D(sj ) are invertible with inverses given by (1.81) and (1.82). The operators B(sj ) and C(sj ) are in the Schmidt class. The principal result of this section is our next theorem: Theorem 1.3.1 Suppose that T < Tc . Let Alt(W+ ) be the alternating tensor algebra over W+ carrying the Q Fock representation of Cliff(W ) with normalized vacuum vector |0 = 1 ⊕ 0 ⊕ 0 ⊕ · · · . Write V = Ŵ(T+ (V )) =

∞ * n=0

T+ (V ) ⊗ · · · ⊗ T+ (V ), n

where T+ (V ) is the restriction of T (V ) to W+ . Let σj denote the element of the spin representation with T (σj ) = sj and normalized so that σj2 = 1 and 0|σj |0 > 0. Suppose that all the sites in A have second coordinates between −j and j . Then the thermodynamic limit of the Ising correlations (1.48) (with 0|·|0M now standing + for the normalized vacuum expectation in Alt(WM )) is given by lim 0|σAj Ŵ(TM+ )σAj −1 · · · Ŵ(TM+ )σA−j |0M

M→∞

= 0|σAj V σAj −1 · · · V σA−j |0. The rest of this section is devoted to the functional analysis used to prove this result. For correlations that involve an even number of spins, the theorem can


37

also be demonstrated by rewriting products of spin operators as finite products of elements of the Clifford algebra. This is examined in more detail when we look at formulas for the critical correlations. However, the spin operators themselves are not finite products in the Clifford algebra but limits of finite products. To handle these limits easily and also because we want formulas that behave well in the scaling limit, we use the infinite Pfaffian formulas in Appendix A to prove Theorem 1.3.1. ± + − Let Q± M denote the orthogonal projection on WM and write QM = QM − QM . There is a natural isometric embedding WM → W given by M− 12

WM ∋

k=−M+ 21

M− 21

xk qk + yk pk →

k=−M+ 21

xk qk + yk pk ∈ W,

⊥ and we write W = WM ⊕WM for the associated orthogonal splitting of W . In what ±1 ±1 follows we write TM for TM ⊕ 0 and QM for QM ⊕ 0, extending these operators ⊥ by 0 on WM to act on all of W .

Lemma 1.3.2 The operators TM±1 converge strongly to T (V )±1 on W as M → ∞. The operator QM converges strongly to Q on W as M → ∞. Proof. Let v be any finite linear combination of the vectors qk and pk . Then it is clear that for all sufficiently large M we have TM v = T (V )v. Since TM is uniformly bounded in operator norm by the maximum of exp γ (z) for z ∈ S 1 , strong convergence on this dense set implies strong convergence on all of W . Now let PM denote the orthogonal projection of W on WM . Observe that PM T (V )−1 PM converges strongly to T (V )−1 on W as M → ∞ since PM converges strongly to the identity. However, PM T (V )−1 PM − TM−1 = PM TM−1 (TM − T (V ))T (V )−1 PM ,

(1.83)

and since TM − T (V ) tends strongly to zero as M → ∞ and TM−1 is uniformly bounded (again by the maximum of exp γ (z) for z ∈ S 1 ), it follows that the right-hand side of (1.83) tends strongly to 0 on W . It follows that T (V )−1 − TM−1 = T (V )−1 − PM T (V )−1 PM + PM T (V )−1 PM − TM−1 also tends strongly to 0 on W as M → ∞. To show that QM converges to Q, observe that if XM is a sequence of self-adjoint operators that converges strongly to X and λ is a point of continuity for the spectral resolution E(λ) associated with X, then the strong limit s − limM→∞ EM (λ) is equal to E(λ), where EM (λ) is the spectral resolution of XM . This is a special case of Theorem 1.15 in Kato [71]. For XM = TM and X = T (V ) we have Q+ M = EM (1) and Q+ = E(1). Since T < Tc , the estimate (1.68) shows that T (V ) has a gap in its spectrum about 1. Thus 1 is a point of continuity for the spectral resolution of T (V ) and we proved that Q+ M tends strongly to Q+ . Since QM = 2Q+ M − 1 and Q = 2Q+ − 1, the proof of the lemma is finished.

38


For the convenience of the reader we mention the two tools that we use in proving Schmidt class convergence. Proofs can be found in [148], where the reader can also find an extensive discussion of the more general Schatten classes. The first result is the quantitative version of the fact that the Schmidt class is an ideal in the algebra of bounded operators on a Hilbert space. If A and B are bounded operators on a Hilbert space and B is a Schmidt class operator, then AB is in the Schmidt class and AB2 ≤ A · B2 , where A is the operator norm of A and B2 is the Schmidt norm of B. The second result is, Lemma 1.3.3 Suppose that An is a sequence of bounded operators on a Hilbert space that converges strongly to A. Suppose that Bn is a sequence of Schmidt class operators that converges in Schmidt norm to B. Then An Bn converges in Schmidt norm to AB. The convergence result at the core of our treatment of the thermodynamic limit is the following. Theorem 1.3.2 Suppose that T < Tc and write, Aj := A(sj ) = Q+ sj Q+ , Bj := B(sj ), Cj := C(sj ), and Dj := D(sj ). Define + + − AM,j := Q+ M sj QM , BM,j := QM sj QM , + − − CM,j := Q− M sj QM , DM,j := QM sj QM .

Then lim Bj − BM,j 2 = 0 and

M→∞

lim Cj − CM,j 2 = 0,

M→∞

and ±1 ±1 s − lim A±1 and s − lim DM,j = Dj±1 , M,j = Aj M→∞

A−1 M,j

where to act by 0

−1 and DM,j ⊥ on WM .

M→∞

are the inverses of AM,j and DM,j on WM , each extended

Proof. We start with the convergence result for BM,j . Since PM tends strongly to the identity, it follows from Lemma 1.3.3 that Bj − PM Bj PM 2 tends to 0 as M → ∞. Thus to show that Bj − BM,j tends to 0 in Schmidt norm as M → ∞ it is enough to show that PM Bj PM − BM,j = PM (Bj − BM,j )PM tends to 0 in Schmidt norm. Now write [A, B] = AB − BA for the commutator of A and B. Then one has Bj = Q+ sj Q− = (Q+ sj − sj Q+ )Q− = [Q+ , sj ]Q− , and for the same reason, − BM,j = [Q+ M , sj ]QM .


39

Thus − + − Bj − BM,j = [Q+ , sj ]Q− − [Q+ , sj ]Q− M + [Q+ , sj ]QM − [QM , sj ]QM , + − = [Q+ , sj ](Q− − Q− M ) + [Q+ − QM , sj ]QM .

But [Q+ , sj ] = Q+ sj Q− − Q− sj Q+ is in the Schmidt class and Q− − Q− M tends strongly to 0 as a consequence of Lemma 1.3.2. Thus [Q+ , sj ](Q− − Q− M ) tends to 0 in Schmidt norm. To see that PM (Bj − BM,j )PM tends to 0 in Schmidt norm it is enough to show this for − + − PM [Q+ − Q+ M , sj ]QM PM = PM [Q+ − QM , sj ]PM QM .

Since Q− M is strongly convergent it suffices to prove that PM [Q+ − Q+ M , sj ]PM tends to 0 in Schmidt norm as M → ∞. Let RM (λ) := (λ − TM )−1 and R(λ) := (λ − T (V ))−1 denote the resolvents for TM and T (V ). When dealing with resolvent identities ⊥ keep in mind that TM extends to act as 0 on WM . Let Sǫ denote the counterclockwise 1 1 oriented circle of radius 2 centered at 2 + ǫ in the complex plane, where ǫ > 0 is a small postitive number chosen as follows. Choose ǫ > 0 such that Sǫ encloses and stays a positive distance from the part of the spectrum of T (V ) that is contained in the interval (0, 1). Since T (V ) has a gap in its spectrum about 1 and has spectrum bounded away from 0, this is always possible. The curve Sǫ also encloses the restriction of the spectrum of TM to the interval (0, 1); the shift by ǫ is done so that Sǫ does not enclose 0, which happens to be in the spectrum of TM . The standard contour integral representation for spectral projections is [71] ) ) 1 1 + QM = RM (λ) dλ, and Q+ = R(λ) dλ. 2πi Sǫ 2π i Sǫ An elementary calculation shows that [RM (λ), sj ] = RM (λ)[TM , sj ]RM (λ), [R(λ), sj ] = R(λ)[T (V ), sj ]R(λ). Furthermore, for M − 1 > |j | we have [TM , sj ] = [T (V ), sj ]. To see this, observe that TM v = T (V )v for v ∈ WM−1 , and because WM−1 is invariant under sj it follows that [TM , sj ]v = [T (V ), sj ]v for v ∈ WM−1 . Also T (V ) is a finite difference operator that is a (matrix-valued) linear combination of translations t 1 , t 0 = id, and t −1 . This implies that [T (V ), sj ]pk = [T (V ), sj ]qk = 0 whenever ⊥ |k − j | > 1. Thus [T (V ), sj ] is zero on WM−1 when |j | < M − 1. Since TM = 0 ⊥ ⊥ on WM it is straightforward to check that [TM , sj ] = 0 on WM−1 for |j | < M − 1.

40


Thus for M − 1 > |j | the commutator of TM with sj agrees with the commutator of T (V ) with sj , so that ) 1 [Q+ − Q+ , s ] = R(λ)[T (V ), sj ]R(λ) dλ M j 2πi Sǫ ) 1 − RM (λ)[T (V ), sj ]RM (λ) dλ. 2πi Sǫ + Subtract and add 2π1 i Sǫ RM (λ)[T (V ), sj ]R(λ) dλ on the right in this last equality and then make use of the resolvent identities R(λ) − RM (λ) = RM (λ)(T (V ) − TM )R(λ), RM (λ) − R(λ) = R(λ)(TM − T (V ))RM (λ), to obtain [Q+ − Q+ M , sj ] =

) 1 RM (λ)(T (V ) − TM )R(λ)[T (V ), sj ]R(λ) dλ 2πi Sǫ ) 1 + RM (λ)[T (V ), sj ]R(λ)(TM − T (V ))RM (λ) dλ. 2πi Sǫ

Multiply this last expression on the left and on the right by PM and then use RM (λ)PM = PM RM (λ) to get ) 1 RM (λ)XM (λ)R(λ)PM dλ PM [Q+ − Q+ , s ]P = j M M 2πi ) Sǫ (1.84) 1 + RM (λ)YM (λ)RM (λ) dλ, 2πi Sǫ where XM (λ) := PM (T (V ) − TM )R(λ)[T (V ), sj ], YM (λ) := [T (V ), sj ]R(λ)(TM − T (V ))PM . We estimate the Schmidt norms of XM (λ) and YM (λ). Observe that [T (V ), sj ] is zero on the basis elements {qk , pk } except for k = j ± 12 , and (T (V ) − TM )PM is zero on the basis elements {qk , pk } except for k = ±(M − 21 ). Thus XM (λ) has only 16 nonvanishing matrix elements in the basis {qk , pk }. Each of these nonvanishing matrix elements can be written in the form ) 2π 1 1 u, (Tθ (V ) − λ)−1 ve±i(M− 2 )θ+i(j ± 2 )θ dθ, (1.85) 0

for vectors u, v ∈ C2 that do not depend on M, and employing a slight abuse of notation, Tθ (V ) is the 2 × 2 matrix Tz (V ) (1.70) evaluated at z = eiθ . Since θ → Tθ (V ) is a smooth function on the circle and λ ∈ Sǫ stays away from the


41

spectrum of T (V ), integration by parts in (1.85) shows that these matrix elements tend to 0 as M → ∞ uniformly for λ ∈ Sǫ . Hence XM (λ)2 tends to 0 as M → ∞ uniformly for λ ∈ Sǫ . Since R(λ) and RM (λ) are uniformly bounded for λ ∈ Sǫ , we see that the first integral on the right-hand side of (1.83) tends to 0 in Schmidt norm as M → ∞. A completely analogous argument works for the second term on the right-hand side of (1.83), and we have finished the proof that Bj − BM,j 2 tends to 0 as M → ∞. Making only obvious changes, the same proof works to show that Cj − CM,j 2 goes to 0 as M → ∞. ±1 We turn to the strong convergence results for A±1 M,j and DM,j . Since AM,j = + + + QM sj QM and QM converges strongly to Q+ , it follows that AM,j converges strongly to Q+ sj Q+ = Aj . The same argument shows that DM,j converges −1 strongly to Dj . Strong convergence for the inverses A−1 M,j and DM,j requires a little more argument. In particular, we first need to establish that AM,j and DM,j are invertible (on WM , of course) for all sufficiently large values of M. Introduce Ej := Aj + Dj and EM,j := AM,j + DM,j , Fj := Bj + Cj and FM,j := BM,j + CM,j . Evidently sj = Ej + Fj with Ej being the “part” of sj that commutes with Q and Fj the “part” of sj that anticommutes with Q. Observe that PM (Ej + Fj )PM = PM sj PM = EM,j + FM,j , so PM Ej PM − EM,j 2 = PM Fj PM − FM,j 2 .

In the first part of this proof we showed that PM Fj PM − FM,j 2 tends to 0 as M → ∞, so we also have lim PM Ej PM − EM,j 2 = 0.

M→∞

(1.86)

We now use this to prove that EM,j is invertible (for M sufficiently large) by showing that PM Ej PM is invertible on WM with a uniform bound on the operator −1 norm of the inverse PM Ej PM . Write PM⊥ = 1 − PM . Then, since PM sj PM⊥ = PM⊥ sj PM = 0,

it follows that where

PM Ej PM + PM⊥ Ej PM⊥ = Ej + δM,j , δM,j = −PM Ej PM⊥ − PM⊥ Ej PM = PM Fj PM⊥ + PM⊥ Fj PM .

(1.87)

But Fj is in the Schmidt class and PM⊥ tends strongly to 0 as M → ∞. Thus δM,j tends to 0 in Schmidt norm as M → ∞. Since Ej is invertible it follows that the right-hand side of (1.87) is invertible with a uniformly bounded inverse for all sufficiently large values of M. But the left-hand side of (1.87) is a direct sum, so the

42


inverse of PM Ej PM (on WM ) exists and is uniformly bounded for all sufficiently large values of M. Since EM,j = AM,j + DM,j is a direct sum, it follows that AM,j and DM,j are invertible (on WM ) with uniformly bounded inverses for all sufficiently large values of M. Now suppose that v ∈ Wm for m < M and M sufficiently large so that the right-hand side of (1.87) is invertible. Then −1 v (Ej + δM,j )−1 v = PM Ej PM and

lim

M→∞

−1 v = Ej−1 v. PM Ej PM

−1 −1 Since EM,j differ by an operator that tends to 0 in operator and PM Ej PM norm on WM as M → ∞, it is easy to see that −1 −1 lim EM,j v = lim PM Ej PM v = Ej−1 v. (1.88) M→∞

M→∞

−1 ⊥ ) is uniformly bounded for The sequence of operators EM,j (acting by 0 on WM all sufficiently large M, so the strong convergence on a dense set given by (1.88) promotes to strong convergence −1 s − lim EM,j = Ej−1 . M→∞

Since Q+ M

−1 + converges strongly to Q− as M → ∞, it follows that A−1 M,j = EM,j QM −1 −1 tends strongly to Ej−1 Q+ = Aj−1 . For the same reason, DM,j = EM,j Q− M tends −1 −1 strongly to Ej Q− = Dj . This finishes the proof of Theorem 1.3.2.

This result and the formulas for vacuum expectations of products in spin representations that are developed in the appendix now make it possible to control the thermodynamic limit for Ising correlations below the critical temperature. We start with a result for the one-point function. , Theorem 1.3.3 Suppose that T < Tc , j ∈ Z and write σj M for the QM Fock expectation 0|σj |0M , where as above, |0 is the normalized vacuum vector in + Alt(WM ). Then lim σj M = σ , M→∞

where σ = 0|σ |0 is the Q Fock expectation of the spin operator σ = σ0 , and |0 is the normalized vacuum vector in Alt(W+ ). Also we have |σ |2 = det |D(s)|.

(1.89)

+ Proof. The action of σj in Alt(WM ) is unitary since σj2 = 1 and σj∗ = σj . We also know from Theorem 1.3.2 that DM := DM (sj ) is invertible for all large enough values of M. Hence Theorem A.7.2 implies that , ∗−1 ∗ −1 − 21 BM BM DM ) . | σj M |2 = det |DM | = det(1 + DM


43

∗ Since sj is self-adjoint, DM = DM . Also, for a Schmidt class operator B on a ∗ Hilbert space, B 2 = B2 [148]. Thus Theorem 1.3.2 and Lemma 1.3.3 imply ∗−1 ∗ −1 that DM BM converges in Schmidt norm to D ∗−1 B ∗ , and BM DM converges in −1 Schmidt norm to BD , where we write B := B(sj ) and D := D(sj ). The ∗−1 ∗ −1 product DM BM BM DM then converges in trace norm [147], and the continuity of the determinant in the trace norm implies that

, 1 lim | σj M |2 = det(1 + D ∗−1 B ∗ BD −1 )− 2 = det |D(sj )|,

M→∞

where the last equality follows again ,from - Theorem A.7.2 and the fact, that - sj is unitary. Since σj is unitary we have | σj |2 = det |D(sj )|, and since σj M > 0 , , , and σj > 0 it follows that σj M → σj as M → ∞. Translation invariance shows that D(sj ) is similar to D(s), and the invariance of the determinant under unitary similarity implies that det |D(sj )| = det |D(s)|, and this finishes the proof.

In Chapter 2 we diagonalize D(s) by an elliptic substitution and use (1.89) to calculate the spontaneous magnetization σ (this is close in spirit to Yang’s original derivation [161]). In the final section of Chapter 2 we calculate σ using results for the asymptotics of Toeplitz determinants using an observation originally due to Montroll, Potts, and Ward [97]. Next we use the Pfaffian formula in Theorem A.7.4 for vacuum expectations of products in the spin group to control the convergence of the thermodynamic limit for the multispin correlations below Tc . In using this formula to investigate the thermodynamic limit we can deal with the product of spin operators σ (an ) · · · σ (a1 ) directly. A straightforward application of Theorem A.7.4 leads to matrix elements in the Pfaffian formula for the n-point correlations that involve factors erγ for r > 0. This is not a problem for the thermodynamic limits since these factors remain bounded in this limit. The advantage of the Theorem A.7.4 for the consideration of the scaling limit considered in Chapter 2 is that by a very mild refactorization of the product σ (an ) · · · σ (a1 ) we can eliminate the appearance of all factors erγ in the resulting formula. This is useful since erγ becomes an unbounded operator in the scaling limit, and it will also produce formulas for which the clustering of the scaling functions is manifest. We are now ready to prove Theorem 1.3.1. Proof. [Proof of Theorem 1.3.1] Suppose that A consists of the n sites {a1 , a2 , . . . , an } in Z2 and write aj = (xi , yi ). Suppose that the second coordinates yi are in natural order, yi+1 ≥ yi .

44


Define σ (aj ) = Ŵ(TM+ )−yj σxi Ŵ(TM+ )yj .

Then Theorem 1.2.1 implies

σA + M = σ (an )σ (an−1 ) · · · σ (a1 )M .

Remark 1.3.1. Note that we write the sequence of points a1 , a2 , . . . , an in “natural order.” However, assuming that the second coordinates of the aj are increasing as j increases means that the product σ (an )σ (an−1 ) · · · σ (a1 ) must be ordered from right to left because of the way in which the transfer formalism works. If we suppose instead that the second coordinates of the points a1 , a2 , . . . , an occur in decreasing order, the appropriate product is σ (a1 )σ (a2 ) · · · σ (an ). I have adopted the first convention because I find it easier to think of the transfer formalism working from “bottom” σ (a1 ) to “top” σ (an ). This convention inspires the backward labeling of products in the last theorems in the appendix, and we return to this matter yet again when we discuss time ordering in Chapter 4. Theorem A.7.4 applies to this representation for the spin correlation, and we obtain n , 1 −UM BM σ (an )σ (an−1 ) · · · σ (a1 )M = , (1.90) σ (aj ) M Pf LM CM 1 j =1

where BM , CM , LM , and UM are computed from the induced rotation s(aj ) = T (σ (aj )) acting on WM . Theorem 1.3.2 and Lemma 1.3.2 show that D(s(aj ))−1 converges in the strong operator topology as M → ∞. The map s(aj ) is complex orthogonal, so that s(aj )s(aj )τ = 1, and since s(aj )2 = 1 it follows that s(aj )τ = s(aj ). Thus D(s(aj ))τ = A(s(aj )), and so D(s(aj ))−τ = A(s(aj ))−1 . Theorem 1.3.2 and Lemma 1.3.2 show that D(s(aj ))−τ converges in the strong operator topology as M → ∞. Since UM and LM have entries that are products of D(s(aj ))−τ and D(s(aj ))−1 , it follows that U M and LM converge in the strong operator topology to limits that we denote by U and L. The same results together with Theorem 1.3.3 show that the B(s(aj )) and C(s(aj )) converge in Schmidt norm. Theorem 1.3.2 and Theorem 1.3.3 then imply that BM and CM converge in Schmidt norm to limits that we denote by B and C. Thus using Theorem 1.3.3 again we see that UM BM and LM CM converge in Schmidt norm to UB and LC. As explained in the appendix, the Pfaffian on the right-hand side of (1.90) is continuous in the Schmidt norm for UB and LC, and the one-point functions converge by Theorem 1.3.3. Thus the right-hand side of (1.90) converges as M → ∞ to 1 −UB n σ Pf . LC 1

Theorem A.7.4 (now applied in reverse) implies that the resulting formula is the infinite-volume vacuum expectation in Theorem 1.3.1, and this finishes the proof of that theorem.

1.4 Duality and the Thermodynamic Limit Above Tc

45

When we prove the convergence of the scaling limit we change the representation of the spin correlations by altering the factorization of σ (an )σ (an−1 ) · · · σ (a1 ) to take advantage of the smoothing properties of the transfer matrix. We will also write out the Pfaffian formula more explicitly when we examine the scaling limits, because the precise version of the formulas are used to establish the τ function connection that is taken up in Chapter 5.

1.4 Duality and the Thermodynamic Limit Above Tc In this section we introduce a duality for the Ising model that translates spin correlation functions with open boundary conditions above Tc into correlation functions for “disorder” variables with plus boundary conditions below Tc . The results we established for the thermodynamic limit below Tc are then applied to deal with the thermodynamic limit above Tc . As above, let denote the (2M +1)× (2N +1) lattice of ordered pairs of integers (m, n) with |m| ≤ M and |n| ≤ N. The partition function for the Ising model with open boundary conditions is obtained by omitting all terms in the energy that arise from coupling to the boundary spins. Following [58] we now describe a notion of duality that applies to a slightly wider class of interactions than the translation-invariant Ising model. Suppose that σ : → {−1, +1} is a spin configuration. It is convenient to identify this configuration with the subset of consisting of those points (m, n) with σ (m, n) = −1. Thus we identify the space of spin configurations C() with the power set P(). The power set P() is naturally an abelian group with the group operation given by symmetric difference, P() ∋ A, B → A • B := A ∪ B\A ∩ B, the union of A and B minus the intersection of A and B, with the empty set ∅ acting as the identity. It is not hard to see that this operation is associative. The reader is invited to check that both (A • B) • C and A • (B • C) are given by the complement of the black region in Figure 1.5 in the union A ∪ B ∪ C. More generally, one can see by induction that . A1 • A2 • · · · • An = A1 ∪ A2 ∪ · · · ∪ An \ Ai ∩ Aj . i=j

For Y a subset of define σY : P() → {−1, +1} by σY (X) = σi (X) = (−1)|X∩Y | . i∈Y

It is easy to check that the map P() ∋ Y → σY is a group homomorphism from P() onto the characters of P().

46


B

A

C

Figure 1.5: The set A • B • C is the complement of the dark region in the union A ∪ B ∪ C. Next we introduce the subset B of P() consisting of B = {{r, s}|r and s are nearest-neighbor sites in } ∪ ∂, where ∂ = {{s}|s = (s1 , s2 ) and either |s1 | = M or |s2 | = N}.

The set B is the set of bonds in our model and evidently consists of two-point and one-point bonds (all the boundary sites are one-point bonds). The one-point bonds are included so we can force plus boundary conditions by a suitable limiting procedure. Now suppose that K is a complex-valued function on B. This function defines the interaction energy for our model. We define a partition function Z(, K) by Z(, K) := exp K(b)σb (X) . X⊂

b∈B

Note that if b is a two-point bond, then σb (X) = 1 if the configuration X has the same spin assignment for both points in b, and σb (X) = −1 if the configuration X makes different spin assignments for the points in b. We obtain the nearestneighbor Ising model partition function with the following specialization,   if b is a horizontal two-point bond, K1 K(b) = K2 (1.91) if b is a vertical two-point bond,   H > 0 if b is a one-point boundary bond.

Plus correlations are obtained in the limit H → ∞ (where ratios of partition functions are involved). For the present we do not need to specialize K. To understand duality we next introduce some notions that are useful in writing down the high- and low-temperature expansions for the partition function. Let P(B) denote the group of subsets of the set of bonds B under symmetric difference. Define a group homomorphism π : P(B) → P() by π {b1 , b2 , . . . , bn } = b1 • b2 • · · · • bn .


47

We write K for the kernel of this homomorphism and we refer to K as the set of closed paths in P(B). The reader can check that closed paths in are either closed paths in the usual sense in the interior of or paths that terminate with a point bond on the boundary of . Next define a group homomorphism γ : P() → P(B) by γ (X) := {b ∈ B|σX (b) = −1}. Evidently, γ (X) consists of all those two-point bonds only one end of which is in X and all the one-point bonds in X that are boundary sites of . We write Ŵ for the image Im(γ ). The duality that we are interested in exchanges low- and high-temperature expansions for the partition function, so it is natural to begin by explaining the low-temperature expansion of the partition function. Observe that the partition function can be rewritten e−2K(b) eK(b) Z(, K) = X⊂ b∈γ (X)

= =

eK(b)

e−2K(b)

X⊂ b∈γ (X)

b∈B

b∈B

e

K(b)

e−2K(b) .

γ ∈Ŵ b∈γ

b∈B

The factors e−2K(b) are small when the temperature is near 0, and so it is easy to identify the dominant contributions to the sum on the right in the limit T → 0. Hence the term low-temperature expansion. To obtain the high-temperature expansion first note that eK(b)σb (X) = ch K(b) + σb (X) sh K(b), since σb (X) is either +1 or −1. Substituting this in the expression for the partition function and factoring out b ch K(b) to take advantage of the fact that thK(b) is small at high temperature, we obtain Z(, K) = = However,

i

j =1

ch K(b)

ch K(b)

(1 + σb (X)thK(b))

X⊂ b∈B

b∈B

b∈B

i

σbj (X)thK(bj ).

X⊂ {b1 ,...,bi }∈P(B) j =1

σbj (X) = σb (X), where b = b1 • b2 • · · · • bi and

X⊂

σb (X) =

0 2||

if b = ∅, if b = ∅.

48


To see that this sum is 0 when b = ∅ suppose that b = ∅ and s ∈ b. If X ⊂ is a subset of that does not contain s, then it is clear that σb (X) = −σb (X ∪ {s}). Thus in the sum over all subsets of , each term σb (X) for a subset X that does not contain s is exactly canceled by a term σb (X ∪ {s}) for a subset that does contain s. Since σ∅ (X) = 1 for all X, it follows that X⊂ σ∅ (X) is just the number of subsets of , or 2|| . Recalling that the subsets {b1 , b2 , . . . , bi } with b1 • b2 • · · · • bi = ∅ are just the closed paths K, we see that the high-temperature expansion becomes Z(, K) = 2||

b∈B

ch K(b)

i

thK(bj ).

{b1 ,b2 ,...,bi }∈K j =1

We now introduce a dual system (∗ , K ∗ ) as follows. Let Z2 21 := Z + 12 × Z + 21 denote the lattice of half odd integers and write ∗ for the subset of this lattice ∗ := {(k, ℓ) ∈ Z2 12 : |k| < M, |ℓ| < N} ∪ ∂∗ ,

where

∂∗ := {(k, ℓ) : |k| = M + 21 , |ℓ| < N }

∪ {(k, ℓ) : |k| < M, |ℓ| = N + 12 }.

The integers M and N are the same that appear in the definition of . We define the set of dual bonds B ∗ := {{i, j } : i and j are nearest neighbors in ∗ } ∪ C, where C = ∪α,β=±1 C(α, β) and the corner bonds C(α, β) are the two-point bonds 4 3 C(α, β) = α(M − 21 ), β(N + 12 , α(M + 21 ), β(N − 12 ) .

There is a map ∗ : B → B ∗ that is important for us defined as follows. If b is a pair bond in B then ∗b is the bond in B ∗ that “crosses” b at right angles. If b is a point bond in B then ∗b is the two-point bond in B ∗ with elements in ∂∗ that lies “closest” to b. Some examples will make clear what we have in mind: ∗{(m, n), (m, n + 1)} = {(m − 12 , n + 12 ), (m + 21 , n + 12 )}, ∗{(−M, N)} = C(−1, 1),

∗{(M, n)} = {(M + 12 , n − 21 ), (M + 21 , n + 12 )} for |n| < N .

The property of the map ∗ that we wish to exploit is that it establishes a bijection of Ŵ onto K∗ . Note that since the bonds in B ∗ are all two-point bonds, the closed paths in K∗ are closed paths in the usual sense. In particular, each path b∗ ∈ K∗


49

Figure 1.6: Left: The set of bonds, γ (X) associated with the set of sites X (the solid circles). The solid circles on the boundary are also point bonds. Right: The associated path b∗ = ∗γ (X) in the dual lattice Z2 21 . encloses a set of sites X in the dual lattice Z2 . One sees that ∗γ (X) = b∗ . The situation is pictured in Figure 1.6. The map ∗ sends Ŵ into K∗ and we introduce a dual interaction K ∗ such that this exchange relates the low-temperature expansion of the partition function for interaction K with the high-temperature expansion for the interaction K ∗ . Note that there is an extra factor of 2|| relating the two partition functions. However, this is not of significance for the correlation functions that are ratios of partition functions. Comparing the high- and low-temperature expansions, we see that we should define thK ∗ (∗b) := e−2K(b) .

The reader can check that K ∗ is equivalently defined by the more symmetrical relation sh 2K ∗ (∗b) sh 2K(b) = 1.

The reader should note that this is a generalization of the relation between Kj and Kj∗ already introduced above. One possible source of confusion should be kept in mind. In the generic interior case, if the bond b is a horizontal bond then K ∗ (∗b) is a vertical coupling, and if b is a vertical bond then K ∗ (∗b) is a horizontal coupling. Following Kadanoff and Ceva [69] we identify correlations with ratios of partition functions. Suppose that B = {b1 , b2 , . . . , bi } is a set of bonds in . Define a modified interaction, K(b) if b ∈ / B, KB (b) = −K(b) if b ∈ B.

50


The dual interaction KB∗ is seen to be K ∗ (b∗ ) ∗ ∗ KB (b ) = K ∗ (b∗ ) + Since iσ

iπ ∗ σb , = exp 2

6

= (−i)|B|

b∗

it follows that

5

b∗ ∈∗B

iπ 2

if b∗ ∈ / ∗B, ∗ if b ∈ ∗B.

σb ∗

(∗ ,K ∗ )

Z(∗ , KB∗ ) , Z(∗ , K ∗ )

where |B| = | ∗ B| is the number of bonds in ∗B. A judicious choice of ∗B allows us to write any even-spin correlation function in this form, and this is essentially the reason for introducing the modified interaction KB∗ . On the other hand, if we compare the low-temperature expansion for the ratio Z(∗ ,K ∗ ) Z(,KB ) with the high-temperature expansion for Z(∗ ,KB∗ ) by making use of the Z(,K) bijective correspondence between Ŵ and K∗ , we obtain −2K (b) Z(, KB ) KB (b)−K(b) γ ∈Ŵ b∈γ e B = e −2K(b) Z(, K) γ ∈Ŵ b∈γ e b∈B ∗ ∗b∈∗γ thKB (∗b) ∗γ ∈K∗ = e−2K(b) ∗ ∗b∈∗γ thK (∗b) ∗γ ∈K∗ b∈B 7 8 ch K ∗ (∗b) Z(∗ , K ∗ ) B = e−2K(b) ∗ ∗, K ∗) ch K (∗b) Z( B b∈B ∗b∈B∗ 8 7 ch K ∗ (∗b) Z(∗ , K ∗ ) B −2K(b) = e ∗ (∗b) ∗, K ∗) i sh K Z( b∈B ∗b∈∗B = (−i)|B|

Z(∗ , KB∗ ) . Z(∗ , K ∗ )

We used ch(K ∗ (∗b) + iπ2 ) = i sh K ∗ (∗b) to obtain the fourth equality and the duality relation e−2K(b) cthK ∗ (∗b) = 1 to obtain the fifth equality. We are ready to specialize to the Ising model interaction (1.91). To describe the dual interaction we say that ∗b ∈ B ∗ is an interior bond if at least one of its endpoints is an interior site in ∗ ; we say that ∗b ∈ B ∗ is a boundary bond if both its endpoints are on the boundary of ∗ . The reader can easily check that the dual interaction is given by  ∗  if *b is an interior horizontal bond, K2 ∗ K (∗b) = K1∗ if *b is an interior vertical bond,   1 − 2 ln(thH ) if *b is a boundary bond.


51

Note the exchange of horizontal and vertical couplings. Next we consider the equality 5 6 Z(∗ , KB∗ ) Z(, KB ) σb∗ = (−i)|B| = (1.92) ∗, K ∗) Z( Z(, K) ∗ ∗ ∗ b ∈∗B

( ,K )

in the limit H → ∞. In the configuration sums that define both Z(, KB ) and Z(, K) those configurations in which all the boundary spins are +1 have associated weights exp (H |∂|); configurations in which at least one boundary spin is −1 are smaller by at least a factor exp(−H ). We see then that the limiting behavior for the ratio of partion functions is Z + (, KB ) Z(, KB ) = + , H →∞ Z(, K) Z (, K) lim

where the configuration sums for Z + (, KB ) and Z + (, K) are restricted to “plus” boundary conditions. On the other hand, since lim − 21 ln (thH ) = 0,

H →∞

it follows that lim (−i)

H →+∞

, KB∗ ) = ∗ Z( , K ∗ )

|B| Z(

∗

5

σb ∗

b∗ ∈∗B

6open

,

(∗ ,K ∗ )

where open boundary conditions refer to configuration sums in which all couplings of boundary spins are set to 0. Thus we obtain 5 6open Z + (, KB ) = + σb∗ . (1.93) Z (, K) ∗ ∗ ∗ b ∈∗B

( ,K )

We now use this result to rewrite even spin correlations above Tc as vacuum expectations of disorder variables for plus boundary conditions below Tc . Note that if A ⊂ ∗ is a collection of sites in the dual lattice ∗ and the number |A| of sites in A is odd, then 5 6open open σA (∗ ,K ∗ ) = σi = 0. i∈A

(∗ ,K ∗ )

The reason is that the product i∈A σi (X) changes sign under the tranformation that changes the configuration X to −X (all spins are flipped), whereas the weight given to the configurations X and −X is the same for open boundary conditions. Suppose now that A ⊂ ∗ and |A| is even. We choose a collection of bonds ∗B such that σA = σb∗ . b∗ ∈∗B

52


Figure 1.7: An even site correlation on ∗ realized with paths of bonds connecting pairs of sites. A simple way to do this is to pair up the sites in A and join each pair by a path of bonds. The situation is pictured in Figure 1.7. The only stipulation we make regarding these paths is that they stay away from the boundary of ∗ . The reader may check that a bond on the boundary of ∗ would flip the associated dual-point bond in to produce a local “minus” boundary condition (in the limit H → ∞). A horizontal bond {(k − 1, ℓ), (k, ℓ)} in the lattice ∗ crosses the vertical bond {(k − 21 , ℓ − 21 ), (k − 21 , ℓ + 12 )} in the lattice (its image under the duality map ∗). We want to understand how to incorporate a change in the sign of the vertical interaction constant K2 for this bond in the transfer matrix formula for the partition function on with plus boundary conditions. It clarifies matters for our initial discussion if we return to the “presymmetrized” version of the transfer formalism in which both the factors V1 and V2 appear. Furthermore, it is of use to label the instances of V1 and V2 in the dual transfer formalism to reflect their position in the integer lattice . Thus V1 (j ) is the instance of V1 in the transfer formalism that incorporates the bonds of the j th row of the lattice , and V2 (j, j − 1)) is the instance of V2 that incorporates the vertical bonds that join the (j − 1)th and j th rows. Recalling the transfer matrix construction in (1.13), one sees that the vertical bonds all occur in the factor M−1

eK2 e−K2 V2 = . e−K2 eK2 j j =−M+1

To change the bond strength from K2 to −K2 at the (k − 21 )th site (which is the appropriate site for the bond {(k −1, ℓ), (k, ℓ)}), one need only multiply this factor in the transfer matrix by 01 10 k− 1 . Recalling the formula for the Brauer–Weyl 2


53

representation (1.19), (1.20), we see that 0 1 = rk−1 rk , 1 0 k− 1 2

where rk := pk σk+ 1 . 2

Thus replacing the factor V2 (ℓ− 21 , ℓ+ 12 ) in the transfer formalism by rk−1 rk V2 (ℓ− 1 , ℓ + 12 ) has the desired effect of incorporating a vertical bond flip for the 2 (k − 21 )th vertical bond. To reproduce the open boundary condition spin correlation for σ(k−1,ℓ) σ(k,ℓ) it suffices to replace V2 (ℓ − 21 , ℓ + 12 ) by rk−1 rk V2 (ℓ − 12 , ℓ + 12 ) for the dual interaction with plus boundary conditions on . Next we calculate rk V1 rk ,

which is a similarity transform of V1 since rk2 = 1. Because the spin operator σk+ 1 2 commutes with V1 , this is equivalent to pk V1 pk . However, σi σi+1 if k ∈ / (i, i + 1), pk σi σi+1 pk = −σi σi+1 if k ∈ (i, i + 1). It follows that the similarity transform of V1 by rk introduces a bond flip (K1 → −K1 ) for the horizontal bond (k − 21 , k + 12 ). Thus to reproduce the spin correlation σ(k,ℓ) σ(k,ℓ−1) for open boundary conditions on the lattice ∗ it suffices to replace V1 (ℓ − 21 ) with rk V1 (ℓ − 12 )rk for the dual interaction on the lattice with plus boundary conditions. From the representation (1.13) for V2 and k− 21

0 rk = 1 j =−M

1 , 0 j

it is obvious that rk V2 = V2 rk . We now assemble the preceding observations to produce a prescription for the calculation of the correlation , -open σ(k2 ,ℓ2 ) σ(k1 ,ℓ1 ) ∗ ,K ∗ for ℓ2 ≥ ℓ1 in the dual description. The prescription is to replace the factor

V2 (ℓ2 + 21 , ℓ2 − 12 )V1 (ℓ2 − 21 ) · · · V1 (ℓ1 + 12 ) for ℓ2 > ℓ1 (respectively V2 (ℓ2 + 21 , ℓ2 − 21 ) when ℓ2 = ℓ1 ) by rk2 V2 (ℓ2 + 12 , ℓ2 − 12 )V1 (ℓ2 − 21 ) · · · V1 (ℓ1 + 12 )rk1

for ℓ2 > ℓ1

(respectively rk2 rk1 V2 (ℓ2 + 12 , ℓ2 − 12 ) for ℓ2 = ℓ1 ) in the transfer-matrix representation for the partition function on with plus boundary conditions. One can see

54


this by joining the lattice sites (k2 , ℓ2 ) and (k1 , ℓ1 ) with a path of bonds that stays away from the boundary. The concatenation of the simple bond representations we obtained above leads to cancellations of all the intermediate factors rk2 = 1 that arise in the dual description. It is convenient at this stage to incorporate the symmetrized transfer matrix VM since we are then able to make direct use of the infinite-volume convergence results we have already established for T < Tc . This can be achieved by replacing each of the factors V1 (ℓ + 21 )rk

that occur in the transfer formalism below Tc by 1

1

V1 (ℓ + 12 ) 2 µk V1 (ℓ + 21 ) 2 , where

1

− 12

µk := V12 rk V1

(1.94)

= (ch(K1 )pk + i sh(K1 )qk ) σk+ 1 . 2

1

The action of similarity transformation by V12 on pk is found to give ch(K1 )pk + i sh(K1 )qk using (1.30). The dual representation (1.93) of the two-point function is then N −ℓ2 − 21

, -open +, VM σ(k2 ,ℓ2 ) σ(k1 ,ℓ1 ) ∗ ,K ∗ =

N+ℓ1 + 21

ℓ −ℓ

µk2 VM2 1 µk1 VM +, VM2N +

+

.

The semi-infinite limit (N → ∞) for the right-hand side is calculated in precisely the same fashion as was done for correlation functions below Tc . We obtain , -open lim σ(k2 ,ℓ2 ) σ(k1 ,ℓ1 ) ∗ ,K ∗ = 0|µk2 Ŵ(TM+ )ℓ2 −ℓ1 µk1 |0M , N →∞

where the notation on the right is the same as in Theorem 1.2.1. It is clear that this result extends to all even correlations, and we have the following result.

Theorem 1.4.1 Suppose that A is a collection of sites in ∗ with an even number of elements. For K ∗ = (K2∗ , K1∗ ) define open

open

σA M,K ∗ := lim σA ∗ ,K ∗ . N →∞

Suppose that for (k, ℓ) ∈ A, the second coordinate satisfies |ℓ| ≤ n + 12 , and write µAℓ = µk . (k,ℓ)∈A

Then open

σA M,K ∗ = 0|µA

n+ 12

Ŵ(TM+ )µA

n− 12

· · · Ŵ(TM+ )µA

−n− 12

|0M ,

(1.95)

where |0M is the vacuum for the transfer matrix evaluated at coupling K = (K1 , K2 ).


We turn to the thermodynamic limit. For k ∈ Z disorder operator

1 2

55

define the infinite-volume

µk = (ch(K1 )pk + i sh(K1 )qk ) σ

k+

1 2

(1.96)

acting on Alt(W+ ), where W+ is figured at interaction constants (K1 , K2 ) (remember that Q depends on these constants). The interaction constants Kj correspond to a temperature below the critical temperature. The spin operator σ is characterized as in Theorem 1.3.1 in terms of its induced rotation and normalized so that σ 2 = 1 and σ > 0. In order to apply Theorem A.7.5 we want to convert µk to normal ordered form. Let δ denote an element of W and suppose that op(δσ

k+

= δ+ σ

1) 2

k+

1 2

+σ

k+

1 δ− 2

= µk .

(1.97)

Then δ is the solution of (Q+ + s

k+

1 Q− )δ 2

= ch(K1 )pk + sh(K1 )qk .

This equation has a solution since the D matrix element of the induced rotation for s 1 is invertible. Use the semi-infinite volume version of (1.97) to convert (1.95) k+

2

to normal ordered form. Write the disorder correlation as a product of disorder operators: ## $ $ y −y −y µ(x, y) = VM op δσ 1 VM = op TM δ σ (x + 12 , y) . x+

2

Apply Theorem 1.30 to write the result as a product of a spin correlation and a finite-dimensional Pfaffian. In order to use this formula we do need to know that the semi-infinite volume version of 1 −UB (1.98) LC 1 is invertible. However, Griffith’s inequality (or the GKS inequality) [74] implies σ (an ) · · · σ (a1 )+ σ (aj )+ M ≥ M. j

It follows that Pf

1 LC

−UB ≥ 1, 1

so that (1.98) is indeed invertible and remains so in the infinite-volume limit. The convergence of the spin correlation is guaranteed by Theorem 1.3.1. A routine application of Theorem 1.3.2 and Lemma 1.3.2 shows the finite-dimensional Pfaffian factor converges. Apply Theorem A.7.5 in reverse to the resulting formula and one obtains the following theorem.

56


Theorem 1.4.2 Suppose that T > Tc is associated with the interaction constants K ∗ = (K2∗ , K1∗ ). Let Q denote the (infinite-volume) polarization associated with the dual interaction constants K = (K1 , K2 ) (note the horizontal–vertical exchange). Suppose that A is a collection of sites in the dual lattice Z2 12 with second coordinates between −j and j . Then : 9 open lim σA K ∗ = µAj V µAj −1 · · · V µA−j , M→∞

Q

where V is also evaluated at (K1 , K2 ).

As before, we have not bothered to write down the explicit Pfaffian formula for the correlations since we will return to this issue when we discuss the scaling limits, and there we will use a slightly different factorization that has advantages for scaling. One consequence of the Fock representations for Ising correlations in Theorems 1.3.1 and 1.4.2 that is more or less immediate is that the plus state correlations below Tc and the open correlations above Tc cluster. Suppose that A = {a1 , a2 , . . . , am }, and B = {b1 , b2 , . . . , bn } are two finite collections of sites in Z2 arranged in order of increasing second coordinates. Suppose that r ∈ Z2 and let A + r = {a1 + r, a2 + r, . . . , am + r}. Then we have lim σA+r σB + = σA + σB +

r→∞

and

lim σA+r σB open = σA open σB open .

r→∞

To see why this works suppose that r = (r1 , r2 ) and r2 ≥ |r1 |. In this circumstance if |r| is sufficiently large, the second coordinates in A + r will all be greater than the second coordinates in B. Thus Theorem 1.3.1 implies that σA+r σB + = 0|σam · · · σ1 V (r1 , r2 )σbn · · · σ1 |0

(1.99)

where V (r1 , r2 ) = Ŵ(z)r1 V r2 . Because Ŵ(z) is unitary, Ŵ(z)|0 = |0 and V is a strict contraction on the orthogonal complement of the vacuum state, it follows that V (r1 , r2 ) tends uniformly to the orthogonal projection |00|. This is enough to show that (1.99) tends to 0|σam · · · σ1 |00|σbn · · · σ1 |0 = σA + σB + as r tends to ∞ in the domain r2 ≥ |r1 |. One can deal with the domain r2 ≤ −|r1 | in the same way with only obvious changes. In order to deal with r1 ≥ |r2 | and r1 ≤ −|r2 | just employ the representation for the correlations that one gets using


57

the horizontal transfer matrix formalism. The same argument works for the disorder representation of the open correlations above Tc . The cluster decomposition property is the signature of pure states in the thermodynamic limit. Thus plus state boundary conditions lead to a pure state below Tc , as do open boundary conditions above Tc . It is known [6] for the Ising model in two dimensions that there are only two pure states below Tc (corresponding to “plus” and “minus” boundary conditions) and just one pure state above Tc . In three dimensions the Ising model is known to have non-translation-invariant states that are not convex combinations of the plus and minus states. This finishes our general discussion of the convergence of the thermodynamic limit for the Ising correlations above and below Tc . Before we move on, however, we will sketch how to obtain an alternative representation of the open correlations above Tc that follows from Theorem 1.4.2 but that is a direct analogue of Theorem 1.3.1. The two elementary matrix identities 0 1

0 1

is1 c1

∗ z c2 0 is2∗ z−1

−is2∗ z c2∗

z 0

c1 −is1

z 0

−1

−is1 z c1

−1 ∗ z c2 = 0 −is2∗

is2∗ c2∗

c1 = is1 z−1

and

0 1

0 1

have the following consequence for the induced rotations of the presymmetrized transfer matrices −1 0 z 0 z T (V1 (K1 )) T (V2 (K2 )) = T V2 (K1∗ ) T V1 (K2∗ ) . (1.100) 1 0 1 0

Now introduce the multiplication operator 1 ∗ 2 0 M(z) = T V1 (K2 ) 1

1 z T (V1 (K1 )) 2 , 0

M11 (z) = i sh K2∗ ch K1 − i ch K2∗ sh K1 z, M12 (z) = − sh K2∗ sh K1 + ch K2∗ ch K1 z,

M21 (z) = ch K2∗ ch K1 − sh K2∗ sh K1 z, M22 (z) = i ch K2∗ sh K1 − i sh K2∗ ch K1 z. Then the symmetrized version of (1.100) is MT(K1 ,K2 ) M −1 = T(K2∗ ,K1∗ ) , where we have temporarily introduced the notation 1

1

T(K1 ,K2 ) = T (V1 (K1 )) 2 T (V2 (K2 ))T (V1 (K1 )) 2 .

(1.101)

58


Write + W(K = Q+ (K1 ,K2 ) W. 1 ,K2 )

Then M induces a map + + M+ : W(K → W(K ∗ ,K ∗ ) . 1 ,K2 ) 2

1

# $ # $ + + The map Ŵ(M+ ) : Alt W(K → Alt W ∗ ,K ∗ ) defined by ,K ) (K 1 2 2

1

Ŵ(M+ ) = I ⊕ M+ ⊕ M+ ⊗ M+ ⊕ · · ·

carries the vacuum vector in the first space to the vacuum in the second space and the symmetrized transfer matrix V(K1 ,K2 ) into V(K2∗ ,K1∗ ) . Another elementary calculation shows that MT (µ)M −1 = T (σ ).

It follows from this that Ŵ(M+ ) carries µ into σ (up to a sign, since µ2 = σ 2 = 1). Thus we have the following theorem. Theorem 1.4.3 Suppose that (K1 , K2 ) are interaction constants associated with T > Tc . Let T (V ) denote the infinite-volume induced rotation for the transfer matrix evaluated at (K1 , K2 ) and let Q denote the associated polarization with W+ = Q+ W . Define σ on Alt(W + ) as the element of Spin(W, Q) with induced rotation s (1.78) and such that σ 2 = 1 (this determines σ only up to a sign, but in this case the sign is irrelevant). Suppose that A is a collection of sites on Z2 with second coordinates between −j and j . Then the infinite-volume limit for correlations with open boundary conditions is open

σA (K1 ,K2 ) = σAj V σAj −1 · · · V σA−j Q . It is automatic that odd correlations vanish in this case. One could more simply state this result by saying that Theorem 1.3.1 applies to open boundary conditions above Tc as well as plus boundary conditions below Tc . However, in calculations we will find it simpler to use the disorder representation in Theorem 1.4.2. The reader might also observe that the action of V on the even subspace of Alt(W+ ) is all that is relevant for the correlations below Tc , but the action of V on all of Alt(W+ ) is relevant for the correlations above Tc .

1.5 Even Correlations at All Temperatures In this section we develop formulas for even-spin correlations in the thermodynamic limit at all temperatures. In particular, we obtain formulas for the critical correlations that have been used elsewhere to analyze the two-point function at Tc [89]. We will confine our application of these formulas to writing the two-point

1.5 Even Correlations at All Temperatures

59

function below Tc as a Toeplitz determinant, a formula that is due to Potts and Ward [125]. In the next chapter we will use this to reproduce the Montroll, Potts, and Ward analysis of the spontaneous magnetization using Szego’s theorem. The simplicity of the analysis compared to the proof of Theorem 1.3.1 is striking, but the author is unaware of any way to promote this to a treatment that includes the odd-spin correlations below Tc . Also, the formulas we arrive at here are not well suited for analyzing the scaling functions that are considered in Chapter 2, but the formulas used to prove Theorem 1.3.1 serve this purpose well. The basic observation is that the product of spin operators at adjacent sites is an algebraic element in the Clifford algebra. Return for a moment to the situation before the semi-infinite-volume limit. Then we have already seen in (1.25) that σ(i,0) σ(i+1,0) = iq

i+

1. 1p i+ 2 2

Conjugating by the j th power of the transfer matrix (i.e., X → V −j XV j is the order appropriate for the transfer formalism), we obtain σ(i,j ) σ(i+1,j ) = iq

i+

p , 1 , j i+ 12 , j 2

where −j

pk,j := TM pk , −j

qk,j := TM qk . We want a similar formula for the product of vertically adjacent spin operators. Since V1 and σ := σ0,0 commute, −1

−1

1

1

−1

1

σ0,1 σ0,0 = V1 2 V2−1 V1 2 σ V12 V2 V12 σ = V1 2 V2−1 σ V2 σ V12 . Consulting (1.24) and (1.26), one sees that σ V2 σ = V2 e

−i2K2∗ p 1 q 1 − 2

2

,

since σpk qk+1 σ = pk qk+1 unless k = 21 , in which case σp 1 q− 1 σ = −p 1 q− 1 , 2 2 2 2 and the pk qk+1 commute among themselves. Thus $ −1 1 # 1 −i2K ∗ p 1 q 1 1 2 2 − 2 V − 2 = V 2 ch 2K ∗ − i sh 2K ∗ p 1 q 1 V 2 , σ0,1 σ0,0 = V12 e 2 2 1 1 − 2

2

2

since (ipq) = 1. In what follows we suppose that the size parameters for the lattice, M and N, are chosen large enough that the exceptional relations (1.28) don’t come into play. Then (1.30) implies 1

− 21

V12 pk V1

−1 V1 qk V1 2 1 2

= ch K1 pk + i sh K1 qk := pk (K1 ), = −i sh K1 pk + ch K1 qk := qk (K1 ).

60


Thus σ0,1 σ0,0 = ch 2K2∗ − i sh 2K2∗ p 1 (K1 )q− 1 (K1 ). 2

2

From this it follows that σi,j +1 σi,j = ch 2K2∗ − i sh 2K2∗ pi+ 1 ,j (K1 )qi− 1 ,j (K1 ), 2

(1.102)

2

where −j

pk,j (K1 ) := TM pk (K1 ), −j

qk,j (K1 ) := TM qk (K1 ). It is pretty clear that any product of two spin operators σa σb can be obtained as the product of adjacent spin operators by connecting the sites a and b by a path of bonds. The formulas we have for products of adjacent spin operators thus make it possible to write even correlations as vacuum expectations of finite products in the Clifford algebra generated by qk and pk . We will concentrate on one simple applications of this idea. This is a famous formula (obtained by Potts and Ward [125]) for the two-point function below Tc as a Toeplitz determinant. Suppose that T < Tc . Then the Pfaffian formula (A.31) and the convergence results for QM show that σ00 σN 0 = lim i N q 1 p 1 · · · qN − 1 pN − 1 QM = i N q 1 p 1 · · · qN − 1 pN − 1 Q . M→∞

2

2

2

2

2

2

2

2

Since q and p anticommute, this last expression can be rewritten as i N (−1)

N(N +1) 2

q 1 · · · qN − 1 p 1 · · · pN− 1 . 2

2

2

2

Recalling that the Fourier representations for qk and pk are 0 1 zk− 12 1 qk → √ and pk → √ 1 , 0 π π zk− 2 the map Q− is the multiplication operator 1 1 Q− = 2 w¯

w , 1

and that the distinguished complex bilinear form is ) dz (f1 (z)g1 (z−1 ) + f2 (z)g2 (z−1 )) , (f, g) = 1 iz S a little calculation shows that (Q− qk , qℓ ) = 0, (Q− pk , pℓ ) = 0, for k = ℓ, and ) 1 k−ℓ dz (Q− qk , pℓ ) = w(z)z ¯ . 1 2π S iz

1.5 Even Correlations at All Temperatures

Thus σ00 σN 0 = i N (−1)

N(N +1) 2

Pf

0 −BNτ

61

BN = (−i)N det BN , 0

where BN is the N × N matrix with entries ) 1 k−ℓ dz bkℓ = . w(z)z ¯ 2π S 1 iz

To convert the Pfaffian to a determinant we used N(N −1) 0 AN = (−1) 2 det AN . Pf −AτN 0 Theorem 1.5.1 Suppose that T < Tc . Then the infinite-volume two-point correlation function for sites separated by a horizontal translation is given by σ00 σN 0 = det(AN ), where AN is the N × N matrix with matrix elements ) 1 k−ℓ dz w(z)z ¯ . akℓ = − 2π S 1 z

2 The Spontaneous Magnetization and Two-Point Spin Correlation

We use the results of the first chapter to obtain more-explicit formulas for the oneand two-point spin correlations of the Ising model in the infinite-volume limit. The one-point function (for plus boundary conditions) is also known as the spontaneous magnetization. The formula in Theorem 2.4.2 we obtain below was written down by Onsager in 1949 [106]. Yang [161] published the first derivation of this result in 1952. A different account was later given by Montroll, Potts, and Ward [97]. We eventually explain the ideas behind both of these calculations, but to begin we make use of the formula (valid for T < Tc ) σ 2 = det |D(s)| from Theorem 1.3.3. To calculate the determinant on the right-hand side we diagonalize the operator D(s). This diagonalization of D(s) is essentially the method employed by Yang. We diagonalize D(s) in stages. An elliptic substitution reduces D(s) to a convolution operator. This convolution operator is then diagonalized by the introduction of the appropriate Fourier series. Although this method is a bit more involved than the Toeplitz determinant calculation of Montroll, Potts, and Ward (which we present at the end of the chapter), it gives more information. In particular, the convolution representations for B(s) and D(s) are also used to simplify the Fredholm expansions of the Pfaffian formulas for the two-point correlations obtained later in this chapter. The diagonalization of D(s) is also used in [108] to compute the one-point function for lattice approximations to holonomic quantum fields and to explicitly compute the inverse D(s)−1 . These calculations and related estimates give control of the scaling limit for these lattice fields.

64

2. The Spontaneous Magnetization and Two-Point Spin Correlation

We have another reason for considering Yang’s derivation in more detail. The induced rotation Tz (V ) for the transfer matrix is a matrix-valued multiplication operator whose “diagonalization” leads to the consideration of a holomorphic line bundle E over an elliptic curve M. The curve M is just the spectral curve det(λI − Tz (V )) = 0, and the line bundle E is obtained by considering the onedimensional null space of λI − Tz (V ) for each point (z, λ) in M. The diagonalization of Tz (V ) takes place in the L2 sections of a restriction of the line bundle E to a pair of distinguished cycles on the complex curve. The introduction of the full complex curve M adds little to the understanding of the action of the transfer matrix, although it does shed some light on matters such as the π2 rotational symmetry. However, the matrix elements of the induced rotation s relative to the spectral splitting polarization derived from Tz (V ) are convolution operators in the uniformization parameter for the curve M. Working in the line bundle E → M makes the substitutions that produce this result for the A, B, C, and D matrix elements of s appear fairly naturally. That these substitutions produce convolution operators remains a mystery; one can see that it happens but not why it happens. No doubt this is a reflection of a more intimate relation between s and Tz (V ) than is apparent. If this relation was understood it might make possible a simplified account of the diagonalization; we leave the contemplation of this matter to the interested reader.

2.1 The Spectral Curve for the Transfer Matrix The spectral curve associated with the induced rotation Tz (V ) for the transfer matrix is the set of (z, λ) for which det(λI − Tz (V )) = 0. This determinant has been calculated in (1.71) above. It is interesting that this relation between z and λ can be rewritten in the more symmetrical form s1

λ + λ−1 z + z−1 + s2 = c1 c 2 . 2 2

(2.1)

In this form the “exchange symmetry” K1 ↔ K2 , z ↔ λ is manifest. For the spectral analysis of Tz (V ) the relevant parts of M are the two cycles (i.e., closed curves) 3 4 M± = (z, λ) = eiθ , e∓γ (θ ) .

We think of these cycles as parametrized curves [−π, π ) ∋ θ → eiθ , e∓γ (θ ) , where we have introduced the shorthand notation γ (θ) = γ (eiθ ).

2.1 The Spectral Curve for the Transfer Matrix

65

The function γ (z) > 0 is defined by (see (1.65)) ch γ (z) = c1 c2∗ − s1 s2∗

z + z−1 . 2

It is helpful for us to think of (2.1) as defining a projective curve M in P2 . To do this we introduce projective coordinates (ζ0 , ζ1 , ζ2 ) such that z = ζ1 /ζ0 and λ = ζ2 /ζ0 (this allows us to deal simply with the points at ∞). Then (2.1) becomes the homogeneous relation s1 (ζ12 ζ2 + ζ02 ζ2 ) + s2 (ζ22 ζ1 + ζ02 ζ1 ) = 2c1 c2 ζ0 ζ1 ζ2 .

(2.2)

To understand this curve we consider the map from M to P1 induced by (ζ0 , ζ1 , ζ2 ) → (ζ0 , ζ1 ).

(2.3)

Since (2.2) is quadratic in ζ2 , this is a twofold covering ramified at the zeros of the discriminant for (2.2). This discriminant is 2 s1 (ζ12 + ζ02 ) − 2c1 c2 ζ0 ζ1 − 4s22 ζ12 ζ02 .

The value ζ0 = 0 makes the discriminant equal to s12 ζ14 . This is zero only when ζ1 = 0. However, (0, 0) is not in P1 , so this root does not contribute to the ramification points of the map (2.3). Dividing the discriminant by ζ04 , the ramification points are found to be the roots of 2 s1 (z2 + 1) − 2c1 c2 z − 4s22 z2 = 0, which are where

z = α1±1 , α2±1 , α1 = (c1∗ − s1∗ )(c2 + s2 ), α2 = (c1∗ + s1∗ )(c2 + s2 ).

The reader should note that we already encountered these roots in (1.62), where they played a role in factoring Q so that the Wiener–Hopf technique could be used to invert the D matrix element of s. Thus M is a two-sheeted covering of P1 ramified at z = α1±1 , α2±1 . The rubber sheet geometry that shows why such a curve is a topologically a torus can be seen in Figure 2.1. This means that the curve M is an elliptic curve (genus 1). The vector space of holomorphic differentials on M is thus one-dimensional [145], and it is well known that is2 dz ω= ; (2.4) (z − α1 )(z − α1−1 )(z − α2 )(z − α2−1 ) is a local representation for such a holomorphic one-form on M (the reason for the factor s2 appears later).

66


Figure 2.1: A sphere is unzipped between two pairs of branch points and glued to a second copy to see that the topology of the two-sheeted ramified covering is that of a torus. Remark 2.1.1. The reader may find it strange that (2.4) is supposed to be holomorphic given the zeros in the denominator. Keep√in mind, however, that near the branch point α = αj± , j = 1, 2, the square root z − α is introduced as a local parameter (coordinate) on the twofold branched covering and this local parameter has holomorphic differential dz . √ 2 z−α

With a little care one can interpret (2.4) as a globally defined holomorphic oneform on M by gluing together copies of ω with opposite choices of the sign for the square root on the two sheets of the covering. However, we won’t do this here. We use the local representation of the holomorphic differential only as a heuristic device to suggest a suitable uniformization of the curve M in a manner that is explained shortly. Observe also that the factorization of the discriminant already noted implies

λ − λ−1 = ±

; s1 (z − α1 )(z − α1−1 )(z − α2 )(z − α2−1 ) s2 z

.

Thus at least formally, ω=±

is1 dz . z(λ − λ−1 )

(2.5)

s1 dθ , 2 sh γ (θ)

(2.6)

The restriction of ω to M± is given by ω=±

2.1 The Spectral Curve for the Transfer Matrix

67

where the ± in the equation for ω is a reflection of the fact that our “definition” of ω is incomplete. This won’t matter for us since we use the “hyperbolic” measure s1 dθ on M± without worrying about how it connects with the one-form ω. This 2 sh γ (θ ) measure plays a role in arriving at natural hyperbolic formulas for the kernels of matrix elements of s relative to the Q polarization. These formulas are presented in Chapter 3, where they are the foundation for an analysis of the scaling limit. We next introduce a transformation in the z-plane by the fractional linear transformation z − α2−1 1 − α2 z = x= , (2.7) z − α2 1 − α2−1 z which has the inverse

z(x) = Define

1 + α2 x . x + α2

α2 − α1 = s1∗ s2∗ < 1 for T < Tc . α1 α2 − 1 Then the fractional linear transformation (2.7) maps the branch points k=

(2.8) (2.9)

α2−1 < α1−1 < α1 < α2 → 0 < k < k −1 < ∞ into “standard position.” Next we calculate what happens to the holomorphic differential in the xcoordinates. For this calculation as well as a later calculation it is useful to record the following elementary results: 1 − kx , x + α2 1 − k −1 x , z − α1−1 = α1−1 (α1 − α2 ) x + α2 1 − α22 , z − α2 = x + α2 x z − α2−1 = α2−1 (α22 − 1) . x + α2 z − α1 = (1 − α1 α2 )

Differentiating (2.8), we obtain α22 − 1 dz . = dx (x + α2 )2 Using (2.10) we find that the product (z − α1 )(z − α1−1 )(z − α2 )(z − α2−1 ) becomes −

(α1 α2 − 1)2 (α22 − 1)2 kx(1 − kx)(1 − k −1 x) . α1 α2 (x + α2 )4

(2.10)

68


Thus the holomorphic differential (2.4) is 1 dx . ω=± ( −1 2 (1 − kx)(1 − kx ) ix

(2.11)

There is an overall sign ambiguity, which can’t be resolved in the x-coordinates of the base, and we used √ α1 α2 1 = , α1 α2 − 1 2s2

which follows from α1 α2 = e4K2 .

2.2 The Uniformization of M We are ready to introduce an elliptic substitution that leads to a “uniformization” of the whole complex curve M. Because M is an elliptic curve, it is holomorphically equivalent to C modulo a lattice L [145]. The isomorphism from C/L to M is called a uniformization of M. The uniformization we are interested in arises from the substitution x = k sn2 (u, k), (2.12)

where sn(u, k) is a Jacobian elliptic function of u with modulus k. This function and its relatives cn(u, k) and dn(u, k) are defined in Chapter XXII of [157]. We refer the reader to this work for all the results that we use concerning these functions. The functions sn(u, k), cn(u, k), and dn(u, k) are all doubly periodic (with somewhat different periods) meromorphic functions of u, and

and

sn2 (u, k) + cn2 (u, k) = 1, dn2 (u, k) + k 2 sn2 (u, k) = 1

(2.13)

d sn(u, k) = cn(u, k)dn(u, k). du

(2.14)

Because we fix k = s 1s throughout, it is convenient to drop this argument hence1 2 forth, writing sn(u) for sn(u, k) etc. The reason for the substitution (2.12) can be seen by computing the holomorphic differential (2.11) using the relations (2.13) and (2.14). One obtains (up to the usual sign ambiguity) ω = du. Now fix a point p0 in M and a holomorphic differential ω on M. Suppose that γ (p, p0 ) is a path in M that joins p0 to p ∈ M. The map ) ω∈C (2.15) M∋p→ γ (p,p0 )

is not independent of the choice of path. However, if γ1 and γ2 are generators for the based homotopy group π1 (M, p0 ), then the image of (2.15) is well defined up

2.2 The Uniformization of M

69

+ to the lattice L in C generated by the period integrals γj ω for j = 1, 2 (L is the + + set of linear combinations n1 γ1 ω + n2 γ2 ω, where n1 and n2 are integers.) The map from M to C/L induced by (2.15) is the inverse of the uniformization map. Since ω = du, this suggests that u is an appropriate uniformization parameter. We now work out such a uniformization. First we want to express z as a function of u. Evidently, z=

1 + α2 x 1 + α2 k sn2 (u) = . x + α2 k sn2 (u) + α2

(2.16)

This can be written in a simpler and more helpful form by introducing a suitable elliptic parametrization of the Boltzmann weights. We want to parametrize the Boltzmann weights e2Kj subject to the restriction k sh(2K1 ) sh(2K2 ) = ks1 s2 = 1. Solving for e2K1 and e2K2 in terms of s1 and s2 , one obtains ; e2Kj = sj + 1 + sj2 .

Using s2 = (ks1 )−1 to eliminate s2 as an independent variable, we obtain ; e2K1 = s1 + 1 + s12 , ; (2.17) 2K2 −1 2 2 e = (ks1 ) 1 + 1 + k s1 .

The identities (2.13) suggest the following substitution for s1 : s1 = −i sn(2ia, k).

Then s1 ranges from 0 to +∞ as 2a ranges from 0 to K ′ . The constant K ′ is an elliptic integral that is defined on page 501 of [157]; 2iK ′ is the imaginary period of sn(u, k). The factor 2 in 2a is not necessary but is convenient for a later calculation. Again dropping the dependence on the modulus k, which is fixed, the appropriate parametrization of the rest of the Boltzmann weights follows from (2.17) and the fact that cn(2ia) and dn(2ia) are positive for 0 < 2a < K ′ (see page 506 of [157]). It is s1 = −i sn(2ia), c1 = cn(2ia), (2.18) s2 = ik −1 ns(2ia), c2 = ik −1 ds(2ia), where we used the standard notation ns(u) :=

1 , sn(u)

ds(u) :=

dn(u) . sn(u)

nx(u) :=

1 , xn(u)

xy(u) :=

xn(u) , yn(u)

In general,

70


where x and y are chosen from among s, c, and d. It is also easy to check that s1∗ = ins(2ia), c1∗ = ics(2ia), s2∗ = −ik sn(2ia), c2∗ = dn(2ia).

(2.19)

For the purposes of calculating z from (2.16) we need to know that α2 = (c1∗ + s1∗ )(c2 + s2 )

= ik −1 (ics(2ia) + ins(2ia))(ds(2ia) + ns(2ia)) dn(2ia) + 1 −1 cn(2ia) + 1 = −k sn(2ia) sn(2ia) dn(2ia) + 1 = −k −1 sn−2 (ia) . = −k −1 1 − cn(2ia)

(2.20)

The last equality follows from the formulas 1 − cn(b) = sn( 12 b)dc( 12 b), sn(b) 1 + dn(b) = ds( 12 b)nc( 21 b), sn(b) which can be found at the bottom of page 508 of [157]. Now substitute (2.20) into (2.11) and make use of the addition formulas for sn that follow (see page 496 of [157]): sn(u)cn(v)dn(v) + sn(v)cn(u)dn(u) , 1 − k 2 sn2 (u) sn2 (v) sn2 (u) − sn2 (v) . sn(u − v) = sn(u)cn(v)dn(v) + sn(v)cn(u)dn(u)

(2.21)

z = z(u, a, k) = k sn (u − ia) sn (u + ia) .

(2.22)

sn(u + v) =

One obtains This formula makes it easier to locate the poles and zeros of z in the u variable, and this is very helpful in obtaining a suitable elliptic parametrization for λ. Suppose that we have a parametrization λ = λ(u, a, k) such that s1 (z(u, a, k) + z(u, a, k)−1 ) + s2 (λ(u, a, k) + λ(u, a, k)−1 ) = 2c1 c2 .

(2.23)

Under translation by iK ′ , the Jacobian elliptic functions transform (see page 503 of [157]) thus: sn(u + iK ′ ) = k −1 ns(u),

cn(u + iK ′ ) = −ik −1 ds(u), dn(u + iK ′ ) = −ics(u).

(2.24)

2.2 The Uniformization of M

71

Using these formulas, the reader can check that sending 2a to 2a +K ′ in the elliptic parametrization of the Boltzmann weights sends s1 to −s2 , s2 to −s1 , and c1 c2 to −c1 c2 . This suggests that the substitution 2a←⊣ 2a + K ′ should interchange the roles of z and λ in the spectral curve (2.23). Guessing that λ(u, a, k) is given by z(u, a + K ′ /2, k) does not work, since the poles don’t match up in (2.23), but the next simplest guess, λ(u, a, k) = z(u + b, a + K ′ /2, k),

(2.25)

can be made to work by choosing b appropriately. The function sn(u) has a simple zero at u = 0 and a simple pole at u = ±iK ′ . The function sn(u) changes sign under translation by 2K (K is the real quarter-period defined on page 498 of [157]), so z(u, a, k) is 2K periodic in u. The function sn(u) is 2iK ′ periodic, and so this is true for z(u, a, k) as well. In the period rectangle 0 < ℜu ≤ 2K, −K ′ < ℑu ≤ K ′ , z(u, a, k) has simple zeros at u = ±ia and simple poles at u = ±iK ′ ∓ ia. To make (2.23) work we need to be able to choose b in (2.25) such that the poles and zeros of u → λ(u, a, k) match those of u → z(u, a, k). Choosing b = K ′ works at this level. Substituting this in (2.25) and making use of (2.24) one obtains the guess sn (u − ia) λ(u, a, k) = . (2.26) sn (u + ia)

To see that this works substitute (2.22) and (2.26) into (2.23). The left-hand side of (2.23) is then a doubly periodic meromorphic function in u. It is a constant function if we can show that it does not have any poles. The residue of sn(u) at u = iK ′ is k −1 and the residue of ns(u) at u = 0 is 1. This is enough to check that the substitution we are considering produces no poles on the left-hand side of (2.23). One can determine the constant that results by evaluating z(0, a, k) = α2−1 and λ(0, a, k) = −1. Since s1 (α2 + α2−1 ) − 2s1 = 2c1 c2 , we have confirmed that (2.22) and (2.26) provide a uniformization of the curve M. We have proved the following result.

Theorem 2.2.1 The map, [0, 2K] × i[−K ′ , K ′ ] ∋ u → (z(u, a), λ(u, a)) with and

z(u, a) = k sn (u + ia) sn (u − ia) λ(u, a) =

is a uniformization of the spectral curve

sn (u − ia) sn (u + ia)

s1 (z + z−1 ) + s2 (λ + λ−1 ) = 2c1 c2 .

(2.27)

72


Here the elliptic modulus k at which the Jacobian elliptic function sn is evaluated is k = s 1s , and 0 < 2a < K ′ is defined by 1 2

s1 = −i sn(2ia). Next we wish to determine where the cycles M± lie in the period parallelogram 0 < ℜu ≤ 2K, −K ′ < ℑu ≤ K ′ . We show that & % iK ′ , M± = u|0 < ℜu < 2K, ℑu = ± 2

where we mean by this that the cycles M± are the image of the set on the righthand side under the uniformization map (2.27). To see this, first note that on page 506 of [157] one finds the following values for the Jacobian elliptic functions at ′ u = iK2 : ′ iK 1 = ik − 2 , sn 2 ′ iK 1 1 cn = (1 + k) 2 k − 2 , (2.28) 2 ′ iK 1 dn = (1 + k) 2 . 2 Together with the addition theorem for sn one obtains (1 + k) sn(u) ± icn(u)dn(u) iK ′ 1 2 k sn u ± . = 2 1 + k sn2 (u)

(2.29)

This is especially interesting for us when u is real. In this case the functions sn(u), cn(u), and dn(u) are all real, and a straightforward calculation shows that the right-hand side of (2.29) lies on the unit circle. Since x (given by (2.12) with u ′ replaced by u ± iK2 ) is just the square of (2.29), it follows that x, and hence also z, is on the unit circle. One can check without difficulty that iK ′ λ u+ ,a < 1. 2 u=0 Hence M+ is the image of ℑu = ′ M− is the image of ℑu = − iK2 .

iK ′ . 2

Translation by −iK ′ inverts λ(u, a), so

2.3 Trivializing E over M± Living on the curve M is a line bundle E whose fiber over (z, λ) is the onedimensional null space of λI − Tz (V ). It is natural to realize the direct integral decomposition of L2 (S 1 , C2 ) that diagonalizes Tz (V ) as the direct sum of L2 sections of E lying over the cycles M± . We make the transformation to this direct integral decomposition explicit by choosing trivializations of the line bundle over M± . In order to do this, it is useful to recall some facts about the polarization Q

2.3 Trivializing E over M±

73

that were developed in Chapter 1. Consulting (1.72) and (1.76), we see that for z ∈ S1, ch γ w sh γ Tz (V ) = = e−γ Q+ + eγ Q− , (2.30) w¯ sh γ ch γ where Q± = 21 (1 ± Q) and

Q=− with w(z) = −i

1, each of these maps has winding number −1, so that the ratio S1 ∋ z →

z − α1 1 − α 2 z ∈ S1 1 − α1 z z − α 2

has winding number 0 about z = 0. Thus this function has a continuous logarithm, iθ e − α1 1 − α2 eiθ i2α(θ) := log , (2.33) 1 − α1 eiθ eiθ − α2 which we normalize so that α(0) = 0. It is simple to check that α(−θ) = −α(θ) with this normalization. The function w(z) is defined (1.76) so that w(1) = −i. Thus the appropriate square root in (2.32) is w(eiθ ) = −ieiα(θ ) . Next we consider what happens to Q under the map (2.7). Equation (2.10) makes it easy to check that 1 − kx z − α1 1 − α2 z = . (2.34) 1 − α1 z z − α2 1 − kx −1

Since k < 1 and |x| = 1, it follows that 1 − kx has winding number 0 for x ∈ S 1 , and hence we can define a continuous logarithm log(1 − kx) for x ∈ S 1 , normalized so that log(1 − k) < 0. Then we define √ q(x) = 1 − kx := exp 21 log(1 − kx)

for x ∈ S 1

74


and q(x −1 ) =

(

1 − kx −1 = q(x) ¯ for x ∈ S 1 .

For later purposes it is useful to note that we can also define ( q(x) := exp 14 log(1 − kx) for x ∈ S 1 .

Equations (2.32) and (2.34) and the normalization w(1) = −i show that in the x variables, < q(x) z − α 1 1 − α2 z w = −i = −i for x ∈ S 1 , 1 − α1 z z − α2 q(x) ¯

so that the matrix-valued function Q becomes  q(x)  0 i  q(x) ¯  Q(x) =  q(x)  ¯ 0 −i q(x)

for x ∈ S 1 .

From this representation for Q it is easy to check that the map   < q(x)   1 q(x) ¯  ∓γ (z)   < M± ∋ (z, e ) → e± (x) :=   2 q(x) ¯  ∓i q(x)

(2.35)

(2.36)

is a section of E over M± , the vector on the right being an appropriate eigenvector for Tz (V ). It is, of course, understood that x is given by (2.7).

2.4 The Matrix Elements of s for the Q Polarization We want to understand the matrix elements of the induced rotation s for the spin operator relative to the splitting of L2 (S 1 , C2 ) associated with the polarization Q. Recalling (1.78), we see that those functions in L2 (S 1 , C2 ) that have analytic continuations into the exterior of the unit disk that vanish at ∞ are eigenvectors for s with eigenvalue 1. Those functions in L2 (S 1 , C2 ) that have analytic continuations into the interior of the unit disk are eigenfunctions for s with eigenvalue −1. Thus the Cauchy integral representation for s is ) 1 1 1 + −1 f (z′ ) dz′ , (2.37) sf (z) = lim r→1− 2πi S 1 rz − z′ r z − z′ where S 1 is given the standard counterclockwise orientation and the limit is in the L2 sense (see [129] for a more precise account of Hardy space theory). Following the uniformization of the spectral curve for the transfer matrix described above,

2.4 The Matrix Elements of s for the Q Polarization

75

we now make a change of variables in stages to diagonalize the matrix elements of s. It is useful to observe that (2.7) maps S 1 in an orientation-preserving manner onto S 1 . It is also a holomorphic bijection from the interior of S 1 to the interior of S 1 and a holomorphic bijection from the exterior of S 1 (including the point at ∞) to the exterior of S 1 . Because we are interested in spectral theory, we want the change of variables to be a unitary transformation. A straightforward calculation using |x|2 = 1 shows that the measure dθ on the unit circle transforms

Thus the map

α 2 − 1 dx dz x dz dx = = 2 iz z dx ix |x + α2 |2 ix

for z ∈ S 1 .

; α22 − 1

(2.38) x + α2 defines a unitary map on L2 (S 1 , C2 ). Under the unitary map (2.38) the map s becomes ) 1 1 1 + F (x ′ ) dx ′ . (2.39) sF (x) = lim r→1− 2πi S 1 rx − x ′ r −1 x − x ′ f → F (x) := f (z(x))

One can see this by explicit substitution of (2.7), (2.8), and (2.38) in (2.37), but it is more instructive to focus on what happens to the eigenfunctions of s under the transformation (2.38). If f (z) has a holomorphic extension into the interior of S 1 , then because α2 > 1 we see from (2.38) that F (x) has a holomorphic extension into the interior of S 1 . If f (z) has a holomorphic extension into the exterior of S 1 that vanishes at z = ∞, then f (z(x)) is holomorphic in the exterior of S 1 with a zero at x = −α2 (which is the x value at which z = ∞). Dividing by x + α2 as in (2.38) moves the zero from x = −α2 to x = ∞. Thus under the map (2.38) the −1 (respectively +1) eigenfunctions for s are mapped into the functions F (x) that have analytic extensions into the interior of the disk S 1 (respectively the functions with analytic extentions into the exterior of S 1 that vanish at ∞). The Cauchy integral representation for s in the x variables is thus (2.39). √ ¯ (x) maps functions on S 1 square integrable Observe that F (x) → q(x)q(x)F dx with respect to ix unitarily onto functions on S 1 square integrable with respect to

We write

1 dx 1 dx =( −1 q(x)q(x) ¯ ix (1 − kx)(1 − kx ) ix du(x) =

1 dx 2q(x)q(x) ¯ ix

for x ∈ S 1 . (2.40)

for the associated measure on S 1 . Note the factor of 21 in the definition of the measure. This is convenient for the elliptic substitution that follows. A short calculation now shows that for x ∈ S 1 , ( q(x)q(x)Q ¯ + (x)F (x) = (q(x)F ¯ 1 (x) + iq(x)F2 (x)) e+ (x), (2.41) ( q(x)q(x)Q ¯ − (x)F (x) = (q(x)F ¯ 1 (x) − iq(x)F2 (x)) e− (x),

76


where e± (x), is the trivialization of the bundle E over M± given by (2.36). This suggests the following definition: ¯ 1 (x) + iq(x)F2 (x), F+ (x) = q(x)F F− (x) = q(x)F ¯ 1 (x) − iq(x)F2 (x).

(2.42)

The reader can check that the map F F1 → + F− F2 is a unitary map from L2 S 1 , dx to L2 S 1 , du(x) . The inverse to (2.42) is ix 1 (F+ (x) + F− (x)) , 2q(x) ¯ 1 F2 (x) = (F+ (x) − F− (x)) . 2iq(x) F1 (x) =

(2.43)

Now we transform the integral representation (2.39) for s to the F± representation to obtain ) 1 q(x)s(x, ¯ x ′ )q(x¯ ′ )−1 (F+ (x ′ ) + F− (x ′ )) dx ′ (sF )± (x) = 4πi S 1 (2.44) ) 1 ′ ′ −1 ′ ′ ′ ± q(x)s(x, x )q(x ) (F+ (x ) − F− (x )) dx , 4πi S 1 where s(x, x ′ ) is shorthand for the singular kernel 1 1 + . s(x, x ′ ) = lim r→1− rx − x ′ r −1 x − x ′ Replacing integration with respect to dx ′ by integration with respect to du(x ′ ), we obtain ) 1 q(x)s(x, ¯ x ′ )q(x ′ )x ′ (F+ (x ′ ) + F− (x ′ )) du(x ′ ) (sF )± (x) = 2π S 1 ) (2.45) 1 ± q(x)s(x, x ′ )q(x¯ ′ )x ′ (F+ (x ′ ) − F− (x ′ )) du(x ′ ). 2π S 1 We are now prepared to make the elliptic substitution, x = k sn2 (u) in the preceding formula. In fact, to maintain the connection with the uniformization of the spectral curve $ transfer matrix discussed earlier, it is useful to substitute # for the ′ x = k sn2 u + iK2 whenever x is the argument of a function that lives on the # $ ′ cycle M+ (e.g., F+ or sF+ ) and substitute x = k sn2 u − iK2 whenever x is the argument of a function that lives on the cycle M− (e.g., F− or sF− ). As noted above, for u ∈ [0, 2K] the first substitution is a counterclockwise parametrization


77

of M+ , and the second is a clockwise parametrization of M− . For convenience, write iK ′ , 2 iK ′ . u′± := u′ ± 2

u± := u ±

It is convenient to introduce functions G± (u) on M± by G± (u) = F± (k sn2 (u± )). A check of signs (at u = K) shows that q(x) = dn(u+ ) and

q(x) ¯ = ics(u+ ) on M+

and q(x) = dn(u− ) and q(x) ¯ = −ics(u− ) on M− .

Making the appropriate substitutions in (2.45) yields ) 2K sn(u′β ) α sGα (u) = − Gβ (u′ ) du′ , S(uα , u′β ) πi 0 sn(uα )

(2.46)

where α = +, −, β = +, − , and the kernel S(u, u′ ) is defined by sn(u)dn(u)cn(u′ ) + cn(u) sn(u′ )dn(u′ ) sn2 (u) − sn2 (u′ ) ′ = ds(u − u ).

S(u, u′ ) : =

(2.47)

The addition theorem for ds, which is the second equality in (2.47), can be deduced from results in [157] in the following way. Example 6 on page 529 of [157] can be written sn(u + v)dn(u − v) =

sn(u)dn(u)cn(v) + sn(v)dn(v)cn(u) . 1 − k 2 sn2 (u) sn2 (v)

Coupled with the addition formula for sn(u + v) already noted in (2.21) above, one obtains dn(u − v) =

sn(u)dn(u)cn(v) + sn(v)dn(v)cn(u) . sn(u)cn(v)dn(v) + sn(v)cn(u)dn(u)

The alternative addition formula for sn, sn(u − v) =

sn2 (u) − sn2 (v) , sn(u)cn(v)dn(v) + sn(v)cn(u)dn(u)

which is Example 3 on page 496 of [157], and the preceding version of the addition formula for dn combine to give (2.47).

78

2. The Spontaneous Magnetization and Two-Point Spin Correlation sn(u′ )

Our final transformation is motivated by the factor sn(uβα ) in (2.46). It is easy to see that this factor can be eliminated by unitary conjugation by the map √ G+ (u) k sn(u+ )G+ (u) √ → . (2.48) G− (u) k sn(u− )G− (u) For the convenience of the reader we now summarize the transformations that lead to the spectral transform S for the transfer matrix that provides a representation of the matrix elements for s as convolution operators. Let z(x) =

1 + α2 x x + α2

and F (x) = f (z(x))

;

α22 − 1

, x + α2 F± (x) = q(x)F ¯ 1 (x) ± iq(x)F2 (x),

G± (u) = F± (k sn2 (u± )), √ k sn(u )G (u) + + Sf (u) := √ . k sn(u− )G− (u)

(2.49)

Except for the last statement concerning the distinguished conjugation (which we leave to the reader) we have proved the following result. Theorem 2.4.1 The A, B, C, and D matrix elements of s relative to the Q polarization are unitarily equivalent under conjugation by S defined in (2.49) to the following operators (acting on L2 [0, 2K]): ) 2K 1 AF (u) = − ds(u − v)F (v) dv, πi 0 ) 2K 1 BF (u) = − ds(u − v + iK ′ )F (v) dv, πi 0 (2.50) ) 2K 1 ds(u − v − iK ′ )F (v) dv, CF (u) = πi 0 ) 2K 1 DF (u) = ds(u − v)F (v) dv, πi 0 where the integrals in the expressions for A and D are to be interpreted in the principal value sense. Note that since ds(u ± iK ′ ) = −ikcn(u), both the B and C operators can be rewritten as convolutions with cn. Since cn(u) does not have poles for real values of u, the kernels for B and C are smooth.


79

The distinguished conjugation ∗ that produces the symmetric bilinear form (1.77) is given in this representation by ∗ g (u) g Sf ∗ := + (u) = − . g− g + (u) We are ready to diagonalize the D matrix element by introducing the appropriate Fourier series. Convolution operators are well known to be diagonalized by Fourier series, but one should keep in mind that the last unitary transformation we introduced in (2.48) above was multiplication by a 2K antiperiodic function (i.e., one that changes sign under the substitution u → u + 2K). It is not surprising then that the eigenfunctions for D are the 2K antiperiodic exponentials iπ ℓu for ℓ ∈ Z + 21 . exp K Define (as is traditional)

πK ′ q = exp − K

.

Then the example at the top of page 512 in [157] can be rewritten as πu iπ iπ ℓu q 2|ℓ| π csc + exp . sgn(ℓ) ds(u) = 2K 2K K 1 + q 2|ℓ| K 1

(2.51)

ℓ∈Z+ 2

πu Now replace csc 2K in this last formula by

π(u − iǫ) π(u + iǫ) 1 , + csc csc 2 2K 2K and call the resulting function dsǫ (u). The principal value prescription for D is equivalent to regarding D as the ǫ → 0 limit of convolution with iπ1 dsǫ (u). Suppose that ℓ ∈ Z + 21 . Then

iπ ℓv dsǫ (u − v) exp dv 0 ) 2K−uK 1 iπ ℓ(u + v) = dsǫ (−v) exp dv iπ −u K ) 2K 1 iπ ℓv iπ ℓu = dsǫ (−v) exp dv exp iπ )0 2K K K 1 iπ ℓv iπ ℓu = dsǫ (v) exp − dv exp . iπ 0 K K The third equality above follows from the fact that dsǫ (−v) exp iπKℓv is a 2Kperiodic function even though this is not true for the separate factors that are 2K antiperiodic. We see from this that the exponentials exp iπKℓu are eigenfunctions 1 iπ

)

2K

80


for D with eigenvalues 1 ǫ→0 iπ lim

)

2K

0

iπ ℓv dsǫ (v) exp − dv. K

A simple residue calculation shows that ) 2K 1 π(v + iǫ) iπ ℓv −2ie−ℓǫ csc exp − dv = 2K 0 2K K 0

(2.52)

for ℓ > 0, for ℓ < 0,

and 1 2K

)

0

2K

π(v − iǫ) iπ ℓv 0 exp − dv = csc 2K K 2ieℓǫ

for ℓ > 0, for ℓ < 0.

Together with the partial Fourier expansion (2.51) we see that the eigenvalues of D are in the limit (2.52) given by q 2ℓ − 1 2q 2|ℓ| −ǫ|ℓ| − lim sgn(ℓ) e . − = ǫ→0 1 + q 2|ℓ| q 2ℓ + 1 It is clear from this formula that the operator D itself does not have a determinant, since as ℓ → +∞ the eigenvalues tend to −1, and as ℓ → −∞ the eigenvalues tend to 1. However, since ' ' 2ℓ ' q − 1 ' 1 − q 2|ℓ| ' ' ' q 2ℓ + 1 ' = 1 + q 2|ℓ| , the determinant of |D| is given by (see [147]) 7∞ 82 1 − q 2|ℓ| 1 − q 2n+1 det |D| = = . 1 + q 2|ℓ| 1 + q 2n+1 1 n=0 ℓ∈Z+ 2

There is one further observation that permits a dramatic simplification in this quantity. On page 493 of [157] it is shown that the complementary modulus k ′ defined by k 2 + k ′2 = 1 can be written as a ratio of theta constants: ϑ4 (0|τ )2 . ϑ3 (0|τ )2 On the other hand, the infinite product formulas for the theta functions ϑ3 and ϑ4 that can be found on page 469 of [157] show that 4 ∞ 1 − q 2n+1 ′ . k = 1 + q 2n+1 n=0 k′ =

Thus we have the famous Onsager–Yang formula for the spontaneous magnetization: Theorem 2.4.2 The spontaneous magnetization for the Ising model σ for T < Tc is given by ∞ 1 − q 2n+1 1 = (1 − k 2 ) 8 , σ = 2n+1 1 + q n=0


81

1

0.8

0.6

M 0.4

0.2

0

0.2

0.4

t

0.6

0.8

1

Figure 2.2: The spontaneous magnetization M = σ as a function of the scaled temperature t = T /Tc . $ # ′ where q = exp − πKK and the elliptic integrals K and K ′ are given by K=

)

0

1

1

1

(1 − t 2 )− 2 (1 − k 2 t 2 )− 2 dt

and ′

K =

)

0

1

1

1

(1 − t 2 )− 2 (1 − k ′2 t 2 )− 2 dt.

The constant k < 1 is given by k=

1 . sh(2K1 ) sh(2K2 )

The graph of the spontaneous magnetization as a function of T is shown in Figure 2.2 for the choice 2Ji /kB = 1. Before we turn to a more explicit calculation of the two-point functions, it is convenient to reconsider the spectral transform (2.49). The multistage character of this transformation is instructive because it reflects the understanding that went into defining the map in the first place. Still, for some purposes it is quite useful to note that the map S can be regarded as a matrix-valued multiplication operator S(z) followed by the change of variables z = z(x) = z(k sn2 (u± )) = k sn(u± + ia) sn(u± − ia).

82


In particular, in our treatment of the scaling behavior of the Ising correlation the action of the multiplication operator S(z) alone will take us to a “hyperbolic” representation that proves technically simpler to work in than the “elliptic” representation of the full transform S. As a first step in determining S(z), observe that as a function of x the matrix-valued multiplication operator is ; α2 − 1 √ q(x) 2 ¯ iq(x) . (2.53) x q(x) ¯ −iq(x) x + α2 To rewrite this as a function of z, recall (2.10), from which one can deduce ; α22 − 1 α2 − z =; (2.54) x + α2 α2 − 1 2

and

α1 − z α1 α 2 − 1 (1 − kx). = 2 α2 − z α2 − 1

(2.55)

√ Since q(x) = 1 − kx, one may use (2.55) and its complex conjugate to rewrite q(x) and q(x) ¯ in terms of z. Recall that we defined ( Aj (z) = αj − z normalized so that Aj (1) > 0. (2.56) Note that

A¯ j (z) =

(

αj − z = Aj (z−1 ),

since αj is real and the normalization condition is invariant under conjugation. We see that 7 82 1 − α2 z A¯ 2 (z) x= = z, z − α2 A2 (z) and from this it follows that √ A¯ 2 (z) √ x= z. A2 (z)

(2.57)

To go from the F± -coordinates to √ we need to multiply each′ √the spectral transform, component F± by the branch of x on M± given by k sn(u± ) for u± = u ± iK2 with 0 < u < 2K. On + this choice of square root is branched along the M ′ ray through x = k sn2 iK2 = −1. The curve (0, 2K) ∋ u → k sn2 (u+ ) winds √ ′ counterclockwise around the unit circle, and since k sn iK2 = i it follows that the appropriate square root for x is defined by choosing the argument π < arg x < 3π, with a branch cut on the ray through x = −1. Equivalently, the real part of taken in the left half-plane.

√ x is

2.5 The Two-Point Function Below Tc

83

On M− the appropriate square root of x is also branched along the ray through ′ x = k sn2 − iK2 = −1. The curve (0, 2K) → k sn2 (u− ) winds clockwise around √ ′ the unit circle, and since k sn − iK2 = −i the appropriate square root for x is again the one with negative real part branched along the ray through √ x = −1. We can thus make (2.57) produce the appropriate square root k sn(u± ) for x on M± by choosing the appropriate branch for ( ( √ z = z(x) = z(k sn2 (u± )).

√ Since x = −1 corresponds to z = −1, we should choose z so that it is branched along the ray through −1. Consulting (2.57), we see that we should pick the branch that is −1 at z = 1. Thus the appropriate square root of z is branched along the ray through −1 and has negative real part. This is minus the usual choice of a square root branched along z = −1, but since this will appear just in a couple of places, we will leave it as is and just flag this odd choice when needed.

Remark 2.4.1. It is precisely because this choice of branch cut for the square root is convenient for a scaling calculation later on, where we do not want functions to branch in a neighborhood of the scaling point z = 1, that we chose the domain (0, 2K) for√the parameter u. It is this choice that determines the location of the branch for k sn(u± ). √ With the understanding that the square root z has negative real part, we now substitute (2.54), (2.55), and (2.57) in (2.53) to find that the multiplication operator associated with S is S(z) =

z α1 α2 − 1

1 2

A¯ 1 A2 (z) iA1 A¯ 2 (z) . A¯ 1 A2 (z) −iA1 A¯ 2 (z)

(2.58)

This is useful both in the section on the two-point function above Tc and in the third chapter, on scaling behavior.

2.5 The Two-Point Function Below Tc In this section we develop a formula for the two-point Ising correlation functions below Tc in the infinite-volume limit. Rather than use the Pfaffian formulas that we employed to control the thermodynamic limit, it is possibly more instructive to use the Fock space characterization in Theorems 1.3.1 and 1.4.2 directly (our method is close in spirit to Abraham [2]). Suppose now that T < Tc and let (M, N) denote a point on the integer lattice Z2 with N ≥ 0. Then Theorem 1.3.1 implies that , σ0,0 σM,N = 0| σ V (M, N)σ |0 , (2.59) where

V (M, N) := Ŵ(z+ )M Ŵ(T+ (V ))N

84


is the combined translation and transfer operator and the spin operator σ acting on Alt(W+ ) characterized by its induced rotation and σ 2 = 1. Since σ is also unitary, it follows that it must be self-adjoint, and we see that to calculate the two-point function (2.59) we need only calculate the inner product of σ |0 with V (M, N)σ |0 in Alt(W+ ). We now proceed to the calculation of σ |0 in the spectral representation for transfer and translation that was worked out in the previous section. Recall from the previous section that the map L2 (S 1 , C2 ) ∋ f → Sf ∈ L2 (0, 2K) ⊕ L2 (0, 2K), which is the composition described in (2.49), identifies W+ (and also W− ) with L2 (0, 2K). Let 1 iℓπ u , for ℓ ∈ Z 21 , u ∈ (0, 2K). exp eℓ (u) := √ K 2K Then {eℓ } is an orthonormal basis of W+ . We write gg+− for a typical element of L2 (0, 2K) ⊕ L2 (0, 2K) and recall from Theorem 2.4.1 that the conjugation ∗ on W that leads to the distinguished symmetric form (1.77) is ∗ g+ g = − . g− g+ Thus {eℓ∗ } with eℓ∗ (u) = eℓ (−u) is the basis of W− that is dual to the basis {eℓ } of W+ with respect to the distinguished bilinear form on W . Let A B s= C D denote the matrix of the induced rotation s for σ in the W+ ⊕ W− splitting of W and write r = BD −1 . Then Theorem A.7.3 in the appendix implies that σ |0 = σ

∞

n=0 α1 Tc . The function λ(u) =

sn(u+ − ia) 0 and hence γ (0) = 2K1 − 2K2∗ > 0 for the “vertical translation mass” at (K1 , K2 ). The reciprocal γ (0)−1 is known as the (vertical) correlation length. The minimum for λ is then λ(K) = e−γ (0) .

2.7 The Correlation Length and Two-Point Asymptotics

Since the product

i<j

97

sn2 (ui − uj )

is less than 1, it is easy to see that the n-fold integrals in the expansion of the two-point function σ0,0 σ0,N are O(e−nN γ (0) ). To leading-order only the first term contributes to the asymptotics of σ0,0 σ0,N as N → ∞ for T > Tc , and only the twofold integral contributes to the leading asymptotics of the difference σ0,0 σ0,N − σ 2 as N → ∞ for T < Tc . We use this observation to prove the following asymptotic result. Theorem 2.7.1 Suppose k = sh(2K1∗ ) sh(2K2∗ ) < 1, so that the interaction constants (K2∗ , K1∗ ) correspond to a temperature T above Tc . Then to leading order, the asymptotics of the two-point function at large vertical separation are given by σ0,0 σ0,N T ∼ (

σ 2T ∗

e−γ (0)N for N → ∞, √ N 2πk sh γ (0)

where σ 2T ∗ is the square of the spontaneous magnetization at interaction (K1 , K2 ) below Tc and γ (0) = 2K1 − 2K2∗ . Suppose that (K1 , K2 ) corresponds to an interaction at a temperature T < Tc . To leading order, the asymptotics of the truncated two-point function at large vertical separation are given by σ0,0 σ0,N T − σ 2T ∼

σ 2T

e−2γ (0)N for N → ∞. 8π 2 sh2 γ (0) N 2

Proof. Both results are obtained by applying the Laplace method to the asymptotic estimation of the relevant integrals. Theorem 2.6.1 implies that to leading order, the asymptotics of the two-point function σ0,0 σ0,N T for T > Tc are governed by the integral ) σ 2T ∗ 2K λ(u)N du, (2.94) π 0

where

λ(u) =

sn(u+ − ia) , sn(u+ + ia)

with u+ = u +

iK ′ . 2

This function has an absolute minimum on the interval [0, K] at u = K. The principal contribution to the N → ∞ asymptotics of the integral in (2.94) is thus determined by the behavior of λ(u) in a neighborhood of u = K. We now investigate this behavior. Let F (u) = log λ(u). Then a short calculation using the derivative formulas for sn, cn, and dn and the quadratic relations among these functions shows that F ′′ (u) = k 2 sn2 (u+ − ia) − ns2 (u+ − ia) − k 2 sn2 (u+ + ia) + ns2 (u+ + ia).

98


Using the uniformization for z and λ, we see that λ(u)z(u) = k sn2 (u+ − ia). This is of particular interest to us at u = K. Since z(K) = 1, we see that iK ′ iK ′ 2 2 2 k sn K + − ia − ns K + − ia = k(λ(K) − λ(K)−1 ) 2 2 = −2k sh γ (0). In a similar fashion, implies

z(u)λ(u)−1 = k sn2 (u+ + ia)

iK ′ iK ′ + ia − k 2 sn2 K + + ia = −2k sh γ (0). ns2 K + 2 2 Thus in a neighborhood of u = K we have log λ(u) = −γ (0) −

4k sh γ (0) (u − K)2 + · · · , 2!

and we wish to determine the large-N behavior of ) σ 2T ∗ −γ (0)N 2 e−αN (u−K) du, e π |u−K| 0 is a small constant chosen to isolate the √ contribution near K. The change of variable αN (u − K)←⊣ u gives ) σ 2T ∗ e−γ (0)N −u2 du. ( √ √ e N π 2k sh γ (0) |u| 0

if s1 s2 = 1.

The leading asymptotics of the modified Bessel function for large order ℓ are 1 2 −1 ℓ 2 2 − ℓ sh exp ℓ + x x Iℓ (x) ∼ . √ 1 2π ℓ2 + x 2 4

If we substitute this asymptotics into (2.97) and make the change of variables x←⊣ rx, then we find that (2.97) becomes s1 s2

)

0

∞

e−r(c1 c2 x−g1,m (x)−g2,n (x)) 1

1

(m2 + s12 x 2 ) 4 (n2 + s22 x 2 ) 4

where 1 gj,ℓ (x) = ℓ2 + sj2 x 2 2 − ℓ sh−1

dx,

ℓ sj x

(2.98)

.

To estimate this last integral using the Laplace method we first calculate the derivative of g(x) = c1 c2 x − g1,m (x) − g2,n (x), to obtain

; ; g ′ (x) = c1 c2 − x −1 m2 + s12 x 2 − x −1 n2 + s22 x 2 .

2.8 The Montroll, Potts, and Ward Calculation of σ

101

The second derivative is g ′′ (x) =

m2 n2 + ; . ; x 2 m2 + s12 x 2 x 2 n2 + s22 x 2

This is manifestly positive, so g ′ (x) is an increasing function that ranges from −∞ at x = 0 to c1 c2 − s1 − s2 , which is strictly greater than 0 as long as s1 s2 = 1. Thus there is a unique critical point x0 at which g ′ (x0 ) = 0. Following McCoy and Wu we define θ1 > 0 and θ2 > 0 so that sh θ1 =

m s1 x0

and

sh θ2 =

n . s2 x0

It follows directly from this that s1 n sh θ1 − s2 m sh θ2 = 0. Also, g ′ (x0 ) = 0 translates into c1 c2 − s1 ch θ1 − s2 ch θ2 = 0. Evaluating g(x0 ) and g ′′ (x0 ), we obtain g(x0 ) = x0 (c1 c2 − s1 ch θ1 − s2 ch θ2 ) + mθ1 + nθ2 = mθ1 + nθ2 and g ′′ (x0 ) = x0−2 (mthθ1 + nthθ2 ). Thus the Laplace method applied to (2.98) yields √ 2π , √ ′′ 1 1 (m2 + s12 x02 ) 4 (n2 + s22 x02 ) 4 rg (x0 ) s1 s2 e−(mθ1 +nθ2 )r

which gives, after a little simplification, √ 2πs1 s2 e−(mθ1 +nθ2 )r . √ √ r m sh θ1 ch θ2 + n sh θ2 ch θ1 This gives the asymptotics as stated in the theorem.

2.8 The Montroll, Potts, and Ward Calculation of σ In [97], Montroll, Potts, and Ward showed how to derive the Onsager–Yang formula for the spontaneous magnetization using the Potts–Ward formula for the two-point function as a Toeplitz determinant and the Szego limit theorem. We will describe this calculation here.

102


Recall that the two-point function is given by σ0,0 σM,0 + = det AM ,

(2.99)

where AM is the M × M matrix with matrix elements ) ) π 1 i j −k dz w(z)z ¯ ϕ(θ)e−i(j −k)θ dθ, = aj,k = − 2π S 1 iz 2π −π where −1

ϕ(θ) = −i w(z ¯ )=

α1 − z α2 − z

1 2

α2 − z−1 α1 − z−1

21

(2.100)

and, of course, z = eiθ . Suppose that T < Tc . Then since |λ(u)| < 1 for u ∈ M+ , it is obvious from Theorem 2.5.1 that limσ00 σ0M + = σ 2 . M

However, since the vertical correlation σ00 σ0M + at interaction constants (K1 , K2 ) is the same as the horizontal correlation σ00 σM0 + at interaction constants (K2 , K1 ), it follows that σ00 σM0 + will tend to σ 2 as M → ∞ (at exchanged interaction constants K1 ⇋ K2 ). Since the calculation will show that σ is invariant under the exchange of K1 and K2 , the exchange will not be important. One simple version of the Szego limit theorem that applies to the asymptotics of (2.99) is the following. Theorem 2.8.1 Suppose that θ → ϕ(θ) is a smooth complex-valued function on the circle with winding number 0 about 0. Let AM denote the M × M Toeplitz matrix with j, k matrix element ) π 1 ϕ(θ)e−i(j −k)θ dθ, and j, k = 0, · · · , M − 1. aj,k = 2π −π Since ϕ has winding number 0 about 0, it has a smooth logarithm with Fourier expansion log ϕ(θ) = bn einθ , n

Let G = e . Then b0

∞ det AM = exp nbn b−n . M→∞ GM n=1

lim

The application of this theorem to the asymptotics of the determinant (2.99) is straightforward. Since α1 and α2 are both greater than 1 for T < Tc , the Fourier expansion of log ϕ is ∞ ∞ 1 α2−n − α1−n inθ α1−n − α2−n −inθ e + e . log ϕ(θ) = 2 n=1 n n n=1

2.8 The Montroll, Potts, and Ward Calculation of σ

Thus b0 = 0, G = 1, and n=1

nbn b−n

∞

1 α2−n − α1−n α1−n − α2−n = n 4 n=1 n −n 7 8 1 − α1−2 1 − α2−2 1 = log . 2 4 1 − (α1 α2 )−1

A little algebra shows that the argument of the log in this last expression is 2 α1 − 1 α22 − 1 (α1 − α2 )2 = 1 − k2, =1− 2 (α1 α2 − 1) (α1 α2 − 1)2 where we used the identity (2.9). Thus 1

limσ00 σM0 + = (1 − k 2 ) 4 = σ 2 . M

103

3 Scaling Limits

3.1 Introduction In this chapter we look at the behavior of the correlation functions for the Ising model in a neighborhood of the critical point. One of the signitures of critical phenomena is the onset of long-range order; at the critical point this is expected to arise from a delicate balance between the short-range forces disposing adjacent spins to line up and the randomizing effects of thermal activity. Below the critical point the short-range forces of order triumph, and above the critical point thermal randomization dominates. One can see the ghost of this effect in a pure phase, such as the plus state below Tc , or the open state above Tc , by looking at the rate at which spin correlations tend to their limiting value at infinite separation. For example, we have seen in Theorem 2.7.1 that for T < Tc , σ0,0 σ0,N T − σ 2T ∼ N −2 e−2γ (0)N

for N → +∞,

and for T ∗ > Tc , 1

σ0,0 σ0,N T ∗ ∼ N − 2 e−γ (0)N

for N → +∞.

(3.1)

The constant γ (0) = 2K1 − 2K2∗ is a function of the interaction constant K = (K1 , K2 ) that tends to 0 as T approaches Tc (where K1 = K2∗ ). The inverse ℓ :=γ (0)−1 is called the (vertical) correlation length, and it represents a scale at which the correlation functions just begin to cluster exponentially.As T approaches the critical point Tc , the correlation length ℓ tends to infinity. At the critical point (where σ = 0) the two-point function (3.1) no longer tends to 0 exponentially fast

106

3. Scaling Limits

but instead tends to 0 as a power N −d+2−η , where the exponent is conventionally written in terms of the dimension d (two in our case) and a critical exponent η. One can find a calculation of the asymptotics of the two-point function at the critical temperature in McCoy and Wu’s book. They confirm that the critical exponent η is equal to 41 . In a neighborhood of the critical point there are three different regimes of interest. The conformal regime looks at the correlations at a scale much larger than the lattice spacing but much smaller than the correlation length. This regime is expected to have rotational and scale invariance, in part because at this scale the correlations decay at less than an exponential rate and the scale is incommensurate with what are expected to be the only two relevant length scales for the problem: the lattice spacing and the correlation length. In two-dimensional models it is often expected that the rotational and scale invariance promotes to a subtly enhanced conformal symmetry [47]. This expectation has been thoroughly exploited in recent years to provide a host of interesting conjectures for the scaling behavior of lattice models, at least some of which have been confirmed by alternative analysis. A second regime considers the correlations at a scale much larger than the correlation length in a neighborhood of the critical point. At this scale the expectation is that the local interaction is no longer felt and the correlations scale to a “noninteracting” Gaussian random field. The Gaussian nature of the scaling reflects the idea that sums of “almost independent random” variables should exhibit central limit theorem behavior. Arguably the most interesting asymptotic regime, and one that is expected to capture much of the behavior of interest in the neighborhood of the critical point, looks at the spin correlation functions at the scale of the correlation length. The leading behavior is given by powers of the one-point function (at least below Tc ), and the correction to this leading behavior is neatly summarized in what are called scaling functions. These turn out to be the correlation functions of a random field that is a special case of a two-dimensional quantum field theory in the so-called Euclidean domain (i.e., imaginary time). In this section we prove the existence of this scaling limit. We also obtain formulas for the scaled correlations that allow us to establish the Sato, Miwa, and Jimbo analysis of these scaling functions as the τ -functions of a holonomic quantum field. This is taken up in Chapters 5 and 6. It is expected that the first two asymptotic regimes are accessible from this intermediate regime by looking at the short-distance and large-distance behavior of the intermediate scaling functions. We don’t take this up in this chapter, but we do note that the Sato, Miwa, and Jimbo analysis of these correlations makes it possible to compute the short-distance asymptotics [114], [115] and confirm the Luther–Peschel formula that also arises from conformal field theory considerations. This will be explained in a little more detail following our discussion of the SMJ deformation theory in Chapter 6. Actually, there are two distinct scaling limits depending on whether T approaches Tc from above or below. However, our approach via duality to the correlations above Tc allows us to focus on constructions below Tc . Recall that the

3.1 Introduction

107

constants Kj are given in terms of the interaction strengths Jj , Kj =

Jj , kB T

(3.2)

where kB is the Boltzmann constant and T is the temperature. For the present we suppose that the horizontal and vertical interaction strengths Jj > 0 are fixed independent of T and that the temperature is allowed to vary 0 < T < Tc . Note that the critical temperature Tc does depend on J1 and J2 through the relation 2J1 2J2 = 1. sh sh kB Tc kB Tc As noted in Chapter 2, the exponential rate at which the plus state correlations cluster for vertical separations is determined by the bottom of the spectrum of γ (θ) (the “mass gap”). Since ch(γ (θ)) = c2∗ c1 − s2∗ s1 cos(θ),

(3.3)

the bottom of the spectrum occurs at m2 := γ (0), where m2 is defined as the positive root of ch(m2 ) = c2∗ c1 − s2∗ s1 = ch(2K1 − 2K2∗ ),

(3.4)

so that m2 = 2K1 − 2K2∗ > 0

Thus m2 → 0 as T ↑ Tc since

K2∗

for T < Tc .

= K1 at T = Tc . Also,

sh(m2 ) = sh(2K1 − 2K2∗ ) = s1 c2∗ − c1 s2∗ .

(3.5)

Exchanging the roles of J1 and J2 , one sees that the exponential rate at which the plus state correlations cluster for horizontal separations is determined by the positive root m1 of ch(m1 ) = c1∗ c2 − s1∗ s2 . (3.6)

As above, and

m1 = 2K2 − 2K1∗ > 0

for T < Tc

sh(m1 ) = s2 c1∗ − c2 s1∗ .

(3.7)

We look at multispin correlations at a horizontal scale determined by m1 (T )−1 (which tends to ∞ as T ↑ Tc ) and at a vertical scale determined by m2 (T )−1 (which also tends to ∞ as T ↑ Tc ). Thus m1 (T ) is the horizontal “mass” and m1 (T )−1 the horizontal correlation length; m2 (T ) is vertical “mass” and m2 (T )−1 the vertical correlation length.

108

3. Scaling Limits

Suppose now that aj = (xj , yj ) ∈ R 2 for j = 1, . . . , n with y1 < y2 < · · · < yn and for T < Tc define , −1 −1 σ m an σ m an−1 · · · σ m−1 a1 T τ (a; T ) := , (3.8) σ nT where a = (a1 , a2 , . . . , an ), m−1 aj := m1−1 xj , m2−1 yj , and the spin correlations are evaluated at temperature T in (3.2) with J1 and J2 held fixed. We typically use ± to index polarizations, but now want to use ± to index quantities or limits from below Tc (−) and quantities or limits from above Tc (+). To avoid confusion we introduce the slightly modified notation [−] ∼ limit from T < Tc and [+] ∼ limit from T > Tc . The way we have defined things, the coordinates m1−1 xj and m2−1 yj are not necessarily integers. However, the Pfaffian formula in Theorem 1.3.1 for the quotient in (3.8) interpolates continuously to noninteger points. It simplifies our analysis at the start to interpret (3.8) in this sense and to define the scaling limit from below Tc as (3.9) τ [−] (a) = lim τ (a; T ). T ↑Tc

Later it will be important to establish that the scaling limit is completely determined by the values of the spin correlation functions at integer points, where we have some probabilistic control over their behavior (i.e., Griffith’s inequalities). The modifications needed to see this are introduced after we obtain Pfaffian formulas for the limit (3.9). To prove the convergence of (3.9) and to find formulas for the limit, we proceed in stages. First we rewrite the Fock representation of the correlation in (3.9) in a way that takes advantage of the fact that we can choose continuous extentions of the horizontal and vertical translation groups from the integers to the reals and the smoothing properties of positive powers of the transfer matrix. Recall the map U in (1.78) of Preface that identifies Q± L2 (S 1 , C2 ) with L2 (S 1 ). On Q+ L2 identified with L2 [−π, π] and for x ∈ R define a multiplication operator eixθ by f (θ) → eixθ f (θ) for |θ| < π. For x ∈ R and Ŵ from (A.82) define σ (x) = Ŵ(eixθ )σ Ŵ(e−ixθ ), and write

A(x) T (σ (x)) := C(x)

B(x) D(x)

(3.10)

3.1 Introduction

109

for the matrix of the induced rotation of σ (x) in the Q polarization. When x is an integer, σ (x) is the spin operator at x. Suppose a = (x, y) ∈ R 2 and define σ (a) = V −y σ (x)V y , where, of course, V = Ŵ(T+ (V )) is the self-adjoint vertical transfer matrix. The reader should keep in mind that the operator V −y is unbounded for y > 0, but as is explained in the appendix it acts on a dense domain that is invariant under the action of the infinite complex spin group. The operator σ (a) is understood to act on this dense domain. Now write aj = (xj , yj ) ∈ R 2 for j = 1, . . . , n, and suppose that y1 ≤ y2 ≤ · · · ≤ yn , so the second coordinates of the points aj increase with j . For T < Tc define σ (a) := 0|σ (an )σ (an−1 ) · · · σ (a1 )|0.

(3.11)

When each aj is in Z2 , this is the representation we obtained for the multispin correlation function in Theorem 1.3.1. For aj ∈ R 2 the quantity σ (a) is a continuous interpolation of the spin correlations at integer sites. For convenience, introduce real numbers y0 and yn+1 such that y0 < y1 and yn+1 > yn , and define ¯

¯

σj := V j +1 σ (xj )V j

for j = 1, 2, . . . , n,

(3.12)

where we have introduced ¯ j := y¯j − yj −1 = and y¯j :=

yj − yj −1 2

yj + yj −1 . 2

Then since V |0 = |0 and 0|V = 0|, it follows that σ (a) = 0|σn σn−1 · · · σ1 |0. The induced rotation T (σj ) :=

Aj Cj

Bj Dj

(3.13)

of σj in the Q polarization is ¯ ¯ e−j +1 γ A(xj )e−j γ ¯ ¯ ej +1 γ C(xj )e−j γ

¯ ¯ e−j +1 γ B(xj )ej γ . ¯ ¯ ej +1 γ D(xj )ej γ

(3.14)

110

3. Scaling Limits

Most important for us is that the operators Dj−1 , Bj Dj−1 , and Dj−1 Cj that appear in the Pfaffian representation for σ −n σ (a) of Theorem A.7.4 are given by ¯

¯

Dj−1 = e−j γ D(xj )−1 e−j +1 γ , ¯ ¯ Bj Dj−1 = e−j +1 γ B(xj )D(xj )−1 e−j +1 γ , ¯ ¯ −1 Dj Cj = e−j γ D(xj )−1 C(xj )e−j γ .

(3.15)

¯

¯ > 0 that appear in these The smoothing properties of the factors e−γ for formulas play an important role in our proof of the convergence of the scaling ¯ j > 0 for the formulas we obtain in the limit. In fact, we need to assume that scaling limit to make straightforward functional-analytic sense. The scaling limit we are interested in is lim

T ↑Tc

0|σ (m−1 an ) · · · σ (m−1 a1 )|0T , σ nT

n where m−1 aj is the element of R 2 with components m1−1 xj , m2−1 yj . Thus the ¯ j in (3.15) are relevant for the scaling ¯ j ←⊣ m2−1 replacements xj ←⊣ m1−1 xj and limit. Next we introduce a “change of variables” that provides a “hyperbolic” representation for the kernals of D −1 , BD −1 , and D −1 C that are ingredients in the Pfaffian formulas for the correlations. Actually, there is one other ingredient in these Pfaffian formulas, namely D −τ . In the representation we work with it will turn out that D −τ = −D −1 at the level of integral kernels. The operator D −1 acts on W− and the operator D −τ acts on W+ , but our identification of both W+ and W− with the same L2 space gives meaning to the equality D −τ = −D −1 . This “identification” allows us to ignore D −τ when we prove convergence results: results for D −τ follow from those for D −1 . The hyperbolic representations allow us to concentrate on the scaling behavior of the function γ (θ) in order to control the scaling limits for D −1 , BD −1 , and D −1 C. Continuity in the Pfaffian representation of the correlations provides Pfaffian formulas for the scaling functions. Theorems 3.4.1 and 3.4.2 are the first goal of this chapter.

3.2 Hyperbolic Formulas for the Kernels of D −1 , BD −1 , and D −1 C We obtain hyperbolic formulas for the kernels of D −1 , BD −1 , and D −1 C through similarity transform by the matrix-valued multiplication operator S(z) introduced in (2.58). However, we first conjugate by U in (1.78) in order to use the standard Wiener–Hopf calculus to calculate D −1 . Our starting point is the representation for the matrix elements A, B, C, and D found in (1.80). From this representation

3.2 Hyperbolic Formulas for the Kernels of D −1 , BD −1 , and D −1 C

111

and the formula (1.81) for D −1 one arrives at the following formulas for D −1 , BD −1 , and D −1 C: −1 −1 D −1 f = 21 w− sw− + w+ sw+ f, BD −1 f =

i 2

(w+ sw− − w− sw+ ) f,

D −1 Cf =

i 2

−1 −1 −1 −1 w− sw+ − w+ sw− f.

(3.16)

These are the operators that appear in the Pfaffian formulas for the multispin correlations. Recall that in this representation each of the vector spaces W± := Q± L2 (S 1 , C2 ) is identified with L2 (S 1 ) and D −1 : W− → W− , BD −1 : W− → W+ , D −1 C : W+ → W− .

(3.17)

We now conjugate the maps in (3.16) by the map S(z)U −1 , which is a unitary map, W+ ⊕ W− ∼ L2 (S 1 , dθ) ⊕ L2 (S 1 , dθ) s1 dθ s1 dθ 2 1 2 1 ⊕L S , . →L S , 2 sh γ (θ) 2 sh γ (θ)

(3.18)

Because we now deal with L2 spaces with respect to different measures, we write L2 (X, dµ) for the Hilbert space of L2 functions on X with inner product ) f¯g dµ. X

After a little calculation and using Aj (z−1 ) = A¯ j (z), we obtain S(z)U

−1

1 2 2z A¯ 1 A2 f+ f+ = , f− −iA1 A¯ 2 f− α1 α2 − 1

(3.19)

which is a unitary map of type (3.18). Conjugating the maps in (3.16) by this multiplication operator (keeping in mind (3.17)) and using w+ = A1 A−1 2 , w− = A¯ 2 A¯ −1 1 , which is just a transcription of (1.74) and (1.75), we obtain −1 ¯ −1 ¯ −1 ¯ D −1 = 12 A1 A¯ 1 sˆ A−1 , 1 A1 + A2 A2 sˆ A2 A2 −BD −1 = D −1 C =

1 2

−1 ¯ −1 ¯ −1 ¯ , A1 A¯ 1 sˆ A−1 1 A1 − A2 A2 sˆ A2 A2

(3.20)

(3.21)

112

3. Scaling Limits

√ √ where sˆ is s conjugated with multiplication by z. Recall that z has negative real part in the formula for S(z). However, in the following formula for sˆ we take √ z branched along z = −1 with positive real part since the two minus signs will just cancel, √ √ −1 (3.22) sˆ := zs z . Of course, BD −1 and D −1 C are maps defined on different spaces; the equality in (3.21) is understood to hold after the#change of coordinates that identifies each sub$ . Note the minus sign in −BD −1 . space in the polarization Q with L2 S 1 , 2 shs1 dθ γ (θ ) Recalling (2.81), we see that this can be rewritten D −1 = 12 sh 12 γ sˆ (sh 21 γ )−1 + ch 12 γ sˆ (ch 21 γ )−1 , (3.23) −BD −1 = D −1 C = 12 sh 12 γ sˆ (sh 21 γ )−1 − ch 12 γ sˆ (ch 21 γ )−1 .

This representation is useful for understanding the scaling limit of D −1 . It is also worthwhile, at this point, to write down explicit formulas for these operators as integral operators. Making use of sh γ = 2 sh γ2 ch γ2 , the action of D −1 , BD −1 , and D −1 C is rewritten 1 , D −1 = sh 21 γ sˆ ch 12 γ + ch 21 γ sˆ sh 21 γ sh γ −BD

−1

−1

= D C = sh

Since

1 γ sˆ ch 21 γ 2

− ch

1 γ sˆ sh 21 γ 2

1 . sh γ

(3.24)

sh 12 γ (θ) ch 21 γ (θ ′ ) ± ch 12 γ (θ) sh 21 γ (θ ′ ) = sh 21 (γ (θ) ± γ (θ ′ )) and the kernel of sˆ is found to be the principal value √ 1 1 1 1 1 z + −1 , := lim √ r↑1 2πi 2 z′ 2π i sin 12 (θ − θ ′ ) rz − z′ r z − z′ where

√ θ z = ei 2 with |θ| < π , we see that ) π 1 dθ ′ D −1 f (θ) = XT+ f (θ) := XT+ (θ, θ ′ )f (θ ′ ) , 2πi −π sh γ (θ ′ ) ) π 1 dθ ′ D −1 Cf (θ) = XT− f (θ) := XT− (θ, θ ′ )f (θ ′ ) , 2πi −π sh γ (θ ′ )

and −1

BD f (θ) =

−XT− f (θ)

1 := − 2πi

)

π

−π

XT− (θ, θ ′ )f (θ ′ )

dθ ′ , sh γ (θ ′ )

(3.25)

(3.26) (3.27)

3.3 Convergence Estimates

where XT± (θ, θ ′ )

:=

sh

γ (θ) ± γ (θ ′ ) , sin 12 θ − θ ′

1 2

113

(3.28)

and the integral in (3.26) is understood in the principal value sense (3.25). At this point it is useful to note that the distinguished bilinear form on W+ ⊕ W− ∼ L2 (S 1 , C) ⊕ L2 (S 1 , C) is ) π s1 dθ . (f, g) = (f1 (θ)g2 (−θ ) + f2 (θ)g1 (−θ )) 2 sh γ (θ) −π It is then a simple matter to see that D −τ f (θ) = −XT+ f (θ).

(3.29)

Convergence results for XT+ will apply to D −τ as well as D −1 .

3.3 Convergence Estimates ¯ j ←⊣ m2−1 ¯ j in (3.15) produce scaled horThe substitutions xj ←⊣ m1−1 xj and −1 ¯ ixj m1−1 θ izontal and vertical “translations” e and e−j m2 γ (θ ) . This suggests the introduction of the scaling variable p = m1−1 θ and the examination of the behavior of m2−1 γ (m1 p) as T ↑ Tc . With this in mind, we introduce a scaled version of the function γ that plays a central role in the analysis of the continuum limits. For T < Tc define γT (p) := γ (m1 p) for p ∈ [−π/m1 , π/m1 ],

(3.30)

where it is understood that m1 = m1 (T ) and the function γ (·) in this formula is also evaluated at temperature parameter T . In preparation for a closer look at γT , it is useful to examine the ratio of m1 to m2 in the limit T ↑ Tc . Note that (3.5) and (3.7) imply that lim

T ↑Tc

c1 − c2∗ s22 sh(m1 ) m1 = lim = lim · . T ↑Tc sh(m2 ) T ↑Tc c2 − c∗ s 2 m2 1 1

(3.31)

The linear approximations for cj and cj∗ near T = Tc are 2Jj (T − Tc ) + O(T − Tc )2 , kB Tc2 2Jj (T − Tc ) + O(T − Tc )2 . cj∗ = cj (Tc ) + sj (Tc )−2 kB Tc2 cj = cj (Tc ) − sj (Tc )

Substituting these approximations in (3.31) and noting that s1 (Tc )s2 (Tc ) = 1, we obtain m1 1 2J1 −1 lim . (3.32) = = sh T ↑Tc m2 s1 (Tc ) kB Tc

114

3. Scaling Limits

The following lemma is the principal tool in our convergence proof for the scaling limits. Lemma 3.3.1 Write ω(p) :=

(

1 + p2 .

The function γT (p) has the following properties:

(1) There exists an interval [T0 , Tc ] (with 0 < T0 < Tc ) and constants c > 0 and C > 0 independent of p and T such that cω(p) ≤ m2−1 γT (p) ≤ Cω(p)

for T ∈ [T0 , Tc ] and |m1 p| ≤ π

and 2 − ǫ ω(p) ≤ m2−1 sh γT (p) ≤ Cω(p) for T ∈ [T0 , Tc ] and |m1 p| ≤ π, π where ǫ > 0 can be made arbitrarily small by choosing T0 sufficiently close to Tc . (2) The limits lim m2−1 γT (p) = ω(p)

T ↑Tc

and lim m2−1 sh γT (p) = ω(p)

T ↑Tc

are uniform for p in any compact subset of R. The explicit lower bound for m2−1 sh γT (p) is useful for a later integrability estimate. Proof. Recall (3.3) and the definition of ch (m2 ) in (3.4) to see that ch(γT (p)) = c1 c2∗ − s1 s2∗ cos m1 p = ch(m2 ) + s1 s2∗ (1 − cos m1 p).

(3.33)

Now define the auxillary function ωT (p), ωT2 (p) := 2m2−2 (ch(m2 ) − 1) + 2s1 s2∗ m2−2 (1 − cos m1 p).

(3.34)

With the introduction of ωT (p), (3.33) becomes ch(γT (p)) = 1 +

m22 ωT2 (p) . 2

(3.35)

Now we need some elementary estimates for the inverse hyperbolic cosine defined on the positive reals. Suppose that x > 0. Then $ # $ # ( ( ch−1 (1 + x) = log 1 + x + (x + 1)2 − 1 = log 1 + x + x 2 + 2x .

The estimates

( √ √ # √ $ 2x ≤ x + x 2 + 2x ≤ 2x 1 + 2x


115

and the estimate for the log, u ≤ 1+u

)

1+u

1

1 dt ≤ u, t

imply that √ √ √ −1 2x(1 + 2x)−1 ≤ log(1 + √2x) ≤ ch √ (1 + x) √ √ ≤ log(1 + 2x(1 + 2x)) ≤ 2x(1 + 2x).

(3.36)

Combining (3.35) with (3.36), one sees that ωT (p) ≤ m2−1 γT (p) ≤ ωT (p)(1 + m2 ωT (p)). 1 + m2 ωT (p)

(3.37)

To finish the proof we need to analyze ωT (p). First observe that since mj (T ) tends to zero as T ↑ Tc , we have lim 2m2−2 (ch(m2 ) − 1) = 1

T ↑Tc

and lim m1−2 (1 − cos m1 p) = p2 .

T ↑Tc

Also, (3.31) shows that lim

T ↑Tc

m12 s2 (Tc ) = , 2 s1 (Tc ) m2

since s1 (Tc )s2 (Tc ) = 1. Together with (3.34) these limits imply, lim ωT2 (p) = 1 + p 2 .

T ↑Tc

The estimate

θ2 2θ 2 ≤ cos(θ) ≤ 1 − 2 for |θ| ≤ π 2 π

(3.39)

1 − cos m1 p 2 1 ≤ ≤ for |m1 p| ≤ π. 2 2 2 π 2 m1 p

(3.40)

1− translates to

(3.38)

Since ωT (p)2 = 2

s1 m12 1 − cos m1 p 2 ch m2 − 1 p , + 2 m22 s2 m22 m12 p 2

the power series expansion for ch m2 and the lower bound (3.40) imply that ωT (p)2 ≥ 1 +

s1 m12 4 2 p . s2 m22 π 2

116

3. Scaling Limits

But because

s1 m12

s2 m22

tends to 1 as T → Tc and ωT (p) ≥

4 π2

< 1, we have the lower bound

2 − ǫ ω(p), for T ∈ [T0 , Tc ], π

where ǫ > 0 can be made arbitrarily small by choosing T0 sufficiently close to Tc . The upper bound for ωT (p) is simpler, since we are not concerned with a precise value for the constant, and we see that there exists a constant B such that 2 − ǫ ω(p) < ωT (p) < Bω(p) for |m1 p| ≤ π. (3.41) π Together with (3.37) we see that there exist constants 0 < c < C such that cω(p) ≤ m2−1 γT (p) ≤ Cω(p)

for |m1 p| ≤ π.

To obtain upper bounds for sh γT (p) note that sh2 γT (p) = (ch γT (p) + 1)(ch γT (p) − 1) = Taking square roots yields = m2−1 sh γT (p) =

1 + ch γT (p) 2 2 m2 ωT (p). 2

1 + ch γT (p) ωT (p) = ch 12 γT (p)ωT (p). 2

(3.42)

But ch 21 γ ≥ 1 and ch 21 γ ≤ ch γ ≤ c1 c2∗ + s1 s2∗ is uniformly bounded for T ∈ [T0 , Tc ], provided T0 > 0. Choose T0 such that 0 < T0 < Tc ; we see that there is a constant C independent of p and T for T ∈ [T0 , Tc ] such that ωT (p) ≤ m2−1 sh γT (p) ≤ CωT (p). Combined with (3.41), this finishes the proof of part 1 of the lemma. To finish the proof we show that as T ↑ Tc , ωT (p) converges uniformly to ω(p) for p in a compact subset of R. Up to a convergent constant multiplier the d 2 ωT (p) is given by derivative dp m1−1 sin m1 p, which converges as T ↑ Tc in L1 to p on compact subintervals of R by dominated convergence. This implies that ωT2 (p) converges uniformly to ω2 (p) on compact subsets of R, and since ωT (p) is uniformly bounded away from zero, it is also true that ωT (p) converges uniformly to ω(p) on compact subsets of R as T ↑ Tc . The estimate (3.37) then shows that m2−1 γT (p) converges uniformly as T ↑ Tc to ω(p) on compact subsets of R. It is obvious that ch 21 γT (p) converges uniformly to 1 as T ↑ Tc on bounded subsets of R. With (3.42) this finishes the proof of part (2), showing that m2−1 sh γT (p) converges uniformly to ω(p) as T ↑ Tc on compact subsets of R.


117

Next we introduce the scale transformation (3.43)

f (θ) → f (m1 p) as a unitary map WT := L2 [−π, π],

s1 dθ 2 sh γ (θ)

→ L2

−

s1 m1 dp π π , , . (3.44) m1 m1 2 sh γT (p)

However, in order to use standard tools from functional analysis to discuss scaling behavior, we incorporate in this scale transformation an isometric embedding of the “temperature-dependent” Hilbert space in (3.44) above into the “temperatureindependent” Hilbert space dp W := L2 R, . 2ω(p) The isometric embedding oT of WT into W that we use is WT ∋ f (θ) → oT f (p) := oT (p)f (m1 p) ∈ W, where

Observe that

<   m1 s1 ω(p) oT (p) := sh γT (p)   0 lim oT (p) = lim

T ↑Tc

T ↑Tc

=

m1 s1 m2

(3.45)

for |m1 p| ≤ π , for |m1 p| > π.
0 and β > 0, each of the operators in (3.15) can be written e−αγ eixθ XT± e−ixθ e−βγ . To obtain a Pfaffian formula for τ (a; T ) (T < Tc ) we first replace x by m1−1 x and α and β by m2−1 α and m2−1 β to obtain −1

e−αm2

γ ixm1−1 θ

e

−1

−1

XT± e−ixm1 θ e−βm2

γ

.

(3.48)

118

3. Scaling Limits

Then we substitute the resulting operators into Theorem A.7.4 to obtain such a Pfaffian formula. However, to understand the T ↑ Tc limit we replace each operator in (3.48) by its isometric image under conjugation by oT (i.e., X → oT XoT∗ ). Conjugation of (3.48) by oT gives −1

e−αm2

γT ixp

e

−1

oT XT± oT∗ e−ixp e−βm2

γT

(3.49)

.

A simple calculation shows that oT∗ f (θ) = oT∗ (θ)f (m1−1 θ ) for |θ| ≤ π, where oT∗ (θ)

=

π. Change variables θ ′ = m1 p ′ in the integral and observe that oT∗ (m1 p) = oT (p)−1 for |m1 p| ≤ π to find that oT XT± oT∗ f (p) is given by ) π/m1 1 m1 dp ′ oT (p)XT± (m1 p, m1 p ′ )oT (p ′ )−1 f (p ′ ) . (3.50) 2π i −π/m1 sh γ (m1 p ′ ) ′

dp We rewrite this integral operator with respect to the measure ω(p ′ ) . Introduce the kernels XT ,± (p, p ′ ) = oT (p)s1−1 XT± (m1 p, m1 p ′ )oT (p ′ ). (3.51)

Recall that oT (p ′ ) vanishes for |m1 p ′ | > π, so (3.50) becomes ) 1 dp ′ oT XT± oT∗ f (p) = XT ,± (p, p ′ )f (p ′ ) . 2πi R ω(p ′ )

(3.52)

The pointwise limit of the kernel XT ,± (p, p ′ ) is easy to determine. Both oT (p) and oT (p ′ ) tend to 1 as T ↑ Tc . The limits (3.32) and lim m2−1 sh γT (p) = lim m2−1 γT (p) = ω(p),

T ↑Tc

T ↑Tc

together with the definition (3.28), imply that lim s1−1 XT± (m1 p, m1 p ′ ) = lim

T ↑Tc

T ↑Tc

ω(p) ± ω(p ′ ) m2 m1 m2−1 XT± (p, p ′ ) = . s1 m1 p − p′

Define X± (p, p ′ ) :=

ω(p) ± ω(p ′ ) . p − p′

(3.53)


119

Then lim XT ,± (p, p ′ ) = X± (p, p ′ ).

(3.54)

T ↑Tc

Our next result promotes this pointwise convergence for XT ,− (p, p′ ) to a more useful Schmidt class convergence when the exponential factors in (3.15) are included. Lemma 3.3.2 Suppose that α, β > 0 and write −1

α,β

−1

XT ,− (p, p ′ ) = e−αm2 γT (p) XT ,− (p, p′ )e−βm2 ′ α,β X− (p, p ′ ) = e−αω(p) X− (p, p ′ )e−βω(p ) .

γT (p ′ )

,

(3.55)

Then lim

T ↑Tc

)

R

dp ω(p)

)

R

'2 dp ′ '' α,β α,β ′ ′ ' (p, p ) − X (p, p ) ' = 0. 'X − T ,− ω(p ′ )

(3.56)

This L2 convergence for the kernels is Schmidt norm convergence for the associated operators. Proof. We give simple estimates to that show that dominated convergence applies α,β in (3.56). We start with an estimate for X− (p, p′ ). Since ' ' ' '' ' ' dω ' ' p ' ' ' = '( ' ≤ 1, ' dp ' ' p2 + 1 ' the mean value theorem implies that ' ' ' ω(p) − ω(p ′ ) ' ' ' ≤ sup |ω′ | = 1, ' ' p − p′ and it follows that

' ' ′ ' α,β ' 'X− (p, p ′ )' ≤ e−αω(p)−βω(p ) .

(3.57)

α,β

We prove a similar estimate for XT ,− (p, p ′ ). Choose T0 such that 0 < T0 < Tc . Then for T ∈ [T0 , Tc ] there exists a constant C independent of p, p ′ and T such that |XT− (p, p ′ )| ≤ C.

(3.58)

We show this by obtaining an upper bound for sh 12 (γ (θ) − γ (θ ′ )) sin 21 (θ − θ ′ )

,

for θ, θ ′ ∈ (−π, π ). Some slight complication in the estimate arises from the need to take into account the vanishing of the denominator not only for θ = θ ′ but also

120

3. Scaling Limits

when θ = ±π and θ ′ = ∓π . To simplify notation write γ = γ (θ) and γ ′ = γ (θ ′ ) and write C for a generic constant independent of θ, θ ′ , and T ∈ [T0 , Tc ]. First we estimate | sh 21 (γ − γ ′ )| ≤ | sh(γ − γ ′ )| = | sh γ ch γ ′ − sh γ ′ ch γ | ≤ | sh γ || ch γ ′ − ch γ | + | ch γ || sh γ − sh γ ′ | ≤ C(| ch γ ′ − ch γ | + | sh γ − sh γ ′ |).

(3.59)

The calculation ' ' ' ' ' ch γ (θ) − ch γ (θ ′ ) ' ' ′' ' ' ∗ ' cos θ − cos θ ' ' ' = s1 s2 ' ' = 2s1 s2∗ | sin 12 (θ + θ ′ )| ' sin 12 (θ − θ ′ ) ' ' sin 21 (θ − θ ′ ) '

(3.60)

combine to give the estimate ' ' ' ' ' sh γ (θ) − sh γ (θ ′ ) ' ' sin 1 (θ + θ ′ ) ' ' ' ' ' 2 ' '. '≤C' ' sin 12 (θ − θ ′ ) ' ' sh γ (θ) + sh γ (θ ′ ) '

(3.61)

and the identities ' ' ' ' ' sh γ (θ) − sh γ (θ ′ ) ' '' sh γ (θ) − sh γ (θ ′ ) '' ' ch γ (θ) − ch γ (θ ′ ) ' ' ' ' ' ' '' ' '= ', ' sin 12 (θ − θ ′ ) ' ' ch γ (θ) − ch γ (θ ′ ) ' ' sin 21 (θ − θ ′ ) ' ' ' ' ' ' sh γ (θ) − sh γ (θ ′ ) ' ' ch γ (θ) + ch γ (θ ′ ) ' '=' ' ' ' ch γ (θ) − ch γ (θ ′ ) ' ' sh γ (θ) + sh γ (θ ′ ) ' ,

Since (see (3.39))

ch γ (θ) − 1 = ch(m2 ) − 1 + s1 s2∗ (1 − cos θ ) ≥ 2π −2 s1 s2∗ θ 2 we see that

; sh γ (θ) = ch2 γ (θ) − 1 ≥ C|θ|

for |θ| ≤ π,

for |θ| ≤ π.

Together with (3.61) this implies ' ' ' ' ' ' ' sh γ (θ) − sh γ (θ ′ ) ' ' sin 1 (θ + θ ′ ) ' ' sin 1 (θ + θ ′ ) ' ' ' ' ' ' ' 2 2 ' '≤C' '≤C' '. ' θ + θ′ ' sin 12 (θ − θ ′ ) ' ' |θ| + |θ ′ | ' '

(3.62)

Now divide the inequality (3.59) by | sin 21 (θ − θ ′ )| and estimate the result using (3.60) and (3.62). The inequality (3.58) is a simple consequence. Lemma 3.3.1 implies < < = m1 s1 ω(p) m1 s1 ω(p) m1 s1 |oT (p)| = ≤ ≤ . sh γT (p) γT (p) m2 m

Since m1 m2−1 converges, we see that (3.58) and Lemma 3.3.1 imply that there exist constants C and m such that −,α,β

|XT

′

(p, p ′ )| ≤ Ce−mαω(p)−mβω(p )


121

for all T ∈ [T0 , Tc ]. Together with (3.57) this establishes dominated convergence for (3.56). Let XT+ = D −1 denote the operator on WT found in (3.26). Define XT ,+ = oT XT+ oT∗ , −1

α,β

XT ,+ = e−αm2

γT

−1

XT ,+ e−βm2

γT

.

Lemma 3.3.3 Let s − lim denote strong operator convergence on the Hilbert space W . Then for α, β > 0 we have α,β

α,β

s − lim XT ,+ = X+ , T ↑Tc

α,β

where X+ is the principal value integral operator ) 1 dp ′ α,β α,β X+ f (p) = X+ (p, p ′ )f (p ′ ) , 2πi R ω(p ′ ) with α,β

X+ (p, p ′ ) = e−αω(p)

ω(p) + ω(p ′ ) −βω(p′ ) e . p − p′

Proof. On L2 (S 1 , dθ) the operator norm of s and hence also of sˆ is 1.#To take ad$ dp . vantage of this it is helpful to work on L2 (R, dp) rather than W = L2 R, 2ω(p) The map ( L2 (R, dp) ∋ f (p) → 2ω(p)f (p) ∈ W α,β

is a unitary equivalence between the two spaces. To show that XT ,+ converges strongly on W it is enough to show that 1 α,β √ √ XT ,+ ω ω

converges strongly on L2 (R, dp). The action of this operator on L2 (R, dp) is given by (see (3.23)) 1 α,β √ √ XT ,+ ωf = a1,T sˆT a2,T + a3,T sˆT a4,T , ω where aj,T for j = 1, 2, 3, 4 are multiplication operators given by & 2m2−1 sh 21 γT (p) a1,T (p) −αm2−1 γT (p) oT (p) =e √ a3,T (p) ω(p) ch 12 γT (p) and

−1 √ & 2m2−1 sh 12 γT (p) a2,T (p) −βm2−1 γT (p) ω(p) =e −1 1 a4,T (p) oT (p) ch γT (p) 2

(3.63)

122

3. Scaling Limits

restricted to the domain |m1 p| ≤ π . The extra factors of 2m2−1 that appear in a1,T (p) and a2,T (p) cancel out but are convenient for the scaling calculations. The operator sˆT is the principal value integral operator ) 1 m1 χT (p)χT (p ′ ) sˆT f (p) = f (p′ ) dp ′ , (3.64) 2πi R sin 21 (m1 p − m1 p ′ ) where

% 1 for |m1 p| ≤ π, χT (p) = 0 for |m1 p| > π.

The operator sˆT has operator norm 1 since it is the image of sˆ under the isometric embedding √ L2 ([−π, π], dθ) ∋ f (θ) → m1 χT (p)f (m1 p) ∈ L2 (R, dp). The pointwise limits of the multiplication operators are simple to compute based on Lemma 3.3.1, 3.46, and 3.47. We obtain lim aj,T (p) = aj (p),

T ↑Tc

where √ √ −1 a1 (p) = e−αω(p) ω(p), a2 (p) = e−βω(p) ω(p) , √ √ −1 a3 (p) = e−αω(p) ω(p) , a4 (p) = e−βω(p) ω(p).

(3.65)

Let aj denote multiplication by aj (p) on L2 (R, dp) and choose T0 such that 0 < T0 < Tc . The estimates in Lemma 3.3.1 imply that for all T ∈ [T0 , Tc ] there exists a constant C such that ( ±1 |aj,T (p)| ≤ C ω(p) e−mrω(p) ,

for the appropriate choice of ± and r = min{α, β}. Dominated convergence now implies that aj,T converges strongly to aj as T ↑ Tc . To finish the proof we show that sˆT converges strongly, as T ↑ Tc , to the principal value integral operator h on L2 (R, dp) given by ) 1 f (p ′ ) hf (p) = dp ′ , (3.66) 2πi R p − p ′

the well-known Hilbert transform. To estimate the difference of the kernels in (3.64) and (3.66) observe that since sin θ ≥ θ − π −1 θ 2 for 0 < θ < π, 0≤ and so

1 1 1 − ≤ sin θ θ π −θ

' ' ' 1 1 '' 1 ' ' sin θ − θ ' ≤ π − |θ |

for 0 < θ < π,

for 0 < |θ| < π.


123

Thus m1 2

' ' ' ' 1 m1 1 ' ' − '≤ ' 1 1 ′ ′ ' sin 2 (m1 p − m1 p ) (m1 p − m1 p ) ' 2π − m1 |p − p ′ | 2

(3.67)

for 0 < m1 |p − p ′ | < 2π . Let χT denote the characteristic function of the interval [−π m1−1 , π m1−1 ]. We use the estimate (3.67) to show that sˆT − χT h tends strongly to 0 as T ↑ Tc . Since (1 − χT ) tends strongly to 0, so does (1 − χT )h. Thus if sˆT − χT h tends strongly to zero as T ↑ Tc , then so does sˆT − h, which is what we want to prove. The operators sˆT − χT h are uniformly bounded, and so to prove strong convergence to 0 it suffices to prove such convergence on a dense subset of L2 (R, dp). Suppose then that f ∈ L2 (R, dp) is in the dense set of compactly supported functions. Then there exists an L > 0 such that the support of f is contained in the bounded interval [−πL, πL], and if T is chosen close enough to Tc that m1 (T )L < 1, then for m1 p ∈ [−π, π] and p′ ∈ [−π L, π L] we see that m1 |p ′ | < π , so that m1 |p − p ′ | < 2π . Thus we can use (3.67) to estimate m1 |(ˆsT − χT h)f (p)| ≤ 4π

)

πL

−π L

|f (p ′ )| dp ′ . 2π − m1 |p − p ′ |

The Schwarz inequality then implies that m12 ˆsT f − χT hf ≤ 16π 2 2

)

π/m1

dp −π/m1

)

πL

−π L

dp ′ (2π − m1 |p − p ′ |)−2 f 2 ,

where · is the L2 (R, dp) norm. The change of variables θ = m1 p, θ ′ = m1 p ′ in the double integral produces the alternative representation for the multiplier, 1 16π 2

)

π

dθ −π

)

πm1 L

−πm1 L

dθ ′ (2π − |θ − θ ′ |)−2 ,

which clearly tends to 0 as T ↑ Tc since m1 → 0. This finishes the proof that sˆT converges to h. Together with the results for the strong convergence of aj,T we see that a1,T sˆT a2,T + a3,T sˆT a4,T → a1 ha2 + a3 ha4

√ in the strong sense on L2 (R, dp) as T ↑ Tc . Conjugating this by ω to translate this into a result for strong convergence on W rather than L2 (R, dp) finishes the proof of the lemma. Remark 3.3.1. With only a little more effort one can show that the convergence in Lemma 3.3.3 takes place in the operator norm topology. We leave this to the reader (but see [117]).

124

3. Scaling Limits

3.4 Pfaffian Formulas for the Scaling Functions We are now ready to use Theorems A.7.4 and A.7.5 to control the scaling limit of the Ising correlations from below Tc . Theorem 3.4.1 Let a1 , a2 , . . . , an denote n distinct points aj = (xj , yj ) in R 2 ordered so that y1 < y2 < · · · < yn , y −y

¯ j = y¯j − yj −1 = j j −1 > 0. Define m−1 aj = (m1−1 xj , m2−1 yj ), and write 2 where m1 = 2K1 − 2K2∗ is the horizontal and m2 = 2K2 − 2K1∗ the vertical inverse correlation length at temperature T :

τ

[−]

(a) := lim

T ↑Tc

, −1 σ m an · · · σ m−1 a1 T

−UB , 1

1 = Pf LC

σ nT

(3.68)

where B and C are the diagonal matrices  Bn Dn−1  . B =  .. 0

··· .. . ···

0 .. . B1 D1−1



 ,

 −1 Dn Cn  .. C= . 0

0 .. .

··· .. . ···

D1−1 C1



 ,

(3.69)

and L and U are the strictly lower triangular and strictly upper triangular matrices   0 0 ··· ··· 0   1 0   −1  Dn−1  1    . .. L =  −1 (3.70) −1 ..   . D D n−2  n−1,n−2    .. ..  . 1 0 0  . −1 · · · D2−1 1 0 Dn−1,2 and 

−τ 0 1 Dn−1  0 0 1    U =  .  ..   0 ···

−τ Dn−1,n−2 −τ Dn−2 .. .

···

···

−τ Dn−1,2

..

.. .

.

1 0 0

D2−τ 1 0

         

(3.71)

3.4 Pfaffian Formulas for the Scaling Functions

125

with Di,j := Di Di−1 · · · Dj for i > j . The operators Bj Dj−1 , Dj−1 Cj , Dj−1 , and # $ dp Dj−τ are integral operators on L2 R, ω(p) given by ) ¯ ¯ 1 dp ′ , Xj −j +1 j +1 (p, p ′ )f (p ′ ) , 2πi R ω(p ′ ) ) ¯ , ¯ dp ′ 1 Xj +j j +1 (p, p ′ )f (p ′ ) , Dj−1 f (p) = 2πi R ω(p ′ ) ) ¯ , ¯ 1 dp ′ Dj−1 Cj f (p) = Xj −j j (p, p′ )f (p ′ ) , 2πi R ω(p ′ ) ) ¯ ¯ dp ′ 1 , Xj +j +1 j (p, p ′ )f (p ′ ) , Dj−τ f (p) = − 2πi R ω(p ′ ) Bj Dj−1 f (p) = −

where α,β

Xj ± (p, p ′ ) = e−αω(p)

(3.72) (3.73) (3.74) (3.75)

ω(p) ± ω(p ′ ) ixj (p−p′ ) −βω(p′ ) e e . p − p′ α,β

The integral operators associated with Xj + are understood in the principal value sense. Proof. We rewrite σ (an ) · · · σ (a1 )T = σn · · · σ1 T , where

¯

¯

σj := V j +1 σ (xj )V j .

(3.76)

As mentioned above in (3.10)–(3.15), we have ¯

¯

D(σj )−1 = e−j γ D(xj )−1 e−j +1 γ , ¯ ¯ D(σj )−τ = e−j +1 γ D(xj )−τ e−j γ , ¯ j +1 γ ¯ − −1 B(σj )D(σj ) = e B(xj )D(xj )−1 e−j +1 γ , ¯ ¯ D(σj )−1 C(σj ) = e−j γ D(xj )−1 C(xj )e−j γ . Modifying these by the substitution xj ←⊣ m1−1 xj and yj ←⊣ m2−1 yj and working in the isometric embedding oT of (3.45), Theorem A.23 gives a Pfaffian formula for the ratio , σ (m−1 an ) · · · σ (m−1 a1 ) T 1 −UT BT . = Pf LT CT 1 σ nT

Lemma 3.3.2 shows that BT converges to B in Schmidt norm as T ↑ Tc , and CT converges to C in Schmidt norm as T ↑ Tc . Lemma 3.3.3 implies that UT converges strongly to U, and as indicated above in (3.29), it also suffices to show that UT converges strongly to U. Since these results together imply that UT BT → UB and UT CT → UC in Schmidt norm as T ↑ Tc , and as is noted in the appendix, the Pfaffian is continuous in the Schmidt norm for the off-diagonal elements, the theorem follows.

126

3. Scaling Limits

One feature of this result that is worth noting right away is that by choosing the horizontal and vertical length scales correctly, no vestige of the relative size of the interaction strengths K1 and K2 survives in the scaling limit. In fact, the final result is rotationally invariant in an appropriate sense. We take this matter up later, in Chapter 6. Before we prove an analogous result for the scaling limit from above Tc we first establish that τ [−] (a) is determined by the correlations of the Ising model on the integer lattice. Griffith’s inequality then implies τ [−] (a) ≥ 1 (see [74]), which in turn implies that 1 −UB LC 1 is invertible, since it has a nonvanishing determinant. This is the only additional technical element needed to deal with the Pfaffian formula above Tc . Suppose that a = (x, y) is a point in R 2 . Let (mT , nT ) ∈ Z2 be a point on the integer lattice such that |m1 (T )−1 x − mT | < 1 and

|m2 (T )−1 y − nT | < 1,

and just to make the point (mT , nT ) unique, we also require that m1 (T )mT ≤ x, m2 (T )nT ≤ y. Define xT = m1 (T )mT , and yT = m2 (T )nT . Then |x − xT | < m1 (T )

and

|y − yT | < m2 (T ).

Thus xT → x and yT → y as T ↑ Tc . Write an,T = (xn,T , yn,T ) and define , −1 σ m an,T · · · σ m−1 a1,T τ (aT ; T ) = . σ n Then we claim that τ [−] (a) = lim τ (aT ; T ). T ↑Tc

(3.77)

To prove this, first observe that the convergence in Lemma 3.3.2 is locally uniform for α, β in compact subsets of R + (the positive reals). The x variable does not appear in Lemma 3.3.2, but since it alters the kernels involved only by the ′ T -independent factor eix(p−p ) , this locally uniform convergence in α, β is globally uniform in x. One can also check that the strongly convergent operators in Lemma 3.3.3 are locally uniformly bounded for α, β in compact subsets of R + and globally uniformly bounded in x. This is enough to show that UT BT and LT CT converge locally uniformly in a to UB and LC in Schmidt norm. This in turn implies (3.77).


127

The result shows that the values of τ [−] (a) are limits of ratios of Ising correlations at integer sites. Griffith’s inequality applies to the limit to show that 1 −UB [−] τ (a) = Pf ≥ 1. LC 1 1 −U B Thus LC is invertible in the scaling limit. 1 As we did for the correlations below Tc we extend the arguments for the correlations above Tc from Z2 to R 2 . For µ = µ− 1 and a = (x, y) ∈ R 2 define 2

µ(x) = Ŵ(e

ixθ

)µŴ(e−ixθ )

and µ(a) = V −y µ(x)V y ,

where µ(a) is understood as a densely defined operator. Suppose that aj = (xj , yj ) ∈ R 2 and y1 < y2 < · · · < yn . Suppose T < Tc and define τ (a; T ) =

, −1 µ m an · · · µ m−1 a1 T σ nT

.

(3.78)

Then the limit of τ (a; T ) as T ↑ Tc from below is, by duality, a scaling function for the spin correlations at T ∗ > Tc evaluated in the limit T ∗ ↓ Tc . Of course, the interaction constants are flipped to (K2∗ , K1∗ ) in the spin correlation version of (3.78). But the reader can see from the final result that as was true for the scaling limit below Tc , the scaling function does not depend on the relative size of K2∗ and K1∗ . As above, it is convenient for us to refactorize the representation (3.69) by introducing ¯ ¯ µj := V j +1 µ(xj )V j , where

¯ j := y¯j − yj −1 =

yj − yj −1 . 2

Then Recall that

µ(an ) · · · µ(a1 )T = µn · · · µ1 T .

(3.79)

µ = op(δσ ).

A little calculation then shows that

µj = op(δj σj ),

(3.80)

where σj is given above in (3.76) and ¯

¯

δj = e−j +1 γ eixj θ δ+ ⊕ e−j γ eixj θ δ− .

(3.81)

128

3. Scaling Limits

The direct sum decomposition is W+ ⊕ W− , and δ± are the components of δ in W± . We are now prepared to state a scaling limit result for the limit from above Tc . Theorem 3.4.2 Suppose that T ∗ > Tc and T is the dual temperature T < Tc . Let aj = (xj , yj ) ∈ R 2 with y1 < y2 < · · · < yn . Define τ (a; T ) =

, µ(m−1 an ) · · · µ(m−1 a1 ) T σ nT

Then

τ [+] (a) := lim τ (a; T ) = Pf T ↑Tc

1 LC

.

−UB Pf(−T ), 1

where T is the n × n matrix with i, j entry above the diagonal given by 7 −1 + 8 δ 1 −UB Uδi+ −Ti,j = , j− , LC 1 δj −Lδi− where

and

(3.82)

i ¯ δj+ (p) = − √ e−j +1 ω(p) eixj p π

(3.83)

i ¯ δj− (p) = √ e−j ω(p) eixj p , π

(3.84)

± and it is understood that δj± is injected into the j th slot in Wn± ⊕ Wn−1 ⊕ · · · ⊕ W1± with 0’s in all the other slots. To be compatible with this reverse labeling the rows and columns of T are indexed starting at the bottom right. Thus for n = 4 we write   0 T4,3 T4,2 T4,1 T3,4 0 T3,2 T3,1  . T = T2,4 T2,3 0 T2,1  T1,4 T1,3 T1,2 0

Proof. We employ the same strategy that we used in proving Theorem 3.4.1. ¯ j in (3.79)–(3.81) and use ¯ j ←⊣ m2−1 Make the substitutions xj ←⊣ m1−1 xj and Theorem A.23 on the result in the image of the isometric embedding oT . One obtains 1 −UT BT τ (a; T ) = Pf Pf (−TT ) , LT CT 1 i,j

where the i, j matrix element −TT of the n × n matrix −TT is given by 7 + 8 −1 δ UT δT+,i 1 −UT BT i,j −TT = , T−,j , (3.85) LT CT 1 δT ,j −LT δT−,i


129

# $ dp and δT±,j is the element of L2 R, ω(p) given by (see (2.82) and recall that √ iθ/2 z = −e for |θ | < π in that formula) = −1 k ¯ + δT ,j (p) := −i oT (p)e−j +1 m2 γT (p) eixj p exp 21 (γT (p) − im1 p) π and δT−,j (p) := i

=

−1 k ¯ oT (p)e−j m2 γT (p) eixj p exp 12 (−γT (p) − im1 p) . π

Of course in (3.85) each vector δT±,j is understood to appear in the j th slot of ⊕ · · · ⊕ W1± . > ?

Wn±

This representation depends on LT1CT −UT1 BT being invertible. As mentioned before, this invertibility follows from Griffith’s inequality for integer sites, but we don’t know in general that the operator is invertible. However, we did show that the limit T ↑ Tc is an invertible operator.> Since the convergence of operators takes ? place in the uniform norm, the operator LT1CT −UT1 BT will be invertible for all T sufficiently close to Tc . Clearly, this suffices for the discussion of the T ↑ Tc limit of τ (a; T ). The fractional power exp −i m21 p is (by design) not branched at p = 0, and dominated convergence ¯

−1

e−m2

γT (p) T ↑Tc

¯

−−→ e−ω(p)

¯ >0 for

is seen to imply that lim δT±,j = δj±

T ↑Tc

$ # dp . Passing to the limit T ↑ Tc in (3.85), one obtains (3.82). in L2 R, 2ω(p)

There is a change of variables for these results that will facilitate a comparison with results in Chapter 4. Let r = ω(p) − p, Thus

so that

r −1 = ω(p) + p.

p(r) = 21 (r −1 − r), ω(r) := ω(p(r)) = 12 (r + r −1 ).

This is a rational parametrization of the “spectral curve” (p, ω(p)). The reader should note that this parametrization differs from (4.70) in Chapter 3 by the inversion r → r −1 . This is deliberate, and as will be explained below, it serves to realign a mismatch of conventions arising from the fact that the “natural” Fourier series conventions on the lattice scale to a transform that differs from the standard Fourier transform on the line by the sign flip p → −p. The last “definition” of ω(r) involves an obvious abuse of notation, but its use should be clear in context.

130

3. Scaling Limits

Since dp dr = , 2ω(p) 2r the map f (p) → f (p(r)) is a unitary map, L R, 2

dp 2ω(p)

2

→ L

dr R , 2r +

.

Writing p = 21 (r − r −1 ), and p ′ = 21 (s − s −1 ), a simple calculation shows that ω(p) − ω(p ′ ) 1 − rs = , p − p′ rs + 1 r +s ω(p) + ω(p ′ ) =− . ′ p−p r −s

(3.86)

α,β

Remark 3.4.1. Theorems 3.4.1 and 3.4.2 remain true if the integral operator Xj + is replaced by the principal value integral operator ) ∞ dr r + s α,β ds L2 R + , ej (r, s)f (s) , ∋ f → − 2r r −s 2π is 0 α,β

the integral operator Xj − is replaced by the integral operator ) ∞ 1 − rs α,β ds + dr ej (r, s)f (s) , ∋ f → L R , 2r rs + 1 2π is 0 2

where α,β

ej (r, s) = e−αω(r) eixj (p(r)−p(s)) e−βω(s) , j

and the vectors δ± (p) are replaced by i dr ¯ ∋ δj+ (r) = − √ e−j +1 ω(r) eixj p(r) L2 R + , 2r π and

i dr ¯ ∋ δj− (r) = √ e−j ω(r) eixj p(r) . L2 R + , 2r π

To see that this is so, it helps to know that the principal value prescriptions for the integral operators are equivalent to taking the limit as ǫ ↓ 0 on the domains |p − p′ | > ǫ and |r − s| > ǫ.

3.5 A Probabilistic Interpretation for the Scaling Limits

131

3.5 A Probabilistic Interpretation for the Scaling Limits Since the work of Dobrushin [48] and Lanford and Ruelle it has been customary to think of the infinite-volume Ising model as a particular case of a Gibbs random field. We briefly review this framework to set the stage for a probabilistic interpretation 2 of the scaling limits of the Ising model. Let = {−1, +1}Z (the space of maps from Z2 into {−1, +1}) denote the configuration space (or sample space) for the Ising model. Let denote the σ -algebra of subsets of generated by the cylinder sets of the form {σ |σ (ai ) = σi for i = 1, . . . , n}, where ai ∈ Z2 are distinct points and σi ∈ {−1, +1}.AGibbs state at temperature T for the two-dimensional Ising model is a probability measure νT whose conditional expectations satisfy the appropriate Dobrushin, Lanford, Ruelle (DLR) equations. The DLR equations are automatically satisfied by measures that arise from limits of Gibbs measures on finite lattices, as is the case for the infinite-volume limits we consider. Because the spin variable σ takes on just the values ±1, it is clear that the distribution functions for σ can be calculated directly in terms of the moments (or correlations) that we have focused on until now. For example, Pr{σ (a1 ) = 1, σ (a2 ) = −1} = 2−2 (σ (a1 ) + 1)(σ (a2 ) − 1)BT , where the boundary conditions B might be ± or “open” in the cases we have explicitly considered above. This allows one to construct measures νTB associated with these different boundary conditions, and of course, the measure νTB is defined so that ) σ (a1 ) · · · σ (an )BT = σ (a1 ) · · · σ (an ) dνTB .

The existence of more than one solution to the DLR equations at a given temperature is expected to capture the possibility of different phases arising in the infinite-volume limit. The set of Gibbs states at temperature T is a Choquet simplex. For the Ising model and T > Tc this set is known to consist of a single point and, for T < Tc there are precisely two extremal Gibbs states νT+ and νT− obtained by taking the thermodynamic limit with + and − boundary conditions [6]. One of the consequences of the infinite wave function renormalization involved in defining the scaling limits is that the scaling functions τ [±] (a1 , . . . , an ) are singular at coincidence ai = aj . In particular, these scaling functions can no longer be the moments of a random variable. What is true, however, is that averaging these scaling functions with suitable smooth functions does give the moments of a probability measure. Suppose fi ∈ S(R 2 ) for i = 1, . . . , n, where S(R 2 ) is the Schwartz space of smooth functions on R 2 that together with all their derivatives

132

3. Scaling Limits

decay faster than any power of the distance from the origin. We will show that there are probability measures ν + and ν on the dual S ′ (R 2 ) such that ) ) f1 f2 · · · fn dν + τ [−] (a) fj (aj )daj = R 2n

S ′ (R 2 )

j

and )

R 2n

τ

[+]

(a)

j

fj (aj )daj =

)

S ′ (R 2 )

f1 f2 · · · fn dν.

On the right-hand side the functions fj ∈ S(R 2 ) are thought of as functions on S ′ (R 2 ) (i.e., random variables) in the usual way, fj (ϕ) = ϕ(fj ) for ϕ ∈ S ′ (R 2 ). The measures we are interested in on S ′ (R 2 ) are defined on the σ -algebra generated by the cylinder sets. A cylinder set S in S ′ (R 2 ) is determined by a finite collection fj ∈ S(R 2 ) for j = 1, . . . , n and a Borel set B ⊂ R n with S = {ϕ|(ϕ(f1 ), ϕ(f2 ), . . . , ϕ(fn )) ∈ B}. Recall that τ [−] is the scaling function obtained by scaling the plus boundary condition correlations from below Tc , hence the association with ν + . The moments of ν − differ from those of ν + by the factor (−1)n . The measure ν is obtained by scaling the correlations for open boundary conditions from above Tc . We will prove this result by verifying the hypothesis of the Minlos theorem [146]. This result is interesting in its own right, but the proof will also afford us the opportunity to meditate on the connection between scaling limits and the central limit theorem in probability, a connection that has inspired hope of detecting a mathematical mechanism for the appearance of “universality” in the study of critical phenomena. To prove this we introduce lattice analogues for the random variables σ (f ) and prove convergence in a strong enough sense to verify the hypothesis of the Minlos theorem. Suppose then that f ∈ S(R 2 ), and for T < Tc define σT (f ) :=

1 σ (n)f (mn)m1 m2 , σ T 2 n∈Z

where σ is the spin random variable in the plus Gibbs state at temperature T , m1 = m1 (T ) is the horizontal mass at temperature T , m2 = m2 (T ) is the vertical mass at temperature T , and mn = (m1 n1 , m2 n2 ). For T > Tc we define σT (f ) :=

1 σ (n)f (mn)m1 m2 , σ T ∗ 2 n∈Z

where σ is the spin random variable for the “open” Gibbs state at temperature T , T ∗ is the Kramers–Wannier dual temperature, and mj = mj (T ) are as above (note


133

that m1 (T ) = m2 (T ∗ ) and m2 (T ) = m1 (T ∗ )). Define the characteristic function ) eiσT (f ) dνT , χT (f ) =

where dνT =

 dνT+

for T < Tc ,

dν open T

for T > Tc .

We will prove convergence of the limits

lim χT (f ) = χ [±] (f ).

T →Tc ±

The conditions that χ [±] (f ) must satisfy in order to be the characteristic function of a random field on S(R 2 ) are a normalization condition χ [±] (0) = 1, a positive definiteness condition α¯ i αj χ [±] (fi − fj ) ≥ 0 i,j

for any finite collection αi ∈ C and fi ∈ S(R 2 ), and finally, strong continuity of the map S(R 2 ) ∋ f → χ [±] (f ).

The normalization and positive definiteness conditions follow directly from the corresponding properties for the lattice fields coupled with the convergence of the limits. Continuity is a direct consequence of the estimates we use to prove convergence and the representation for the limit in terms of the scaling functions. Since σT (f ) is a bounded random variable, the power series expansion of the exponential, eiσT (f ) converges uniformly, and it is permissible to integrate this series term by term to obtain (see (3.8)) χT (f ) =

∞ k i k=1

χk (f, T ),

k!

where χk (f, T ) =

m1k m2k τ (mn1 , . . . , mnk , T )

n1 ,n2 ,...,nk ∈Z2

n

fj (mnj ).

j =1

Given the pointwise convergence theorems we have proved and the Riemann sum structure of this last result, it is natural to conjecture that lim χk (f, T ) = χk[±] (f ) :=

T →Tc ±

)

R 2k

τ [±] (a)

k

j =1

fj (aj )daj .

134

3. Scaling Limits

The following lemma provides an estimate on the two-point function that we use to prove this result. Lemma 3.5.1 There is a positive temperature T0 < Tc such that for all T ∈ [T0 , T0∗ ] there exist constants ǫ > 0 and β > 0 such that τ (a1 , a2 ; T ) ≤ ch

1 + ǫ K0 (βr) , 2

where r = |a1 − a2 | and K0 is the modified Bessel function. The constant ǫ > 0 can be made arbitrarily close to 0 by choosing T0 sufficiently close to Tc . Proof. Let aj = (xj , yj ) and 1 = x2 − x1 and 2 = y2 − y1 . By relabeling a1 and a2 if necessary, we can suppose that 2 ≥ 0. We also suppose to begin with that 2 ≥ |1 |. Now recall Theorem 2.5.1 and make the observations that for T < Tc we have k < 1, for real values of ui − uj we have sn2 (ui − uj ) ≤ 1, and |z(u)| ≤ 1. We see that |τ (a1 , a2 ; T )| ≤

∞ n=0

1 (2n)!

)

1 π

2K

−1

λ(u)m2

0

2

du

2n

.

It is convenient to transform this integral into the “hyperbolic representation” whereby λ(u) becomes e−γ (θ ) and du becomes s1 dθ . 2 sh γ (θ) The integral in this last estimate is then 1 π

)

2K

λ(u)

m2−1 2

0

1 du = π

)

π

−1

e−2 m2

γ (θ )

−π

s1 dθ . 2 sh γ (θ)

Make the change of variables θ = m1 p in this last integral to obtain (recall (3.30)) )

π/m1

−π/m1

−1

e−2 m2

γT (p)

s1 m1 dp . 2π sh γT (p)

Now recall Lemma 3.3.1. There exists a positive temperature T0 < Tc and a constant c > 0 such that for all T ∈ [T0 , Tc ] we have m2−1 γT (p) ≥ cω(p), 2 − ǫ ω(p). m2−1 sh γT (p) ≥ π


135

Here we use ǫ > 0 for a generic constant that can be made arbitrarily small by choosing T0 sufficiently close to Tc (i.e., the actual value of ǫ changes from equation to equation but this property does not). Thus we have the estimate ) ∞ ) π/m1 dp s 1 m1 1 −2 m2−1 γT (p) s1 m1 dp e e−c2 ω ≤ +ǫ . 2π sh γ (p) m 4 ω(p) T 2 −π/m1 −∞ Since

s1 m 1 m2

tends to 1 as T → Tc and 2 ≥ |1 | implies that r 2 ≥ √ , 2

we see that there exist constants ǫ > 0 and β > 0 such that ) ∞ ) ∞ dp dp 1 s1 m1 e−βrω(p) e−c2 ω ≤ +ǫ , 2π m2 c −∞ ω(p) 4 ω(p) −∞ for T ∈ [T0 , Tc ]. But

1 K0 (r) = 2

)

∞

e−rω(p)

−∞

dp , ω(p)

so we have finished the proof in the case 2 ≥ |1 |. To obtain the estimate when 1 ≥ |2 | just observe that we can use the transfer-matrix formalism in the horizontal rather than the vertical direction to obtain a different but completely analogous formula for the two-point function. An estimate of the type we are looking for follows immediately from this alternative representation. This finishes the proof for τ (a; T ) with T < Tc . Now suppose that T < Tc . On a finite lattice for plus boundary conditions, Griffith’s inequalities [74] imply that the spin correlations are decreasing functions of the temperature, so + σ (a1 )σ (a2 )+ T ∗ ≤ σ (a1 )σ (a2 )T .

It is known that the boundary conditions don’t influence the infinite-volume correlations above Tc , so the estimate we proved for τ (a1 , a2 ; T ) with T < Tc works for τ (a1 , a2 ; T ) with T > Tc as well. We can now prove the main theorem of this section. Theorem 3.5.1 Suppose that fj ∈ S(R 2 ). Then the scaling functions τ [±] (a) are locally integrable functions on R 2n , and the correlations ) n τ [±] (a) fj (aj ) daj R 2n

j =1

are the moments of cylinder measures on S ′ (R 2 ). That is, there are cylinder measures ν + and ν such that ) ) n f1 · · · fn dν + τ [−] (a) fj (aj ) daj = R 2n

j =1

S ′ (R 2 )

136

3. Scaling Limits

and

)

R 2n

τ

[+]

(a)

n

j =1

fj (aj ) daj =

)

S ′ (R 2 )

f1 · · · fn dν.

The + in dν + is intended to remind the reader that plus boundary conditions are involved. For T > Tc the boundary conditions don’t influence the correlations, so dν is without modifiers. Proof. We will verify the hypothesis of Minlos’s theorem using Gaussian domination to reduce all the needed estimates to Lemma 3.5.1. A partition {1 , . . . , m } of a set {a1 , a2 , . . . , an } is said to be a pair partition if every set j has two elements when n is even, and when n is odd exactly one set j has one element and all the rest have exactly two elements. If = {a1 , a2 } we write τ (; T ) := τ (a1 , a2 ; T ). For = {a} we write

τ (; T ) = 1 for T < Tc and τ (; T ) = 0 for T > Tc . Theorem 3 in [101] implies the following correlation inequality: τ (a1 , . . . , an ; T ) ≤ τ (1 ; T ) · · · τ (m ; T ),

(3.87)

where the sum is over all pair partitions of {a1 , . . . , an }. This is referred to as Gaussian domination. As above, write m(n1 , n2 ) = (m1 (T )n1 , m2 (T )n2 ) and mZ2 for the scaled lattice consisting of ordered pairs mn with n ∈ Z2 . Suppose that f1 , f2 ∈ S(R 2 ). We first prove that ) 2 2 lim f1 (a1 )f2 (a2 )τ [−] (a)da, (3.88) f1 (a1 )f2 (a2 )τ (a; T )m1 m2 = T →Tc −

R4

a∈mZ4

where a ∈ mZ2 is shorthand for (a1 , a2 ) ∈ mZ2 × mZ2 and da =

2 d a¯ j daj

j =1

2i

is Lebesgue measure. For definiteness we will first look exclusively at τ (a; T ) for T < Tc with plus boundary conditions always understood. First observe that Theorem 2.4.2 implies 1 τ (a, a; T ) = = σ 2T

s1 s2 1 − s1 s2

1

4

# $ 1 = O |T − Tc |− 4 .

Since mj (T ) is O (|T − Tc |) for j = 1, 2, elementary estimates show that the diagonal terms with a1 = a2 in the sum on the left-hand side of (3.88) do not make


137

a contribution in the limit T → Tc . It is not essential to get rid of these terms in the following developments, but it is instructive to see directly that no “delta functions” arise from the diagonal terms. Next define a (T -dependent) function F on R 4 in the following manner. If xj ∈ R 2 (j = 1, 2) is an element of the open rectangle with horizontal side m1 (T ) and vertical side m2 (T ) centered at aj ∈ mZ2 , then we define F (x1 , x2 ; T ) = f1 (a1 )f2 (a2 )τ (a1 , a2 ; T )

for a1 = a2 .

For all other pairs (x1 , x2 ) ∈ R 4 we define F (x1 , x2 ; T ) = 0. We have defined F such that ) F (x1 , x2 ; T ) dx, f1 (a1 )f2 (a2 )τ (a; T )m12 m22 = R4

a1 =a2

where the sum on the left is over noncoincident points in mZ2 × mZ2 and dx is Lebesgue measure on R 4 . Theorem 3.4.1 implies that F (x1 , x2 ; T ) converges almost everywhere to f1 (x1 )f2 (x2 )τ [−] (x1 , x2 ). To finish the proof of (3.88) we need only show that F (x1 , x2 ; T ) is dominated by a family of L1 (dx) functions G(x1 , x2 ; T ) that converges in L1 (dx) as T → Tc . Since fj ∈ S(R 2 ), there exists a constant C > 0 such that |fj (x)| ≤ C(1 + |x|2 )−2 . Define G(x1 , x2 ) = C 2 (1 + |x1 |2 )−2 (1 + |x2 |2 )−2 ch

1 + ǫ K0 (β|x1 − x2 | . 2

The notation is as in Lemma 3.5.1 with ǫ > 0 a constant that can be made arbitrarily close to 0 by choosing T0 sufficiently close to Tc . Since ch(αK0 (βr)) is O(r −α ) as r → 0, it follows that G is locally integrable, provided ǫ < 32 . Henceforth, we assume that T is in a sufficiently small interval [T0 , Tc ] to guarantee that ǫ < 21 ; this makes a later convergence estimate trivial. Now define G(x1 , x2 ; T ) by setting it equal to G(a1 , a2 ) whenever xj is in the rectangle R(aj , m) centered at aj ∈ mZ2 with horizontal side m1 (T ) and vertical side m2 (T ) and a1 = a2 . At all other values of (x1 , x2 ) set G(x1 , x2 ; T ) = 0. The function G was defined so that in conjunction with Lemma 3.5.1 it is clear that F (x1 , x2 ; T ) ≤ G(x1 , x2 ; T ). To finish the proof we need only show that ) lim |G(x1 , x2 ) − G(x1 , x2 ; T )|dx = 0. T →Tc

R4

The integral we want to estimate is a sum of integrals over rectangles, ) |G(x) − G(a)|dx, a1 ∈mZ2 a2 ∈mZ2

R(a1 ,a2 )

(3.89)

138

3. Scaling Limits

where R(a1 , a2 ) := R(a1 , m) × R(a2 , m). But ) ) |m| |G(x) − G(a)| dx ≤ |G′ (x)| dx. 2 R(a1 ,a2 ) R(a1 ,a2 ) However, since K0′ (r) = O(r −1 ) as r → 0, it follows that for ǫ < 12 the norm of the derivative, |G′ (x)|, is still locally integrable in R 4 . It is not hard to see that |G′ (x)| is integrable on all of R 4 , and since ) ) |m| |G(x) − G(x; T )| dx ≤ |G′ (x)| dx, 2 R4 R4 we see that (3.89) is true and hence also (3.88). Now suppose that fj ∈ S(R 2 ) for j = 1, . . . , n. We want to show that f1 (a1 ) · · · fn (an )τ (a, T )m1n m2n lim T →Tc

=

a∈mZ2n

)

R 2n

f1 (x1 ) · · · fn (xn )τ [−] (x) dx.

(3.90)

Using Gaussian domination (3.87) and Lemma 3.5.1, we see that f1 (x1 ) · · · fn (xn )τ [−] (x) is dominated by an integrable function with an integrable derivative. As above, this is all that is needed to prove (3.90). Now we turn to the proof of convergence for the characteristic function χT (f ) = where χj (f, T ) = We have seen that

χj (f, T ) −−→

)

j =0

j!

(3.91)

,

j

a∈mZ2j

T ↑Tc

∞ χj (f, T )

j

f (a1 ) · · · f (aj )τ (a; T )m1 m2 .

R 2j

f (x1 ) · · · f (xj )τ [−] (x) dx,

so there is term-by-term convergence in (3.91). On the other hand, Gaussian domination and Lemma 3.5.1 are enough to show that for j = 2k we have |χj (f, T )| ≤ where M=

)

R4

f (x1 )f (x2 ) ch

j! Mk, 2k k!

1 + ǫ K0 (β|x1 − x2 |) dx. 2


139

An analogous estimate for j odd shows that the series in (3.91) are dominated by a fixed summable series. This implies that χT (f ) converges to χ [−] (f ). Continuity in S(R 2 ) is a simple consequence of the estimates we have already given, and this finishes the proof for subcritical scaling. The supercritical case is completely analogous given the estimate in Lemma 3.5.1 for T > Tc . This probabilistic interpretation of the scaling limits is interesting because the relevant random variables are limits of sums of spin variables, a circumstance reminiscent of the setting for the central limit theorem. The central limit theorem asserts that sums of (suitably scaled) independent identically distributed random variables of finite mean and variance have Gaussian fluctuations: the distributions of the individual random variables “wash out” in a manifestion of probabilistic universality. All that survives are the mean and variance parameters of the Gaussian distribution. An important difference in our circumstance is that the spin variables at different sites are not independent. However, away from the critical temperature the spin correlations decay exponentially fast with the separation of sites, so one is looking at limits of sums of “almost independent” random variables. In certain circumstances, such limiting sums behave in distribution like Gaussian random variables [101]. Fixing the temperature T either above or below the critical value produces such behavior in the sums of spin variables. This is the Gaussian regime mentioned at the beginning of the chapter in which one looks at the correlations at scales that are much bigger than the correlation length. The Ising scaling limit we are considering is not Gaussian, however, because it focuses on the correlation length scale as T tends to the critical value Tc ; this is the scale at which exponential clustering just starts to set in. The scaling functions do cluster exponentially, but the distribution is not Gaussian. Fixing the temperature T at the critical temperature produces the third sort of scaling behavior. The clustering of the spin correlations is expected to take place at a rate given by a fractional power of the spin site separation. It is believed that there are analogues of the central limit theorem for the behavior of sums of such weakly independent random variables. That many of the details of the critical random variables do not directly figure in the large-scale limit of the critical correlations is expected to account for certain “universal behavior” found experimentally in critical phenomena. It is anticipated that large classes of interactions have identical critical scaling, dependent only on symmetries and the spatial dimension. Thus the critical scaling may provide a sort of “stable” signature for critical phenomena a bit like a topological invariant for a manifold. The infinite determinant formulas we have found for the Ising correlations, which manifest clustering behavior at large separation, do not obviously specialize to the critical temperature, and the finite determinant formulas at the end of Chapter 1 that do make sense at the critical temperature do not have transparent behavior at large separation. The principal technical difficulty in our approach is that the induced rotation for the spin operator no longer corresponds to an element of the spin group at T = Tc . The mechanism by which the correlations cluster seems

140

3. Scaling Limits

to change qualitatively at the critical point, and understanding how this takes place might suggest workable formulas for the critical correlations. You have to be a real enthusiast for explicit formulas to appreciate the inversion of priorities implicit in this thought. The long-range asymptotics of the critical two-point function on the diagonal have been rigorously analyzed, and it is known that [89] # $ 1 σ0,0 σN,N T =Tc = O N − 4 as N → +∞.

There are some isolated results for the behavior of the four-point function, but as far as I am aware there are no mathematical results from which one can infer even the existence of the critical scaling limit for the n-point correlations of the Ising model. However, conformal field theory ideas do provide a natural conjecture for this limit [47]. It is expected that lim N n σ (Na1 ) · · · σ (Na2n )2T =Tc = c |ai − aj |2ǫi ǫj , n→∞

|ǫ|=0 i<j

where c is a constant, ǫ is an n-tuple each entry of which is ± 21 , and |ǫ| := ǫj . j

To prove this is an interesting open mathematical problem. There is another route to the critical signature of the Ising model that is accessible to the techniques developed in this book. The short-distance behavior of the scaling functions τ [±] (a) is expected to coincide with the large-scale critical scaling, at least to the extent that this makes sense. The odd correlations vanish at T = Tc , but the short-distance scaling limit for the odd subcritical scaling functions does not. Tracy [153] and in an improved treatment Tracy and Widom [154] show that the two-point scaling function has short-distance behavior that precisely matches the large-scale asymptotics of the critical two-point function. They analyze the explicit formula for the two-point scaling function given at the end of this chapter as a Fredholm determinant. At the end of Chapter 6 in this book we mention some results for the short-distance behavior of the n-point scaling functions that follow from the SMJ characterization of the logarithmic derivative of the tau function as a level-one Fourier coefficient in the expansion of a particular solution to the Dirac equation. For the even scaling functions below Tc and the scaling functions above Tc this does reproduce the expected result from conformal field theory.

3.6 Expansions for the Two-Point Scaling Functions We conclude this chapter with explicit formulas for the two-point functions in the hyperbolic parametrization and also explicit formulas for the two-point scaling functions. The two-point function formula has been used to analyze analyticity properties of the magnetic susceptibility [107], and the scaling limit formula

3.6 Expansions for the Two-Point Scaling Functions

141

(as a Fredholm determinant) has been used to do explicit calculations of the shortdistance behavior of the two-point function in [154]. We will make use of the explicit formulas to prove results for the nonvanishing of the scaling functions above Tc that later figure in our analysis of the scaling functions. We want to introduce new integration variables θj in the series expansions for the two-point functions in Theorems 2.5.1 and 2.6.1 given by iK ′ iK ′ eiθj = ksn uj + − ia sn uj + + ia . 2 2 We already know that the integration differential is given by duj =

s1 dθj , 2 sh γ (θj )

and we have the uniformization result for e−γ , so the one ingredient we need to understand under this change of variables is sn(ui − uj ). Because this is proportional to the kernel of BD −1 and we found that the kernel of BD −1 in the hyperbolic representation is proportional to sh 12 (γ (θi ) − γ (θj )) sin 12 (θi − θj )

,

one might guess that this is what becomes of sn(ui − uj ) under the change of variables. However, BD −1 maps W− to W+ , and so the change of variables that carries the kernel of BD −1 in the hyperbolic representation to the kernel of BD −1 in the elliptic representation involves the different substitution iK ′ iK ′ iθj − ia sn u − + ia e = ksn u − 2 2 on W− . However, this is just the reciprocal of the substitution we are interested in (u←⊣ u + iK ′ just inverts the right-hand side), and we can compensate by sending θj to −θj . We thus examine what happens to (γ = γ (θ) and γ ′ = γ (θ ′ )) sh 12 (γ (θ) − γ (θ ′ ) sin 21 (θ + θ ′ )

=i

e

γ +iθ 2

e−

γ ′ +iθ ′ 2

iθ −γ

−e 2 e eiθ − e−iθ ′

γ ′ −iθ ′ 2

(3.92)

under the elliptic substitution eiθ = ksn(u − ia)sn(u + ia), ′

eiθ = ksn(v − ia)sn(v + ia), ′

where ℑu = ℑv = iK2 and the real part of u and v are in the interval [0, 2K]. The uniformization in Theorem 2.2.1 implies that √ γ +iθ e 2 = ksn(u + ia), √ iθ −γ e 2 = ksn(u − ia).

142

3. Scaling Limits

Thus the numerator of (3.92) becomes sn(u + ia) sn(v + ia) sn(u − ia) − sn(u − ia) sn(v − ia) sn(v + ia) and the denominator becomes ksn(u − ia)sn(u + ia) − (ksn(v − ia)sn(v + ia))−1 . Now we rewrite the numerator and denominator using the identities ksn(u − ia)sn(u + ia) = and

sn2 (u) − sn2 (ia) 1 − k 2 sn2 (u)sn2 (ia)

sn(u)cn(ia)dn(ia) + sn(ia)cn(u)dn(u) sn(u + ia) = , sn(u − ia) sn(u)cn(ia)dn(ia) − sn(ia)cn(u)dn(u)

which follow from the addition formulas (2.21). After some simplification we find that (3.92) becomes isn(2ia)k

sn(u)cn(v)dn(v) − sn(v)cn(u)dn(u) . 1 − k 2 sn2 (u)sn2 (v)

This is just −s1 ksn(u − v) = −s2−1 sn(u − v).

Thus the elliptic substitution results in the replacement sn(u − v) ∼ −k −1 s1−1

sh 21 (γ (θ) − γ (θ ′ )) sin 12 (θ + θ ′ )

.

We find the following expansion for the two-point function σ0,0 σM,N T for T < Tc : n2

σ 2T

k n− 2 n!(2π)n n∈even

)

π

−π

···

where hij = s1−1

)

π

n

−π i<j

h2ij

n

eiMθj −N γ (θj )

j =1

sh 21 (γ (θi ) − γ (θj )) sin 12 (θi + θj )

s1 dθj , sh γ (θj )

.

The reason for the constants s1 in the measure and hij has to do with simplification in the scaling behavior, which we examine in a moment. In a similar fashion the expansion for the two-point function σ0,0 σM,N T ∗ for T ∗ > Tc is k

− 21

σ 2T

2 ) π n n k n− n2 ) π s1 dθj 2 · · · h eiMθj −N γ (θj ) , ij n!(2π)n −π sh γ (θj ) −π i<j j =1

n∈odd


143

with hij as above. The interaction constants in this formula are (K1 , K2 ) if T ∗ corresponds to interaction strengths (K2∗ , K1∗ ). These hyperbolic formulas are better for analyzing the susceptibility, which is σ0,0 σM,N , M,N

than the elliptic formulas. The following result for the two-point scaling functions is also a direct consequence of the these formulas and the scaling results described earlier in this chapter. Theorem 3.6.1 The scaling functions τ [±] (0, a) are given by τ [−] (0, a) = 1 + and τ [+] (0, a) = where r = |a| and Fn (r) =

1 n!

)

0

∞

···

)

0

∞

∞

∞

F2n (r)

n=1

F2n+1 (r),

n=0

n ui − uj 2 − r (u +u−1 ) duj . e 2 j j ui + uj j =1 2π uj i<j

Proof. The scaling behavior of hij is easily found, s1−1

m2 m2−1 sh 21 (γ (m1 pi ) − γ (m1 pj )) m1 m1−1 sin 12 (m1 pi + m1 pj )

since s1−1

→

ω(pi ) − ω(pj ) , pi + p j

m2 → 1. m1

In a similar fashion, s1 m1 dpj −1 m2 m2 sh γ (m1 pj )

→

dpj . ω(pj )

Introducing the scaling variables a1 = m1 M and a2 = m2 N in the hyperbolic formulas for the two-point function and then passing to the scaling limit yields τ [−] (0, a) = 1 +

∞ n=1

F2n (a1 , a2 )

144

3. Scaling Limits

and τ [+] (0, a) =

∞

)

n

where Fn (a1 , a2 ) =

1 n!

∞

−∞

···

)

∞

−∞ i<j

and

F2n+1 (a1 , a2 ),

n=0

Hij2

n

eia1 pj −a2 ω(pj )

j

dpj 2π ω(pj )

ω(pi ) − ω(pj ) . pi + p j

Hij =

Now introduce the change of variables pj = 21 (u−1 j − uj ) for u ∈ (0, ∞), which was discussed earlier in this chapter. It is easy to check that under this change of variables, ui − u j Hij → ui + u j

and

du dp → . ω(p) u Also, if we introduce polar coordinates for a, a = reiφ , then e

ia1 p−a2 ω(p)

=e

− 2r

#

$ # $−1 eiφ u + eiφ u

.

Substituting these results into the formula for Fn and using the analyticity in u to rotate all the u contours to e−iφ R + , we obtain the result of the theorem. This result was first obtained in Palmer–Tracy [117]. Using this formula it is a trivial matter to find the large-r asymptotics of the two-point scaling functions by the Laplace method. Much more interesting is the behavior of the two-point scaling function as r → 0. Tracy [153] and Basor and Tracy [19] have results for not only the short-distance behavior of the Ising model two-point functions but also the twopoint functions that arise for holonomic quantum fields. Tracy and Widom also have a very nice treatment of this in [154]. Their treatment depends on recognizing the formula for the two-point scaling function as a Fredholm determinant. The scaling formulas we found earlier in this chapter are Pfaffians rather than determinants, but it is not hard to see that the two-point scaling function below Tc given in Theorem 3.6.1 is just the Fredholm expansion for det(1 − Kr2 ), du where Kr is the integral operator on L2 R + , 2π with kernel given by u r

Kr (u, v) = e− 4 (u+u

−1 )

u − v − r (v+v−1 ) e 4 . u+v


145

Their result is (for r = |a|) 1

τ [−] (0, a) ≃ Cr − 4 . The constant that appears here has a nice product formula that is related to the zeta determinant of the Laplace operator on the two-sphere [19]. It would be interesting if one could understand the appearance of a determinant formula for this coefficient in a “natural” way (as, say, a determinant of a mass-zero Dirac operator). A second application of the scaling function formula we have obtained is important for our treatment of the T > Tc scaling limit. It is clear from the explicit formula that τ [+] (0, a) > 0. The GKS inequalities imply that for the infinitevolume correlations, σa1 σa2 · · · σa2n ≥ σa1 σa2 σa3 σa4 · · · σa2n−1 σa2n . Writing this inequality for T > Tc , dividing out by the appropriate power of the magnetization at the dual temperature, and passing to the scaling limit, we see that the strict lower bound for the scaled two-point function implies that τ [+] (a1 , . . . , a2n ) > 0.

(3.93)

This will ultimately show the existence of the T > Tc Green function for the Dirac operator that is seen to control the scaling behavior of the Ising correlations.

4 The One-Point Green Function

4.1 Introduction “The two-dimensional Ising model is a free Fermion.” Remarks to this effect are commonplace in the physics literature, although for mathematicians it sounds like a cross species identification. In the previous three chapters we have seen that the spin variables of the Ising model can be realized as operators acting in a projective representation of the orthogonal group. This representation lives on an alternating tensor algebra that is also the arena for the mathematical description of free fermions. The alternating tensor algebra reflects the antisymmetric statistics that are characteristic of many particle states for fermions, and the spin operator is characterized by its action as an automorphism of the canonical anticommutation relations. This is one way to understand the opening remark about free fermions, but in this chapter we begin to develop a different and deeper take on this observation that was first worked out by Sato, Miwa, and Jimbo in a remarkable series of papers that introduced the notion of holonomic quantum fields [137]. The scaling fields of the Ising model are examples of holonomic quantum fields in the Euclidean regime (pure imaginary time) and the Ising scaling functions are instances of tau functions in their theory. The tau functions of the SMJ theory can be expressed in terms of solutions to a nonlinear holonomic system of differential equations. Roughly speaking, a holonomic system of differential equations is maximally overdetermined and has a finite-dimensional space of solutions. The terminology holonomic quantum fields was adopted to reflect this central feature of these models. It is a pleasant surprise that in these models the correlations of a strongly interacting

148

4. The One-Point Green Function

system with infinitely many degrees of freedom are nonetheless governed by the solutions to a differential equation with only a finite-dimensional space of solutions (at least at the level of n-point functions for fixed n). In the Ising model the nonlinear holonomic system arises in the description of a monodromy-preserving deformation of the Euclidean Dirac equation in two dimensions (with mass term). The Minkowski version of this Dirac equation governs relativistic free fermions in two dimensions, and this provides the alternative take on the opening remark of this chapter. Remark 4.1.1. The monodromy in question is related to the holonomy of a Diraccompatible connection that arises in the analysis of such deformations. The adjective holonomic is thus suggestive of more than one feature of these quantum fields. We begin, in the second section, with a sketch of the SMJ ideas for the scaling limit of the Ising model. The principal result explained there is a “free Fermi” representation for a Green function associated with the Dirac operator acting on a domain that includes multivalued functions branched in a special way at the spin sites {ai }. This is not the central player in the SMJ development, but an important focus of this book is that it can be made central in their theory in a way that illuminates the significance of the tau functions. This theme is taken up in the succeeding chapters. The motivational form we choose to introduce these ideas is not the form that we develop with mathematical rigor in subsequent chapters. The ideas of the following section in this chapter are mathematically elaborated in [118], but developing the theory along these lines has a number of disadvantages. Lattice precursors of the objects that are central to the continuum analysis are introduced and their scaling limits are controlled. This is instructive, but there are irritating details associated with making lattice approximations to multivalued functions in the continuum. These details are even more trying for lattice approximations to general holonomic fields [43]. The local operator product expansions that are at the heart of this analysis are also controlled by clumsy lattice approximation, and the significance of the tau function in the associated deformation theory remains obscure. Once past motivational developments, we devote the remainder of the chapter to making an explicit calculation of the Green function for the Dirac operator of interest in the case of just one spin site (or branch cut). A contour integral representation for this Green function is found in Theorem 4.5.1. The formula for the Green function is then used to compute the projection associated with localizing this Dirac operator in the complement of a strip containing the branch cut. This projection has “matrix elements” that are naturally identified with the operators D −1 , D −τ , BD −1 , and D −1 C that appear in the formulas for the scaled Ising correlations developed in Chapter 2. This is the “glue” that allows us in the succeeding chapter to make the connection between the scaling functions for the Ising model and tau functions naturally associated with the “twisted” Dirac operator.

4.2 Free Fermions

149

4.2 Free Fermions There is a sense that much of what we do in this chapter and the next is to restore a two-dimensional picture of the Ising model (i.e., the Pfaffian formalism) that the transfer matrix formalism has squashed into obscurity. We begin with a discussion of the “free fermion” representation of the Green function for the finite difference operator on the two-dimensional lattice associated with the induced rotation of the transfer matrix. This connection is, incidentally, a simple illustration of the reason that physicists often refer to two-point functions as Green functions. Suppose that f and g are maps, f, g : Z 12 × Z → C2 . Let λ denote translation by 1 unit in the second coordinate, λf (ℓ, m) = f (ℓ, m + 1). Recall that theinduced rotation T (V ) for the transfer matrix is a finite difference operator on Z 21 , and so (4.1) (λ − T (V )) f = g

is a finite difference equation. That (4.1) is a finite difference equation is not significant for initial developments, but we do return to it later in detail. In order to solve (4.1) for f , introduce the partial Fourier series fˆ(ℓ, λ) = λ−m f (ℓ, m) m∈Z

(the λ−m in this formula will make multiplication by λ work out to lattice translation by +1) and recall that by (1.72), T (V ) = e−γ Q+ + eγ Q− , where Q+ is the spectral projection for T (V ) associated with the spectrum in [0, 1] and Q− is the spectral projection for T (V ) associated with the spectrum in [1, ∞). Then we rewrite (4.1) as λ − e−γ Q+ fˆ + (λ − eγ ) Q− fˆ = gˆ and solve for fˆ to get

−1 Q+ gˆ + (λ − eγ )−1 Q− g. ˆ fˆ = λ − e−γ

The inversion formula

f (·, m) =

1 2πi

)

S1

dλ fˆ(·, λ)λm λ

150


gives f (·, m) =

1 2π i

) # $ −1 λ − e−γ Q+ g(·, ˆ λ) + (λ − eγ )−1 Q− g(·, ˆ λ) λm−1 dλ. S1

Substitute the Fourier series for g(·, ˆ λ) in this last expression and make use of the contour integrals ) 1 0 for N ≤ −1, −γ −1 N λ−e λ dλ = −N γ 2π i S 1 e for N ≥ 0, and 1 2π i

)

S1

γ −1

(λ − e )

λN dλ =

−eNγ 0

for N ≤ −1, for N ≥ 0,

to find that f (·, m) =

n<m

e−(m−n−1)γ Q+ g(·, n) −

n≥m

e(m−n−1)γ Q− g(·, n).

(4.2)

pk Now write e1 (k) = √qk2 and e2 (k) = √ for the orthonormal basis (1.22) of 2 1 2 ℓ Z 2 . This basis is also complex orthogonal with repect to the complex bilinear form (·, ·) defined in the paragraph following (1.22), and writing

fi (k, m) = (ei (k), f (·, m)) for the coordinate functions of f , we find that (4.2) becomes fi (k, m) = Gi,j (k, m; ℓ, n)gj (ℓ, n), j,ℓ,n

where the sum is over j = 1, 2, ℓ ∈ Z + 21 , and n ∈ Z, and Gij (k, m; ℓ, n) =

ei (k), e−(m−n−1)γ Q+ ej (ℓ) − ei (k), e(m−n−1)γ Q− ej (ℓ)

for m > n, for m ≤ n.

Since T (V ) is complex orthogonal (i.e., T (V )τ = T (V )−1 ) we can rewrite this expression for the Green function G as ei (k, m), Q+ ej (ℓ, n + 1) for m > n, (4.3) Gij (k, m; ℓ, n) = − ei (k, m), Q− ej (ℓ, n + 1) for m ≤ n, where we have introduced the vector

ei (k, m) := T (V )−m ei (k).

(4.4)

4.2 Free Fermions

151

The formula we have obtained for G can be rewritten in the Q Fock representation F . For brevity write ei (k, m) = F (ei (k, m)), and introduce Fermionic “time ordering” for m ≥ n, ei (k, m)ej (ℓ, n) T ei (k, m)ej (ℓ, n) = −ej (ℓ, n)ei (k, m) for m < n. Thus the expression (4.3) for G can be written as a time-ordered vacuum expectation (two-point function) in the Q Fock representation (see Theorem A.31). Proposition 4.2.1 Suppose that f and g are square summable C2 -valued maps 1 on Z 2 × Z. Let {e1 , e2 } denote the standard basis of C2 , and write fi (k, m) for the ith coordinate function for f , so that f (k, m) = f1 (k, m)e1 + f2 (k, m)e2 . Let T (V ) denote the finite difference operator (4.6) acting on C2 -valued square 1 summable functions on Z 2 . The solution f of the finite difference equation f (·, m + 1) − T (V )f (·, m) = g(·, m)

has a Green function representation fi (k, m) = Gij (k, m; ℓ, n)gj (ℓ, n), j,ℓ,n

where ei (k, m), Q+ ej (ℓ, n + 1) for m > n, Gij (k, m; ℓ, n) = − ei (k, m), Q− ej (ℓ, n + 1) for m ≤ n,

and the vectors ei (k, m) are given by

ei (k, m) := T (V )−m ei (k). This Green function has an alternative representation as the time-ordered vacuum expectation , Gij (k, m; ℓ, n) = T ei (k, m)ej (ℓ, n + 1) (4.5)

in the Fock representation associated with the polarization Q.

There is an extension of this formula to the continuum that incorporates a product of spin operators σ (a) (or µ(a)) and represents the Green function for a Dirac operator acting on sections of a vector bundle over R 2 \{spin sites}. It is simplest to discuss this in the continuum limit, and we turn next to a naive scaling analysis for the lattice fermions, (4.4).

152


The vectors (4.4) satisfy a finite difference equation on the lattice that scales to the Dirac equation in the continuum limit. To see this, recall (1.70), where the action of the induced rotation for the transfer matrix in the ℓ2 representation is given by the finite difference operator T (V )f (k) = T1 f (k − 1) + T0 f (k) + T−1 f (k + 1),

(4.6)

where the matrices T1 , T0 , and T−1 are s∗ s1 i(1 + c1 ) T1 = − 2 , s1 2 i(1 − c1 ) c1 is1 , T0 = c2∗ −is1 c1 and T−1

s2∗ i(c1 − 1) s1 . =− s1 2 −i(c1 + 1)

Since T (V ) is complex orthogonal, the action of T (V )−1 is the same as the action of T (V )τ . Thus τ f (ℓ − 1). T (V )−1 f (ℓ) = T1τ f (ℓ + 1) + T0τ f (ℓ) + T−1

(4.7)

We now translate this action in coordinates into the “dual” action on basis vectors ei (k) in the usual fashion. The coordinate function for ei (k) is ℓ → δ(ℓ − k)ei , where e1 and e2 are the standard basis vectors in C2 . One can write the basis vector representation of ei (k) as the delta function sum e(ℓ)τ δ(ℓ − k)ei , ei (k) = ℓ

where e(ℓ)τ is the row vector [e1 (ℓ), e2 (ℓ)] (each component of which is itself a vector in ℓ2 ) and the pairing between row vectors and column vectors is the usual one. From this and (4.7) we see that T (V )−1 ei (k) τ = e(ℓ)τ T1τ δ(ℓ + 1 − k)ei + T0τ δ(ℓ − k)ei + T−1 δ(ℓ − 1 − k)ei ℓ

τ = e(k − 1)τ T1τ ei + e(k)τ T0τ ei + e(k + 1)τ T−1 ei .

Converting this to an equality of column vectors and multiplying the result by the diagonal action of T (V )−m , yields e(k, m + 1) = T1 e(k − 1, m) + T0 e(k, m) + T−1 e(k + 1, m),

(4.8)

4.2 Free Fermions

with the ℓ2 vector e(k, m) :=

153

e1 (k, m) . e2 (k, m)

For the purpose of understanding the scaling limit of this finite difference equation it is convenient to rewrite it in terms of (τ2 − 1) and (τ1 − 1), where τi is lattice translation by +1 in the ith slot. These difference operators will scale to derivatives. The rewritten equation is (τ2 − 1) + T1 (1 − τ1−1 ) − T−1 (τ1 − 1) + (1 − T1 − T0 − T−1 ) e = 0. (4.9) To understand the scaling limit of the Clifford algebra we introduce −1

−1 eT (x1 , x2 ) := m1 2 e(m−1 1 x1 , m2 x2 ),

where m1 and m2 are the temperature (T ) dependent horizontal and vertical mass −1

parameters (3.4) and (3.6). The rescaling by m1 2 , referred to as wave function renormalization in the physics literature, becomes infinite in the T ↑ Tc limit and is important to obtaining nontrivial limits for the appropriate quantities, as we describe below. Now rewrite (4.9) in terms of eT (x1 , x2 ) and divide the resulting equation by m2 . Naively suppose that m−1 2 (F (x1 , x2 ) − F (x1 , x2 − m2 )) tends to ∂2 F (x) as m2 → 0 and m−1 (F (x , x ) − F (x1 − m1 , x2 )) tends to ∂1 F (x) as 1 2 1 m1 → 0. Then supposing some sense for the limit, we write lim eT = e,

T ↑Tc

and the scaling limit of (4.9) becomes 0 i 0 ∂2 − ∂1 + i 0 −i

i e(x) = 0, 0

(4.10)

where we used the fact that m1 i 0 ∗ m1 0 =− lim (T1 − T−1 ) = lim −s2 i 0 i m2 →0 m2 →0 m2 m2

i 0

and (since m2 = 2K1 − 2K2∗ ) −1 m−1 2 (1 − T1 − T0 − T−1 ) = m2

1 − ch(m2 ) −i sh(m2 )

i sh(m2 ) , 1 − ch(m2 )

0 i which has the evident limit −i 0 as m2 → 0. This is a variant of the Euclidean Dirac equation in the plane (with a mass term) that we rewrite as 0 −i ∂e −1 0 ∂e + − e = 0. i 0 ∂x2 0 1 ∂x1

154


However, this is not the most convenient form for the Dirac equation. Introduce 1 ψ1 (x) e1 (x) + e2 (x) ψ(x) = := √ . (4.11) ψ2 (x) 2 −e1 (x) + e2 (x) Then ψ satisfies the equation Dψ − ψ = 0, where

0 D := ¯ 2∂

and the complex derivatives ∂ and ∂¯ are 1 ∂ ∂ ∂ := and −i 2 ∂x1 ∂x2

(4.12)

2∂ , 0 1 ∂¯ := 2

∂ ∂ +i ∂x1 ∂x2

.

This form for the Dirac equation considerably facilitates the local analysis done subsequently. Remark 4.2.1. Mathematicians refer to D (or its chiral components) as the Dirac operator since it is the massless version that has a basic significance for index theory on compact manifolds. We refer to (I − D)ψ = 0 or in the presence of a mass parameter, (mI − D)ψ = 0, as the Dirac equation. Since the Minkowski version of the Dirac equation was introduced by Dirac with a mass term, this usage is reasonable. Unfortunately, the simple-minded scaling analysis we have sketched here has some technical complications: the “operators” ψj (x) do not make sense even as operatorvalued distributions in x = (x1 , x2 ). We illustrate the problem by doing the formal calculation of the T ↑ Tc limit. √ in W+ ⊕ W− is given by The image of e1 (x) = q(x) 2 −1 im p eix1 p oT (p) ex2 m2 γT (p) 0 1 S e . √ −1 0 2π m1 0 e−x2 m2 γT (p)

To determine the limit as T ↑ Tc , observe that since α1 = em1 , we have √ ( m1 A1 e±im1 p 1 − e−m1 (1±ip) 2 =e → 1 ± ip √ √ m1 m1 as m1 → 0. A short calculation then shows that √ √ S eim1 p 1 − ip i √1 + ip √ = lim . √ 1 − ip −i 1 + ip m1 →0 m1

Thus the image of e1 (x) in W+ ⊕ W− has a limit as T ↑ Tc given by = 1 − ip ix1 p ex2 ω(p) lim eT ,1 (x) = . e e−x2 ω(p) T ↑Tc 2π

4.2 Free Fermions

155

In a similar fashion the image of eT ,2 (x) in W+ ⊕W− has a limit as T ↑ Tc given by = 1 + ip ix1 p ex2 ω(p) lim eT ,2 (x) = i . e −e−x2 ω(p) T ↑Tc 2π Of course, these are just pointwise limits; neither limit is in W+ ⊕ W− and the exponential factors also keep the limit from being a tempered distribution in (x1 , x2 ) (smoothing by a C ∞ function of rapid decrease at ∞ does not give an element in W+ ⊕ W− ). This difficulty is circumvented in SMJ IV [136] by introducing “Minkowski” versions of the associated fields; the exponential factors e±x2 ω(p) are replaced by the better-behaved e±ix2 ω(p) . However, the resulting fields are rather far removed from the Euclidean regime. One gets back to the regime relevant to the Ising model by an analytic continuation of vacuum expectations in the time variable (i.e., x2 ). This analytic continuation is well understood for the free Fermi field ψ, but there are technical difficulties that arise when the spin fields are included. For this reason we won’t introduce the Minkowski fields here (but see [136]). The quantities that do behave well in the transfer formalism are vacuum expectations of “time” ordered products. We have already encountered time ordering in the representations of the spin correlations and the free Fermi Green function. The representation of the spin correlations as a vacuum expectation requires that we order the second coordinates of the sequence of sites aj so that they occur in increasing order. We can formalize this somewhat differently as follows. Suppose that a1 , a2 , . . . , an are distinct points in Z2 ; define the time-ordered product of the spin operators σ (aj ) by T σ (a1 ) · · · σ (an ) = σ (aπ(n) ) · · · σ (aπ(1) ),

(4.13)

where the permutation π of {1, 2, . . . , n} is chosen so that the second coordinates of the vectors in the sequence aπ(1) ,

aπ(2) , . . . ,

aπ(n) ,

occur in increasing order. As mentioned in Chapter 1, the time ordering that occurs in (4.13) is increasing from right to left. In some places this is referred to as antitime ordering, but we won’t introduce this terminology here. There is some ambiguity in the choice of the permutation if there are coincidences among the second coordinates, but this does not affect the time-ordered product since the “equal time” spin operators commute with one another (this is obvious at the level of induced rotations and it can also be proved for the spin operators themselves [108]). Theorem 1.3.1 implies that the multispin correlation at sites a1 , a2 , . . . , an below Tc is given by 0|T σ (a1 )σ (a2 ) · · · σ (an )|0. (4.14) It also makes sense to think of time-ordered products of the continuum scaling fields as given by the scaling limits of the spin correlations. This is how we interpret expressions such as (4.14) when the sites are points aj in R 2 . Note that

156


multiplying the “operator” T σ (a1 ) · · · σ (an ) on the left by a sufficiently high positive power of the transfer matrix will always produce a bounded operator. However, because negative powers of the scaling limit of the transfer matrix are unbounded and a wave function renormalization is involved in defining σ , it seems hard to characterize the domain of this time-ordered product (i.e., the vectors that are mapped by the product into an element in the Hilbert space Alt(W+ )). In general these time-ordered products occur for us in vacuum expectations, where the large negative powers of the transfer matrix do not cause a problem because 0|V −n = 0|. For fermions it is natural to include a sign flip for exchanges, and so we define T ψi1 (x1 ) · · · ψin (xn ) = sgn(π)ψiπ(n) (xπ(n) ) · · · ψiπ(1) (xπ(1) ), where the permutation π of {1, 2, . . . , n} is chosen, as above, so that the second coordinates of the vectors (now in R 2 ) in the sequence xπ(1) , xπ(2) , . . . , xπ(n) , are in increasing order. Note that if there are coincidences among the second coordinates, the time-ordering prescription introduces ambiguities. These ambiguities are resolved by treating the vacuum expectations of time-ordered products as “functions” only for noncoincident sets of second coordinates. At coincident second coordinates there are upper and lower boundary values that do not agree in general. The situation is best understood by looking at an example. The two-point function is given by Gij (x, y) := 0|T ψi (x)ψj (y)|0. Our lattice calculation suggests that the 2 × 2 matrix G(x, y) should be closely related to the Green function for the Dirac operator. We sketch an alternative account of this that is useful for understanding why the formula incorporating products of spin operators that is later introduced represents the Green function for a Dirac operator on a modified domain. We do leave it to the reader to fill in the details since this is mostly a motivational discussion for us. Each column of G(x, y) satisfies the Euclidean Dirac equation (4.12) in the x variable away from coincidence x2 = y2 . Similarly, the transpose of each row of G(x, y) satisfies (4.12) in the y variable away from coincidence x2 = y2 . The time-ordering prescription is 0|ψi (x)ψj (y)|0 for x2 > y2 , 0|T ψi (x)ψj (y)|0 = −0|ψj (y)ψi (x)|0 for y2 > x2 . Define upper and lower boundary values for G as follows: G(x, y± ) := lim G(x1 , x2 , y1 , x2 ± ǫ), ǫ↓0

4.2 Free Fermions

157

where ǫ tends to 0 through positive values. Then the difference of the upper and lower boundary values of G is given by, Gij (x, y+ ) − Gij (x, y− ) = −0|ψi (x1 , 0)ψj (y1 , 0) + ψj (y1 , 0)ψi (x1 , 0)|0 = −δij δ(x1 − y1 ), (4.15) since the “equal time” anticommutator for the ψ field is the delta function in the “space variable.” The “time zero” field ψ(x1 , 0) does make sense as an operatorvalued distribution. Because the vacuum |0 is an eigenvector with eigenvalue 1 for the transfer matrix we find that vertical separation is incorporated in the Green function as follows 0|ψi (x1 , 0)V x2 −y2 ψj (y1 , 0)|0 for x2 − y2 > 0, Gij (x, y) = y2 −x2 −0|ψi (y1 , 0)V ψj (x1 , 0)|0 for y2 − x2 > 0. where V is the scaling limit of the transfer matrix. Since V is a strict contraction on the one-particle space and the function Gij (x, y) is a matrix element of V on the one-particle space it follows that Gij (x, y) tends to 0 exponentially fast as |x2 − y2 | tends to ∞. The properties of G(x, y) sketched in this paragraph are enough to show that it is the Green function for I − D with one adjustment that we now turn to. The Dirac operator I − D was obtained from the differential operator with coefficient 1 in front of ∂/∂x2 by multiplying on the left by the matrix 0 i J := . −i 0 Thus to obtain the Green function for the Dirac operator I − D from the Green function with the fermionic representation we need to multiply on the right by J . This is reflected somewhat differently in the following observation. If u is a column vector in C2 , write u and uτ = [u1 , u2 ] u= 1 u2 for the transpose. The operator I − D is neither formally symmetric nor antisymmetric with respect to the symmetric bilinear form ) d x¯ dx f (x)τ g(x) (f, g) = , 2 2i R

¯ = dx1 dx2 . However, where dx = dx1 + idx2 and d x¯ = dx1 − idx2 , so that d xdx 2i I − D is formally symmetric with respect to the bilinear form ) d x¯ dx f (x)τ J g(x) . (4.16) (f, g)J = 2i R2

The infinitesimal expression of this formal symmetry is exactness for the difference {((I − D)f )τ J g − f τ J (I − D)g}

d x¯ dx = d(f1 g1 dx + f2 g2 d x). ¯ 2i

(4.17)

158


The complex representation of the exterior derivative d = dx ∧ ∂ + d x¯ ∧ ∂¯ makes this simple to check and is a good illustration of the advantage of the “complex” version of the Dirac equation (4.12). Now suppose that f (x) is an element of C0∞ (R 2 , C2 ), the smooth functions of compact support on R 2 with values in C2 . Then G(x, y) is a Green function for the Dirac operator in the sense that ) d y¯ dy G(x, y)J (I − D)f (y) f (x) = . (4.18) 2i R2 It is instructive to use the properties of G to confirm this. Since the rows of the Green function are solutions to the Dirac equation for x = y, the infinitesimal symmetry (4.17) implies that for x = y, d y¯ dy = −d (G1 (x, y)g1 (y) dy + G2 (x, y)g2 (y) d y) ¯ , 2i (4.19) where D and the exterior derivative d act on functions of y, and Gj (x, y) is the j th column of G(x, y). Choose R > 0, x ∈ C and define upper and lower half-disks about x by G(x, y)J (I − D)g(y)

DR± (x) = {y : |y − x| ≤ R, ±(y2 − x2 ) ≥ 0}, with DR (x) := DR− (x) ∪ DR+ (x). Stokes’s theorem and the identity (4.19) together imply ) d y¯ dy G(x, y)J (I − D)f (y) 2i DR (x) =−

)

G1 (x, y)f1 (y) dy + G2 (x, y)f2 (y) d y¯

+

)

G1 (x, y)f1 (y) dy + G2 (x, y)f2 (y) d y, ¯

+ ∂DR (x)

− ∂DR (x)

(4.20)

where the boundaries ∂DR± (x) are oriented counterclockwise. In the limit R → ∞, the left-hand side of (4.20) approaches the right-hand side of (4.18). Because f has compact support, the contribution of the half-circles in the boundaries of ∂DR± (x) to the right-hand side of (4.20) vanishes in the limit R → ∞. What remains of the right-hand side of (4.20) in this limit are the contributions coming from upper and lower boundary values on the line y2 = x2 , or ) (G1 (x, y− ) − G1 (x, y+ )) f1 (y) dy1 y2 =x2

+ (G2 (x, y− ) − G2 (x, y+ )) f2 (y) dy1 .

4.2 Free Fermions

159

Consulting (4.15), we see that this is f (x). We now introduce the principal player in our variant of the SMJ analysis. Let a = {a1 , . . . , an } denote a collection of distinct points in C and define Gi,j (x, y; a) =

0|T ψi (x)ψj (y)σ (a1 ) · · · σ (an )|0 . 0|T σ (a1 ) · · · σ (an )|0

(4.21)

The time-ordering applies to all the points x, y, a1 , . . . , an , and as above, there is an extra minus sign if the second coordinates of the pair x, y occur in decreasing order (but ψ and σ formally commute inside the time ordering). The columns of G(x, y; a) satisfy the Dirac equation (4.12) in the variable x away from the horizontal lines where the second coordinate x2 agrees with the second coordinates of one of the other points y, a1 , . . . , an and the transposed rows satisfy the same equation in the y variables away from the horizontal lines where y2 agrees with the second coordinate of one of the points x, a1 , . . . , an . The denominator in the definition of Gi,j (x, y; a) is chosen so that (4.15) implies the corresponding result Gi,j (x, y+ ; a) − Gi,j (x, y− ; a) = −δi,j δ(x1 − y1 ).

Next we consider the behavior of G(x, y; a) for x in a neighborhood of ak . For simplicity suppose that the second coordinate of ak is distinct from the second coordinate of the other points aℓ for ℓ = k. Then in the expression for Gi,j (x, y; a) for x2 > ak,2 (the second coordinate of ak ) one encounters the factor ψi (x)σ (ak ), and in the expression for Gi,j (x, y; a) for x2 < ak,2 one encounters the factor σ (ak )ψi (x). The induced rotation (1.78) for the spin operators suggests that lim ψi (x)σ (ak ) = −sgn(x1 − ak,1 ) lim σ (ak )ψi (x).

x2 ↓ak,2

x2 ↑ak,2

Thus the columns of G(x, y; a) are solutions to the Dirac equation branched at each of the points ak with boundary values that differ by a sign on the horizontal ray to the right of ak . It is not obvious that the boundary values of G(x, y; a) glue together to give a solution to the Dirac equation on the rays to the left of ak , but this is so (continuity promotes via elliptic regularity to smoothness). The transposed rows of G(x, y; a) are branched solutions to the Dirac equation in the y variable of the same sort. We would like to interpret G(x, y; a) as the Green function for a Dirac operator acting on “functions” that are suitably branched at each of the points a1 , a2 , . . . , an . Working with “branched” functions and boundary values is sometimes awkward, so we return to the smooth category by introducing an appropriate vector bundle over R 2 \a in which the “branched” functions become trivializations of smooth sections. This will be taken up in succeeding sections. The properties of G(x, y; a) mentioned so far do not completely characterize this function. We can completely characterize G(x, y; a) by making a closer study of the behavior of this function for x in a neighborhood of each of the points ak and at infinity. In [118] this is done as follows. Converting a time-ordered product on the lattice to normal ordered form produces an equality T ψj (x)σ (ak ) = op(j (x, ak )σ (ak )),

(4.22)

160


where j (x, ak ) is a “function space”-valued function of x and ak . This equality has a scaling limit in which j (x, ak ) goes over into an explicit “function space”valued solution of the Euclidean Dirac equation. This (branched) solution to the Dirac equation has a Fourier expansion in the angular variable θ = arg(x − ak ), which can be substituted into (4.22) and results in a local operator product expansion for T ψj (x)σ (ak ). The explicit form of this local operator product expansion shows that the singularity in T ψj (x)σ (a) as |x − ak | → 0 is restricted; the mostsingular terms behave like |x − ak |−1 . As we will see later, knowledge of the most-singular term completely characterizes G(x, y; a). If we regard G(x, y; a) as a Green function, its behavior for x near ak reflects the behavior of functions in the domain of the associated Dirac operator. This is the point of view that we adopt in the later sections of this chapter. The local operator product expansions for the Green function for x and y near ak produce a whole host of “wave functions” arising as coefficients in these expansions. Incorporating the disorder variables µ(a) increases the variety of such functions. These wave functions are branched solutions to the Dirac equation, which are in turn characterized by their local expansions. Working through various relations among these local expansions, one comes upon a formula for da log0|T σ (a1 ) · · · σ (an )|0 expressed in terms of local Fourier coefficients for appropriate wave functions. These Fourier coefficients are shown to be expressible in terms of solutions to monodromy-preserving deformations of the Dirac operator. This is the principal result of the SMJ analysis. To see this worked out along these lines the reader can consult the original papers of SMJ or [70]. In this chapter we next study spaces of (sometimes multivalued) solutions to the Dirac equation. This allows us to characterize boundary conditions that lead to (Fredholm) Dirac operators acting on sections of the appropriate vector bundle over the punctured plane R 2 \a. For a suitable choice of boundary conditions we find a Green function G(x, y; a) that can be identified with the Green function described above. However, we don’t try to show this directly since our understanding of local operator product expansions via lattice approximation is mathematically awkward. Instead we introduce a natural “tau function” for the Dirac operator and show by a calculation in the “transfer formalism” that this tau function is related to the scaling limit of the Ising correlations. The deformation analysis of the scaling functions follows from this connection.

4.3 Contour Integral Representations In this section we introduce a contour integral representation for the Green function of the Dirac operator. We also introduce contour integral representations for the multivalued solutions to the Dirac equation that are simultaneously eigenfunctions for the infinitesimal rotation 1 1 0 (4.23) R = x∂ − x¯ ∂¯ + 2 0 −1

4.3 Contour Integral Representations

161

that commutes with m − D. These eigenfunctions greatly facilitate the local analysis of boundary conditions for the Dirac operators of interest to us. Note that we introduce a mass parameter m > 0 even though we arranged the scaling analysis so that m = 1 is principally relevant. A later analysis of the short distance behavior of the scaling functions is done by analyzing the m → 0 limit for solutions to the Dirac equation. Roughly speaking, solutions to the Dirac equation (m − D)ψ = 0 have Fourier transforms in R 2 that are sections of a holomorphic line bundle with support on the “real cycles” of a complex “spectral curve.” The uniformization of this spectral curve and a trivialization of the associated line bundle allows us to introduce appropriate contour integral representations. The deformation of the contours using Cauchy’s theorem is a crucial ingredient in later calculations. Although we do not emphasize the connection, the reader might note that the elliptic curve of Chapter 1 pinches off in the scaling limit along one of the fundamental cycles to a copy of P1 . This makes the uniformization in the scaling regime simpler than the elliptic substitution of Chapter 1. In the physics literature the uniformization parameter for the spectral curve of the Dirac equation is called the “rapidity.” Some complication arises because the uniformization parameter that is most convenient for keeping track of contour deformations is not quite the uniformization that is natural for spectral theory. The two-dimensional (unitary) Fourier transform ) 1 fˆ(p) = f (x)e−ix·p dx 2π R2 transforms the Dirac operator m−D into the matrix-valued multiplication operator m −i p¯ . −ip m The complex spectral curve MC is the set of (p1 , p2 ) ∈ C2 such that m −i(p1 − ip2 ) det = p12 + p22 + m2 = 0, −i(p1 + ip2 ) m and the line bundle E has a fiber over the point (p1 , p2 ) ∈ MC given by the one-dimensional null space of m −i(p1 − ip2 ) . −i(p1 + ip2 ) m

or

The Fourier inversion formula suggests ) 1 m (m − D)−1 f (x) = 2π R2 −ip −1

(m − D) f (x) =

)

R2

−i p¯ m

−1

fˆ(p)eix·p dp,

G0 (x − y)f (y) dy,

162


where G0 (x) =

1 (2π)2

)

R2

m ip

eix·p i p¯ dp. m |p|2 + m2

Now suppose that x2 > 0. Then the ; dp2 integral can be closed in the upper half-

plane with a single pole at p2 = i p12 + m2 = iω(p1 ), where ω(p1 ) > 0 is the positive square root, to yield G0 (x) =

) R

m −ip−

i ¯ −) ip+ e 2 (xp+ −xp dp1 , m 4πω(p1 )

(4.24)

where p± = ω(p1 ) ± p1 . Observe that M± := {(p1 , ±iω(p1 ))| p1 ∈ R} ⊂ MC and that the two “real” cycles M± are relevant to the Fourier representation (4.24) ( M− arises for x2 < 0). Now introduce ip− u= . (4.25) m Then ip+ u−1 = − , m and so 2p1 = m(iu − (iu)−1 ),

2ω(p1 ) = −m(iu + (iu)−1 ). Thus the variable u gives a rational parametrization of MC . One can see that the positive imaginary axis iR + is mapped into M+ and the negative imaginary axis, −iR + , is mapped into M− . Substituting the representation for p1 in terms of u in the integral for G0 and noting that the standard orientation for R becomes the negative orientation for iR + , one obtains (for x2 > 0) ) −1 ) m du 1 −u−1 − m2 (xu+xu ¯ G0 (x) = , (4.26) e −u 1 + 4π u iR where the orientation of iR + in the integral is given by the parametrization [0, ∞) ∋ t → it. The choice of uniformization variable u is dictated for us by the form xu ¯ + xu−1 that appears in the exponent in this representation. The exponential factor m

¯ e− 2 (xu+xu

−1 )

is “small” for those values of u with real part ℜ(xu ¯ + xu−1 ) > 0.

(4.27)


163

But ℜ(xu ¯ + xu−1 ) = (1 + |u|−2 )(x · u),

where x · u = x1 u1 + x2 u2 is the standard inner product on R 2 . Thus ℜ(xu ¯ + xu−1 ) > 0 iff x · u > 0. This simple geometric relation is the reason for the particular choice we made for the uniformization parameter u. For x in the upper half-plane (x2 > 0) and u along the positive imaginary axis x · u > 0, one sees that the exponential factor (4.27) forces the convergence of the integral (4.26) at the two ends u → 0 and u → ∞. This same exponential convergence at the ends makes it easy to supply the estimates needed to use Cauchy’s theorem to shift the contour in (4.26) from the positive imaginary axis iR + to the ray through 0 and x. Thus ) −1 ) m du 1 −u−1 − m2 (xu+xu ¯ . G0 (x) = e 1 4π u xR + −u This representation is valid for all x = 0. Let x = eiθ |x| and u = eiθ r. Then ) ∞ −1 m dr 1 −e−iθ r −1 − m|x| e 2 (r+r ) G0 (x) = iθ −e r 1 4π r 0 ′ −iθ m K0 (m|x|) e K0 (m|x|) . = K0 (m|x|) 2π eiθ K0′ (m|x|) The modified Bessel function of the third kind, K0 , is defined in [157]. The reader should be aware that there is more than one definition for the modified Bessel functions. Since we used Whittaker and Watson as a reference for the Jacobian elliptic functions, as a convenience to the reader we also use it as our reference for the properties of Bessel functions. As described above, it is convenient to modify this Green function. Define

i

G(x) := G0 (x)J,

(4.28)

0 where J = −i 0 . For later reference we compute the contour representation of this Green function, −1 ) m ¯ −1 ) im du u 1 . G(x) = e− 2 (xu+xu −1 −u 4π u xR +

Observe that for each u ∈ C\{0} in the punctured plane the vector-valued function m ¯ −1 ) 1 (4.29) x → e(x, u) := e− 2 (xu+xu −u is a solution to the Dirac equation. We rewrite the contour representation above for the Green function as follows: ) im du u−1 e(x, u)e(y, −u)τ . (4.30) G(x − y) = + 4π u (x−y)R By taking suitable superpositions of the functions e(x, u) we can construct solutions to the Dirac equation that are simultaneously eigenfunctions for the infinitesimal rotation R (4.23). The integer eigenfunctions we construct for R are

164


multivalued solutions of the Dirac equation that change sign under a circuit of 0. These are the eigenfunctions relevant for the Ising model. We seek such eigenfunctions Rwℓ = ℓwℓ as contour integrals ) wℓ (x) = gℓ (u)e(x, u) du, Ŵ

where the contour Ŵ and function gℓ are to be determined. Since Re(x, u) = (−u∂u + 12 )e(x, u), we see that at least formally, Rwℓ (x) =

)

Ŵ

gℓ (u)(−u∂u + 21 )e(x, u) du.

Now suppose that the contour is chosen so that none of the boundary terms survive integration by parts for the u derivative in this last integral. Then ) Rwℓ (x) − ℓwℓ (x) = ∂u (ugℓ (u)) + ( 12 − ℓ)gℓ (u) e(x, u) du. Ŵ

The differential equation for gℓ that makes the integrand of the right-hand side of 3 this last equation vanish identically has solutions gℓ (u) = cℓ uℓ− 2 , where cℓ is a constant. Thus we look for eigenfunctions wℓ (x) in the form wℓ (x) = cℓ

)

1

uℓ− 2 e(x, u)

Ŵ

du . u

1

The function in the integrand uℓ− 2 is multivalued, and this prevents us choosing simple closed contours in C\{0} to eliminate the boundary terms in the integration by parts described above. Instead we will choose contours that flow into the singular points at u = 0 and u = ∞ but do so in directions of exponential decay for the factor e(x, u). The boundary terms for integration by parts vanish as a consequence of this exponential decay. Choose a point b ∈ C\{0}. The contours of interest are pictured in Figure 4.1. Each contour Cj (b, ǫ) as depicted depends not only on b but also on a small parameter ǫ > 0. In both cases ǫ represents the “opening angle” in the contour. Thus the two points e±iǫ b are the endpoints of the arc of the circle that appears in Cj (b, ǫ). In later calculations it is convenient to take the limit ǫ ↓ 0. We will write Cj (b) for this limiting contour. In both contours the ray through 0 and b serves as 1 a branch cut for the multivalued function uℓ− 2 . To describe this we introduce a logarithm, logb , branched along bR + and defined by logb (x) = log |x| + iθ,

(4.31)


b

C1(b,ε)

165

b

C2(b,ε)

Figure 4.1: The two contours used to represent the rotational eigenfunctions for the Dirac equation. with the argument θ chosen in the window arg b < θ < arg b + 2π, and the argument of b chosen such that 0 < arg b < 2π. ¯ + , a complex number that is not a positive real number Now suppose that x ∈ C\R ¯ + is the closure of R + or equivalently the set of nonnegative real numbers). or 0 (R Then it is always possible to find b ∈ C in the left half-plane ℜb < 0 such that x · b > 0. For such a b define ) du 1 ℓ+1 wℓ (x) := (−1) uℓ− 2 e(x, u) , (4.32) 2π u C1 (b,ǫ) where the fractional power in the integrand is uα := eα logb u . The constant cℓ = (−1)ℓ+1 /2π is chosen to simplify the “ladder” relations for the derivatives ∂wℓ ¯ ℓ and the Bessel function representation for wℓ (x) in polar coordinates and ∂w found below. The restriction that b lie in the left half-plane is in place to ensure that (4.32) ¯ + . The defines a single branch of the multivalued function wℓ (x) for x ∈ C\R analyticity and decay properties of the integrand make it easy to see that (4.32) does not depend on ǫ (so we can let ǫ ↓ 0). Cauchy’s theorem implies that if b in (4.32) is replaced by αb, then the resulting integral does not depend on α. It is also true that if b in (4.32) is replaced by b(t) = eit b, then the resulting integral is independent of t, for t in the largest connected interval containing 0 on which b(t) · x > 0. This is again a consequence of Cauchy’s theorem and the fact that as

166


ai

R R

ai

Figure 4.2: On each of the contours depicted above, Cauchy’s theorem applies to 1 (different analytic) choices for the integrand uℓ− 2 e(x, u). Provided the exponential term e(x, u) is small on the appropriate sectors, the contribution from the arcs at radius R will vanish in the limit R → ∞. Combined with a careful accounting of the contributions from the various arcs on the circle this suffices to prove that the contour deformations described in the text are legitimate. 1

the branch cut moves it “opens” up on one side to the analytic continuation of uℓ− 2 1 and “covers over” on the other side an analytic “wedge” of uℓ− 2 . The situation is depicted in Figure 4.2. Thus the definition (4.32) is independent of the choice of b in the left halfplane, since any two points b and b′ in the left half-plane for which b · x > 0 and b′ · x > 0 can be rotated and scaled into one another along a path b(t) that maintains positivity b(t) · x > 0. Thus the prescription (4.32) determines a unique ¯ +. branch for wℓ (x) for x ∈ C\R To rewrite wℓ (x) in polar coordinates for x it is convenient to rotate and scale b x = eiθ (for 0 < θ < 2π ). Of course, eiθ need no longer be in in the definition to |x| the left half-plane. Once this is done, an elementary calculation (for the limiting ǫ ↓ 0 contour) shows that iℓ θ e − I (m|x|) , (4.33) wℓ (x) = iℓ+ θ ℓ− e Iℓ+ (m|x|) where

ℓ± = ℓ ± 21 ,

and the Bessel functions Ik (r) have the integral representation [157] ) # $ dz − r z+z−1 z−k e 2 . Ik (r) = ei(k+1)π 2π iz C2 (1,ǫ)


167

The polar form (4.33) makes it easy to see that the solution wℓ (x) is branched along R + with boundary values from above and below that differ by a sign. In the space of solutions to the Dirac equation in the punctured plane C\{0} there is a two-dimensional family of solutions to the “angular” eigenvalue equation Rw = ℓw. A second independent eigenfunction is obtained by changing the ¯ + and choose b in the left contour from C1 to C2 . As above, suppose that x ∈ C\R half-plane ℜb < 0 and such that x · b > 0. Define ) du 1 ∗ uℓ− 2 e(x, u) w−ℓ (x) := (−1)ℓ+1 . (4.34) 2π u C2 (b,ǫ) ∗ The pair {wℓ , w−ℓ } is a basis for the two-dimensional space of solutions to the Dirac equation in the punctured plane that are eigenfunctions for R with eigenvalue ℓ. The reason for the notation is that the Dirac equation (m − D)ψ = 0 has a conjugation symmetry defined by ψ¯ ψ1 ψ= → ¯ 2 := ψ ∗ . (4.35) ψ2 ψ1

The reader can check that if (m − D)ψ = 0 then (m − D)ψ ∗ = 0 as well. The function wℓ∗ (x) defined by (4.34) is given by the conjugate wℓ∗ (x) = wℓ (x)∗ .

(4.36)

Notice that because of the conjugation, wℓ∗ is an eigenfunction for R with eigenvalue −ℓ rather than ℓ. However, both wℓ (x) and wℓ∗ (x) are less and less locally singular at x = 0 as ℓ increases. This aspect of the indexing is quite convenient for the local expansions that are the reason for introducing these functions. Remark 4.3.1. The conjugation * is a symmetry already on the lattice. In (q, p) coordinates {xℓ , yℓ }, the conjugation is ∗ : {xℓ , yℓ } → {−x¯ℓ , y¯ℓ }. A short calculation shows that this conjugation commutes with the action of the induced rotation Tz (V ) acting on W . This “reality condition” will help us to single out an appropriate Green function a little later in this chapter. Deforming the branch in the contour from b to x in the definition of wℓ∗ (x), as was done for wℓ (x), we obtain −iℓ θ e + Iℓ+ (m|x|) ∗ wℓ (x) = −iℓ− θ , (4.37) e Iℓ− (m|x|) where x = eiθ |x|, 0 < θ < 2π , and ℓ± = ℓ ± 12 . This calculation confirms (4.36); the substitution u←⊣ u¯ −1 in definition (4.34) achieves the same end. At this point it is convenient to introduce the rotational eigenfunctions for the Dirac equation that are well behaved at ∞. Define ∗ wˆ ℓ (x) = wℓ (x) − w−ℓ (x).

(4.38)

168


Then for the contour C(x, ǫ) := C2 (x, ǫ) − C1 (x, ǫ) we have ) du 1 ℓ wˆ ℓ (x) = (−1) uℓ− 2 e(x, u) 2πu C(x,ǫ) $ ) # $ # (−1)ℓ+1 i ℓ− 12 θ ∞ ℓ− 1 − m|x| dr 1 r+r −1 = r 2e 2 e . iθ −re π r 0

(4.39)

The last formula is obtained by collapsing C(x, ǫ) onto the ray r → reiθ , where 0 < |θ| < 2π is the argument of x. From this formula one easily sees that wˆ ℓ (x) decays exponentially fast as |x| → ∞. Differentiating the contour representations (4.32) and (4.34), the two additional infinitesimal symmetries ∂ and ∂¯ for the Dirac operator m − D are seen to act on the angular eigenfunctions m ¯ ℓ = m wℓ+1 , (4.40) ∂wℓ = wℓ−1 , ∂w 2 2 and m ∗ ∗ ¯ ℓ∗ = m wℓ−1 ∂wℓ∗ = wℓ+1 , ∂w . (4.41) 2 2 These ladder relations facilitate the analysis of local expansions and essentially fix this choice of a basis for the angular eigenfunctions. Note that iR, ∂/∂x1 , and ∂/∂x2 are infinitesimal generators for a projective representation of the group of rigid motions of the plane R 2 , which commutes with the Dirac operator. The ladder relations (4.40) and (4.41) make it much simpler to work with the complexified generators ∂ and ∂¯ instead of ∂/∂x1 and ∂/∂x2 .

4.4 Local Expansions In this section we suppose that a is a branch point for a solution w to the Dirac equation (m − D)w = 0, which changes sign when followed around a simple circuit of a. We want to show that w(x) has a convergent expansion w(x) =

∞

ℓ=−∞

cℓ (w)wℓ (x − a) + cℓ∗ (w)wℓ∗ (x − a)

for all x in a punctured neighborhood of a. As mentioned earlier, it is awkward to work with branch cuts and boundary values (although later considerations force us to consider some aspects of these matters). For the present there are a number of ways to circumvent this complication and work in the smooth category. The punctured plane C\{a} = {x ∈ C : x = a} has an easily visualized simply connected covering space, and the functions we want to study are smooth maps on this covering space that change sign under the action of the generator of the fundamental group (the loop that winds once around the origin). This point of view does simplify much of the analysis and is adopted in SMJ III. A telling disadvantage for us is that the simply connected covering becomes much more difficult to deal with explicitly when there is more than one branch point aj , and this makes it

4.4 Local Expansions

169

very clumsy to discuss the Green function G(x, y; a). Another strategem is to construct an appropriate twofold ramified covering space of C\a and work with smooth functions on this space. This works well for the Ising model [37] but not as simply for holonomic fields associated with multipliers e2π iλ when λ is not rational. Instead, we introduce an appropriate line bundle over C\a via explicit trivializations and transition functions. This is the point of view found in [115] and works as well for arbitrary multipliers e2π iλ . We explain this for n points a1 , a2 , . . . , an in Chapter 4. For the present we introduce a line bundle E → C\{a} over the punctured disk about a in the simplest case of just one branch point a. Choose θ = arg(x − a) such that 0 ≤ θ < 2π . The line bundle we define is trivial over each “slit domain” ¯ + }. Write Eθ := E|Br for the restriction of E to Brθ and let Brθ := C\{a + eiθ R θ ϕθ denote a trivialization ϕθ : Eθ → Brθ × C that identifies Eθ with the trivial line bundle

Brθ × C → Brθ

with base Brθ . To define the line bundle E we need only give the transition maps ϕθ ′ ϕθ−1 . Let I (θ, θ ′ ) denote the open interval with endpoints θ and θ ′ , and I¯(θ, θ ′ ) the closure of this interval; then ϕθ ′ ϕθ−1 is multiplication in the fiber by the transition function 1 for arg(x − a) ∈ I (θ, θ ′ ), ϕθθ ′ (x) = −1 for arg(x − a) ∈ / I¯(θ, θ ′ ),

where 0 ≤ arg(x) < 2π . The compatibility ϕθθ ′ ϕθ ′ θ ′′ = ϕθ θ ′′ required for these functions to define a line bundle can be seen by drawing a couple of pictures. The vector bundle we are mostly interested in is E ⊕ E ≃ E ⊗ C2 . This bundle is defined in the same way as E but with fiber C2 instead of C. To avoid a proliferation of notation we simply refer to the vector bundle E when we discuss E ⊗ C2 . The functions wℓ (x − a) and wℓ∗ (x − a) defined in (4.33) and (4.37) can clearly be thought of as sections of the vector bundle E in the ϕ0 trivialization. We write wℓ(0) and wℓ∗(0) for these functions thought of as ϕ0 trivializations. Write wℓ(π) (x −a) and wℓ∗(π) (x − a) for the functions defined in (4.33) and (4.37) with the change that the argument θ = arg(x − a) is chosen such that −π < θ < π. The reader can check that

wℓ(0) (x − a) = ϕ0π (x − a)wℓ(π) (x − a) for x ∈ D0 ∩ Dπ .

Thus the pair {wℓ(0) , wℓ(π) } defines an element of C ∞ (E), the smooth sections of E; we will denote this section by wℓ without worrying too much about the possible confusion with wℓ(0) . The same comment applies as well to wℓ∗ . The reader should check that w is a smooth section of E if and only if the function θ

a + reiθ → e−i 2 w (0) (a + reiθ )

for r > 0 and 0 < θ < 2π

defined on Br0 , extends to a smooth function on C\{a}.

170


Since multiplication by −1 commutes with the Dirac operator, it follows that m − D acts on C ∞ (E). The principal result of this section is our next theorem. Theorem 4.4.1 Suppose R > 0 and let ER denote the restriction of E to the base ˙ D(a, R) := {x : 0 < |x − a| < R}. Suppose that w ∈ C ∞ (ER ) and (m − D)w = 0. Then there are constants cℓ and cℓ∗ such that w(x) =

∞

ℓ=−∞

˙ cℓ wℓ (x − a) + cℓ∗ wℓ∗ (x − a) for x ∈ D(a, R).

(4.42)

Note that cℓ∗ is not determined by cℓ ; the notation is simply intended to remind the reader that cℓ∗ is the coefficient of wℓ∗ . The convergence is uniform on compact ˙ subsets of D(a, R), as is the convergence of derivatives. Proof. Let w (0) denote the function that represents w in the ϕ0 trivialization. Then w ∈ C ∞ (E) if and only if for θ = arg(x − a) and 0 < θ < 2π the function iθ

e− 2 w (0) (x) ˙ extends to a C ∞ function on D(a, R). Let r = |x − a|. Then since it is smooth in r and θ, the preceeding function has a convergent Fourier expansion iθ

e− 2 w (0) (x) = or w(0) (x) =

∞

eiℓθ

ℓ=−∞

∞

ℓ=−∞

1

ei(ℓ+ 2 )θ

aℓ (r) , bℓ (r)

aℓ (r) . bℓ (r)

(4.43)

Because this expansion arises from the Fourier expansion of a smooth function on the circle, the Fourier coefficients are rapidly decreasing in ℓ; this makes it easy to deal with technical aspects of convergence for such series, and we leave the details to the reader. The polar representations e−iθ ∂ i ∂ ∂= − , 2 ∂r r ∂θ i ∂ eiθ ∂ ∂¯ = + , 2 ∂r r ∂θ allow one to formally calculate the action of the Dirac operator m − D on (4.43). Now multiply (m − D)w(0) (x) = 0 by e−iℓ+ θ and then integrate from θ = 0


171

to θ = 2π . Integrate by parts in θ and commute differentiation by r with the calculation of the Fourier coefficient to obtain ′ maℓ (r) − bℓ+1 (r) −

mbℓ (r) −

′ aℓ−1 (r)

+

ℓ+ 23 bℓ+1 (r) r ℓ− 12 aℓ−1 (r) r

= 0,

= 0.

(4.44)

It is convenient to alter the first of these equations by the substitution ℓ←⊣ ℓ − 1 to obtain ℓ + 12 maℓ−1 (r) − bℓ′ (r) − bℓ (r) = 0. (4.45) r This equation and the second equation of (4.44) together show that bℓ (r) satisfies the second-order differential equation 7 8 1 2 ℓ + 1 2 bℓ′′ (r) + bℓ′ (r) − m2 + bℓ (r) = 0. r r2 The general solution to this equation is (see [157]) ∗ bℓ (r) = cℓ Iℓ+ (mr) + c−ℓ I−ℓ+ (mr), ∗ for some constants cℓ and c−ℓ . The first equation of (4.44) and the recurrence relations for modified Bessel functions now show that ∗ aℓ (r) = cℓ+1 Iℓ+ (mr) + c−ℓ−1 I−ℓ+ (mr).

An obvious relabling converts the Fourier series expansion (4.43) into the local expansion (4.42). There is also a useful technical result on the behavior of solutions to the Dirac equation at ∞ that is associated with expansions in the sections wˆ ℓ . For simplicity we take a = 0; the modifications for a different choice of a are obvious. Theorem 4.4.2 Suppose that R > 0 and let E>R denote the restriction of E to the base, {x : |x| > R}.

If w ∈ C ∞ (E>R ), (m − D)w = 0, and w ∈ L2 (E>R ), then there exist constants cˆℓ such that ∞ w(x) = cˆℓ wˆ ℓ (x). ℓ=−∞

The convergence of this series and any series obtained from it by taking a finite number of derivatives is absolute and uniform on subsets of the form {x : |x| ≥ ¯ is any polynomial in ∂ and ∂¯ with constant r0 > R}. Furthermore, if p(∂, ∂) coefficients, then for x large, −mr e ¯ p(∂, ∂)w(x) =O √ , where r = |x|. r

172


Proof. The sections wℓ and wˆ ℓ are independent eigenfunctions for the infinitesimal rotation that commutes with the Dirac operator. As in the preceeding theorem, the Fourier series in θ for w(reiθ ) can be reorganized as a convergent series w(reiθ ) =

∞

ℓ=−∞

cℓ wℓ (reiθ ) + cˆℓ wˆ ℓ (reiθ ).

Since we suppose that )

∞

R

w(reiθ )2 rdrdθ < ∞,

we use the orthogonality of the Fourier expansion to conclude that each Fourier coefficient of w must be square integrable with respect to r dr. In particular, the first component of the coefficient of eiℓ− θ has a multiplier cℓ Iℓ− (mr) + cˆℓ Iℓ− (mr), where ℓ− = ℓ− 12 and Ik (z) := Ik (z)−I−k (z). The first term grows exponentially with r, and the second goes to 0 exponentially fast. To be in L2 near ∞ we must have cℓ = 0 for all ℓ. Thus iθ

w(re ) =

∞

ℓ=−∞

iθ

cˆℓ wˆ ℓ (re ) =

∞

ℓ=−∞

cˆℓ

eiℓ− θ Iℓ− (mr) . eiℓ+ θ Iℓ+ (mr)

(4.46)

Remark 4.4.1. Our reference, Whittaker and Watson [157], defines a modified Bessel function of the second kind Kℓ (z) (the solution that is small at infinity) in a way that makes it identically 0 when ℓ is an odd half-integer, the case we are currently interested in. Thus we introduce Ik (z) rather than sort out a suitable alternative definition for Kk (z). For any r > R the function θ

eiθ → e−i 2 w(reiθ ) is a smooth function on the circle. Thus the Fourier series for this function converges absolutely, ∞ |cˆℓ | · |Iℓ± (mr)| < ∞. (4.47) ℓ=−∞

However, |Iℓ± (mr)| is a monotonically decreasing function of r, and it follows that the series ∞ cˆℓ wˆ ℓ (z) ℓ=−∞


173

coverges # −mr $absolutely and uniformly for all |z| ≥ r0 > R. In order to show that w is O e√r as r → ∞ we need a different representation for Iℓ (z). On page 384 of [157] we find the representation ) 1 21−ℓ sin(ℓπ)zℓ ∞ −zt 2 Iℓ (z) = − 1 e (t − 1)ℓ− 2 dt 1 Ŵ( 2 )Ŵ(ℓ + 2 ) 1 for arg(z) < π2 . Suppose z > 0 is real and introduce the change of variables t = u + 1 followed by u←⊣ z−1 u. One obtains 21−ℓ sin(ℓπ) e−z Iℓ (z) = − 1 √ Ŵ( 2 )Ŵ(ℓ + 21 ) z

)

∞

0

e

−u

u2 u+ z

ℓ− 12

du for z > 0.

Because the function z→

)

∞

0

ℓ− 21 u2 e−u u + du z

is a monotonically decreasing function of z, the representation we have just given for Iℓ implies √ e−mr Iℓ (mr) for r ≥ r0 . ≤ mr0 emr0 √ Iℓ (mr0 ) mr It follows from this and (4.47) that iθ

w(re ) ≤

∞

Hence w(reiθ ) is O

±

#

' ' ' Iℓ± (mr) ' e−mr ' ' ≤ C(r0 ) √ |cˆℓ | · |Iℓ± (mr0 )| ' . ' Iℓ± (mr0 ) mr ℓ=−∞

e√−mr mr

$

as r → ∞. We have shown that any smooth section of # −mr $ E>R that has an expansion (4.46) is O e√mr as r → ∞. However, since ∂ wˆ ℓ = m wˆ and ∂¯ wˆ ℓ = m2 wˆ ℓ+1 , we need only justify the term-by-term differentiation 2 ℓ−1 ¯ we have of the expansion (4.46) in ∂ and ∂¯ to see that for any polynomial p(∂, ∂) −mr e ¯ p(∂, ∂)w =O √ . mr To see that the series for the ∂ and ∂¯ derivatives, ∞ m cˆℓ wˆ ℓ−1 2 ℓ=−∞

and

∞ m cˆℓ wˆ ℓ+1 , 2 ℓ=−∞

converge absolutely at r0 > R, it is enough to show that the series ∞

ℓ=−∞

|cˆℓ | · |Iℓ− 3 (mr0 )| and 2

∞

ℓ=−∞

|cˆℓ | · |Iℓ+ 3 (mr0 )| 2

(4.48)

174


are absolutely convergent. Since (see p. 373 [157]) ℓ Iℓ′ (r) ± Iℓ (r) = Iℓ∓1 , r 2ℓ Iℓ−1 (r) − Iℓ+1 (r) = Iℓ (r), r we obtain

ℓ Iℓ′ (r) = Iℓ−1 (r) − Iℓ (r). r Observe that for ℓ > 0, Iℓ (r) is negative and increases monotonically to 0 as r → 0. It follows that ℓ Iℓ−1 (r) ≥ Iℓ (r), r and since dividing by Iℓ (r) reverses the sense of the inequality, ' ' ' Iℓ−1 (r) ' ℓ ' ' (4.49) ' I (r) ' ≤ r for ℓ > 0. ℓ Because

Iℓ−1 (r) − Iℓ+1 (r) = we see that (4.49) also implies ' ' ' Iℓ+1 (r) ' 3ℓ ' ' ' I (r) ' ≤ r ℓ

2ℓ Iℓ (r), r

for ℓ > 0.

(4.50)

Using the estimates (4.49) and (4.50) in (4.48) in an obvious way we see that the convergence of (4.48) follows from the convergence of the series ∞

ℓ=−∞

|ℓ||cˆℓ | · |Iℓ± (mr0 )|.

Since w is a smooth section in the θ variables, term-by-term differentiation in θ preserves the absolute convergence of the Fourier series. Such differentiation introduces an extra factor asymptotic to |ℓ| in the absolute value of the Fourier ¯ coefficients. This finishes the proof that the series for the derivatives ∂w and ∂w converges absolutely at r = r0 and hence uniformly for r ≥ r0 . The derivatives ∂w ¯ are represented by these series, and the first argument of this proof shows and ∂w # $ ¯ are O e√−mr as r → ∞. An obvious induction finishes the that both ∂w and ∂w mr ¯ proof of the analogous asymptotics for p(∂, ∂)w. It is convenient to record a contour integral representation for the local expansion ˙ coefficients cℓ and cℓ∗ . Suppose that v and w are sections of E over D(a, R) that are solutions to the Dirac equation. Then it is easy to check that the form v1 w1 dz + v2 w2 d z¯

4.5 The Green Function for One Branch Point

175

˙ is a single-valued smooth closed form on D(a, R). The Bessel function identity [157] 2 sin ℓ + 21 π Iℓ+ 1 (mr)I−ℓ+ 1 (mr) − Iℓ− 1 (mr)I−ℓ− 1 (mr) = − 2 2 2 2 π mr allows one to simply check that cℓ (w) = and cℓ∗ (w)

)

im(−1)ℓ+1 4

im(−1)ℓ = 4

Sǫ1 (a)

)

Sǫ1 (a)

w−ℓ,1 w1 dz + w−ℓ,2 w2 d z¯

∗ ∗ w−ℓ,1 w1 dz + w−ℓ,2 w2 d z¯ .

(4.51)

(4.52)

Here Sǫ1 (a) is the circle of radius ǫ < R about a. The deformation invariance for the integrals of the closed forms (4.51) and (4.52) will be useful in a later calculation.

4.5 The Green Function for One Branch Point In this section we use local expansions to determine a Green function G(x, y; a) for the Dirac operator mI − D acting on sections of E, the vector bundle over C\{a}. For simplicity we will locate the “branch point” a at a = 0. Our strategy for computing the Green function is straightforward. The columns and rows of the Green function should satisfy the Dirac equation (away from the diagonal x = y). We will incorporate appropriate branching in the domain by expanding G(x, y; 0) in rotational eigenfunctions (eigenfunctions for the infinitesimal rotation R) with appropriate branching. We will then enforce some expected symmetries and a limitation on the local degree of singularity for functions in the domain of mI − D to obtain a unique result. A resummation using the contour integral representation of the rotational eigenfunctions makes it possible to analyze the convergence of the eigenfunction expansion and to confirm that we do have an appropriate Green function. In what follows, wℓ (x) should be thought of as a 2 × 1 column vector and wℓ (x)τ the corresponding 1 × 2 transposed row vector. Expanding the columns of G(x, y; 0) in the appropriate rotational eigenfunctions yields G(x, y; 0) = wℓ (x)αℓ (y)τ + wℓ∗ (x)βℓ (y)τ for |x| < |y|. (4.53) ℓ

Since G(x, y; 0) encounters a “singularity” at x = y, one might expect the convergence of the series (4.53) to reflect this; we have restricted the domain of the expansion to |x| < |y| in anticipation. In our version of the Green function the rows of G(x, y; 0) should be solutions to the Dirac equation. This is simplest to

176


arrange if we suppose that the functions (sections of E really) αℓ (y) and βℓ (y) satisfy the Dirac equation. We want the operator G(a) defined by ) i dy d y¯ G(x, y; a)Jf (y) G(a)f (x) = 2 2 R to commute with the infinitesimal rotation about a. We wrote G(a) in the last display to make the dependence on the parameter a apparent, but we return immediately to the specific choice a = 0. A little calculation shows that RG(0) = G(0)R, provided Rαℓ (y) = −ℓαℓ (y) and Rβℓ (y) = ℓβℓ (y). We also want the Green function G(x, y; 0) to be in L2 in y near ∞ so that the integral defining G(0) is convergent (at least in a neigborhood of ∞) when f is in L2 . We have seen that ∗ wˆ ℓ (y) := wℓ (y) − w−ℓ (y)

is an eigenfunction, R wˆ ℓ = ℓwˆ ℓ , which tends to 0 as |y| → ∞. In fact, up to a multiple this is the unique such eigenfunction. Thus we suppose that αℓ (y) = αℓ wˆ −ℓ (y)

and βℓ (y) = βℓ wˆ ℓ (y),

for some constants αℓ and βℓ . Arguing that x → G(x, y; 0) should be L2 for x in a neighborhood of ∞ we arrive at the ansatz   αℓ wℓ (x)wˆ −ℓ (y)τ + βℓ wℓ∗ (x)wˆ ℓ (y)τ for |x| < |y|,   ℓ G(x, y; 0) =  γℓ wˆ −ℓ (x)wℓ (y)τ + δℓ wˆ ℓ (x)wℓ∗ (y)τ for |y| < |x|.   ℓ

We now introduce the most ad hoc of our assumptions and limit the local singularity as x → 0 = a by restricting the eigenfunction expansion for G(x, y; 0) to terms with ℓ ≥ 0. As mentioned earlier, a more detailed look at the scaling limit for T ψj (x)σ (a) [118] does suggest this. Our ansatz for the Green function becomes  ∞    αℓ wℓ (x)wˆ −ℓ (y)τ + βℓ wℓ∗ (x)wˆ ℓ (y)τ       ℓ=0 for |x| < |y|, G(x, y; 0) = ∞    γℓ wˆ −ℓ (x)wℓ (y)τ + δℓ wˆ ℓ (x)wℓ∗ (y)τ       ℓ=0 for |y| < |x|.

(4.54)

Finally, we determine appropriate values for the coefficients αℓ , βℓ , γℓ , and δℓ by examining the inversion formula ) idyd y¯ , (4.55) G(x, y; 0)J (mI − D)f (y) f (x) = 2 2 R


177

through a variant of the Stokes’s theorem calculation done above in (4.20). Splitting the integration over R 2 , in this last integral into the domains |y| < |x| and |y| > |x| and using Stokes’s theorem yields ) i dy d y¯ G(x, y; 0)J (I − D)f (y) 2 |y||x| ) G1 (x, y− ; 0)f1 (y) dy + G2 (x, y− ; 0)f2 (y) d y¯ =− ) C|x| G1 (x, y+ ; 0)f1 (y) dy + G2 (x, y+ ; 0)f2 (y) d y, ¯ + C|x|

where C|x| is the counterclockwise-oriented circle of radius |x| about 0 and the vector Gj (x, y± ; 0) is the boundary value of the j th column of G(x, y; 0) as y approaches C|x| from the inside (−) or the outside (+). Note that the boundary terms at |y| = 0 and |y| = ∞ are expected to vanish because the function (section) f (y) may be taken with compact support in the punctured plane R 2 . Define Gi,j (x, y; 0) = Gi,j (x, y+ ; 0) − Gi,j (x, y− ; 0) for |y| = |x| = r and ′ write x = reiθ and y = reiθ . Then the inversion formula (4.55) is confirmed by the Stokes’s theorem calculation, provided ′

′

irG1,1 (reiθ , reiθ ; 0)eiθ = δ(θ − θ ′ ), ′ ′ −irG2,2 (reiθ , reiθ ; 0)e−iθ = δ(θ − θ ′ ),

(4.56)

and ′

G1,2 (reiθ , reiθ ; 0) = 0, ′

G2,1 (reiθ , reiθ ; 0) = 0.

Next we use the polar representations (4.33), (4.37) for wℓ and wℓ∗ to examine these jump conditions. It is straightforward to check that δℓ = −αℓ and γℓ = −βℓ for all ℓ ≥ 0 is necessary and sufficient to make the off-diagonal elements in G equal to 0. The Bessel function identity [157] 2 sin ℓ + 21 π Iℓ+ 1 (mr)I−ℓ+ 1 (mr) − Iℓ− 1 (mr)I−ℓ− 1 (mr) = − 2 2 2 2 π mr implies G1,1 =

∞ iℓθ 2(−1)ℓ −i θ+θ ′ e 2 , αℓ e + βℓ e−iℓθ mπ r ℓ=0

G2,2 = −

∞ ℓ=0

αℓ e

iℓθ

+ βℓ e

−iℓθ

2(−1)ℓ mπ r

e

i θ +θ 2

′

(4.57) ,

178


where θ = θ − θ ′ . Thus the δ-function jumps on the diagonal for G arise, provided α0 + β0 = −

im , 4

αℓ = βℓ = (−1)ℓ+1

im 4

(4.58) for ℓ > 0.

At this point we have found a one-parameter family of Green functions determined by the solutions to the relation α0 +β0 = − m4 i. The reason for the ambiguity is that wˆ 0 (mx) is an L2 solution to (m − D)wˆ 0 = 0, so that multiples of wˆ 0 can be freely added to solutions of (m − D)w = f without altering the fact that the solution is in L2 . We can eliminate this ambiguity by insisting that the Green function define an operator G(0) that commutes with the conjugation (4.35). Such symmetry for G(0) is reflected in the kernel by CG(x, y; 0)C = G(x, y; 0). This forces just one additional constraint, α¯ 0 = −β0 , on the coefficients, so that α0 = β 0 = −

im . 8

The distinguished Green function we have found is thus  ∞    αℓ (wℓ (x)wˆ −ℓ (y)τ + wℓ∗ (x)wˆ ℓ (y)τ ),       ℓ=0 for |x| < |y|. G(x, y; 0) = ∞   − αℓ (wˆ −ℓ (x)wℓ (y)τ + wˆ ℓ (x)wℓ∗ (y)τ ),       ℓ=0 for |y| < |x|, where

αℓ =

 im   −   8

(4.59)

for ℓ = 0,

   (−1)ℓ+1 im for ℓ = 0. 4 Remark 4.5.1. The reader might note that the Dirac operator associated with the Green function G(x, y; 0) just defined is formally symmetric with respect to the bilinear form ) i dx d x¯ f (x)τ J g(x) . 2 R2

This is reflected in the kernel by G(y, x; 0)τ = −G(x, y; 0). This symmetry is a reflection of the formal Hermitian skew symmetry of the Dirac operator D for the Ising boundary conditions. The reader is also cautioned that the skew symmetry of


179

the kernel G(x, y; 0) does not imply the skew symmetry of the projections (4.65) that will make an appearance in a moment because these projections are defined by a limiting procedure that is flipped by the transpose (limits from the interior become limits from the exterior). In order to find contour integral representations for this Green function it is convenient to introduce the Green functions Gt (x, y; 0) associated with the choices α0t := −t im and β0t := (t − 1) im for t = 1 and t = 0. The instance t = 12 that is 4 4 of direct interest for us is just the average of these two Green functions. Let aℓ :=

im (−1)ℓ+1 4

for ℓ = 0, 1, 2, . . . .

Then a reindexing of the summations gives  ∞    aℓ (wℓ∗ (x)wˆ ℓ (y)τ − wℓ+1 (x)wˆ −ℓ−1 (y)τ ),     ℓ=0    for |x| < |y|,  G0 (x, y; 0) =  ∞    ∗  − aℓ (wˆ −ℓ (x)wℓ (y)τ − wˆ ℓ+1 (x)wℓ+1 (y)τ ),       ℓ=0 for |y| < |x|.

(4.60)

Also,

G1 (x, y; 0) =

           

∞ ℓ=0

∗ aℓ (wℓ (x)wˆ −ℓ (y)τ − wℓ+1 (x)wˆ ℓ+1 (y)τ ),

for |x| < |y|,

(4.61)

 ∞     − aℓ (wˆ ℓ (x)wℓ∗ (y)τ − wˆ −ℓ−1 (x)wℓ+1 (y)τ ),       ℓ=0 for |y| < |x|.

For the moment concentrate on G0 (x, y; 0) for |x| < |y|. Recall the contour integral representation (4.39) for wˆ ℓ (y) and the contour C(y) = C2 (y) − C1 (y). Substitute this in the expression for G0 (x, y; 0) and interchange the sum and the integral to obtain (for |x| < |y|) im G (x, y; 0) = − 4 0

)

C(y)

∞ $ dv # 3 1 wℓ+1 (x)v −ℓ− 2 + wℓ∗ (x)v ℓ− 2 e(y, v)τ . 2πv ℓ=0

The contour integral representation for wℓ (x) allows us to write ∞ ℓ=0

wℓ+1 (x)v

−ℓ− 32

) ∞ ℓ = (−1) ℓ=0

C1 (x)

du ℓ+ 1 3 u 2 e(x, u)v −ℓ− 2 . 2π u

180


The contour C1 (x) can be replaced by C1 (ǫx), where ǫ > 0 is chosen such that ǫ|x| < |v|. Of course such an ǫ will in general depend on both x and v, but for fixed x and v the integrand of the du integration is a convergent geometric series, and we obtain 1 1 ) ∞ du u 2 v − 2 −ℓ− 32 wℓ+1 (x)v = e(x, u). C1 (ǫx) 2πu u + v ℓ=0 In a similar fashion we obtain ∞ ℓ=0

1

wℓ∗ (x)v ℓ− 2 = −

)

C2 (ǫ −1 x)

1

1

du u 2 v − 2 e(x, u), 2πu u + v

where now ǫ > 0 must be chosen such that ǫ −1 |x| > |v| (note that x = 0 is a branch cut for this one-point Green function). Substituting these results back into the expression above for G0 (x, y; 0), we obtain the contour integral representation of G0 (x, y; 0) for |x| < |y| given below. Precisely analogous calculations determine the representation of G0 (x, y; 0) for |y| < |x|.

Theorem 4.5.1 Let Gt (x, y; 0) denote the Green function associated with the boundary conditions α0t := −t im and β0t := (t − 1) im . Then for t = 0 and t = 1 4 4 we have the following contour integral representations:  1 1 ) )  dv du u 2 v − 2 im   e(x, u)e(y, v)τ ,   4 C(y) 2πv Cǫ (x) 2πu u + v    for |x| < |y|. G0 (x, y; 0) = (4.62) 1 −1 ) )  2v 2 im du dv u  τ  e(x, u)e(y, v) ,    4 C(x) 2πu Cǫ (y) 2πv u + v   for |y| < |x|, where Cǫ (x) = C2 (ǫ −1 x) − C1 (ǫx) and ǫ is chosen in the first representation such that ǫ|x| < |v| < ǫ −1 |x| and in the second representation such that ǫ|y| < |u| < ǫ −1 |y|. Also,  1 1 ) )  dv du u− 2 v 2 im   e(x, u)e(y, v)τ , −   4 C(y) 2πv Cǫ (x) 2πu u + v    for |x| < |y|. (4.63) G1 (x, y; 0) = ) ) − 12 21  du dv u v im  τ  e(x, u)e(y, v) , −   4 C(x) 2πu Cǫ (y) 2πv u + v   for |y| < |x|,

with the same choices made for ǫ. The reader can see the contours C(y) and Cǫ (x) depicted in Figure 4.3. The Green function which is relevant for the Ising model 1 is G 2 (x, y; 0) = 21 G0 (x, y; 0) + G1 (x, y; 0) .

These formulas play an important role in the calculation of the projections associated with localizations of the Dirac operator. We now turn to this matter.

4.6 Localization and Projections

y

181

x v Cε(x) 0

C( y)

–v

Figure 4.3: The contours in the contour integral representation for the Green function for |x| < |y| with y on the edge of a strip about 0 and x inside the strip. Deforming the outer contour in Cǫ (x) to the inner one encounters a pole at u = −v and collapses the contour to C(x).

4.6 Localization and Projections The principal idea we use to relate the two-dimensional formalism associated with Dirac operator and the one-dimensional transfer formalism that produced the scaling formulas in Theorems 3.4.1 and 3.4.2 is the notion of localization. It is simplest to explain this idea in the context of the differential operator mI − D acting on C ∞ (R 2 , C2 ) (i.e., no branching for functions in the domain). Suppose that g ∈ C0∞ (R 2 , C2 ) (the space of compactly supported smooth functions) and consider the function ) i dy d y¯ . f (x) = G(x, y)J g(y) 2 2 R This will of course satisfy the equation (mI − D)f = g. Now suppose that the support of g is contained in the open upper half-plane ℑy > 0. Then we can write ) i dy d y¯ f (x) = G(x, y)J (mI − D)f , (4.64) 2 2 ¯ H ¯ 2 is the closed upper halfwhere H2 is the open upper half-plane (ℑy > 0) and H plane (ℑy ≥ 0). Consider this equation for ℑx < 0. Because y is in the upper half-plane and x is in the lower half-plane, the integrand is an exact differential ¯ 2 = R, Stokes’s theorem implies for y ∈ H2 (see (4.19)), and since ∂ H ) f (x) = − G1 (x, y)f1 (y) dy + G2 (x, y)f2 (y) d y¯ for ℑx < 0. R

182


We can also express this in terms of boundary values, ) f (x) = − lim G1 (x − iǫ, y)f1 (y) dy + G2 (x − iǫ, y)f2 (y) d y¯ := Q− f (x), ǫ↓0

R

(4.65) for x ∈ R. We define a localization of mI − D to the upper half-plane by restricting the differential operator mI − D to a domain that consists of functions f : H2 → C2 that have boundary values on R in the range of Q− , (4.66)

f |R = Q− f |R . Note that if we define ) f (x) = − G1 (x, y)f1 (y) dy + G2 (x, y)f2 (y) d y¯ R

for ℑx ≤ 0,

then because x → Gj (x, y) is a solution of the Dirac equation for x = y we have (at least formally) (mI − D)f (x) = 0 for ℑx < 0, and f (x) for x ∈ R is the boundary value of a solution to the homogeneous Dirac equation in the lower half-plane. The sense in which this boundary condition defines a localization is straightforward. If g ∈ C0∞ has support in the open upper half-plane, then the function f , which satisfies (mI − D)f = g, is in the domain of the localization when restricted to H2 ; conversely, a solution to (mI − D)f = g that is in the domain of the localization extends to a solution of the same equation defined on the whole of R 2 . It is useful for us to make this discussion more precise by introducing appropriate Hilbert spaces for the domain and boundary values. The Hilbert spaces appropriate for us are L2 Sobolev spaces. Suppose that U is an open subset of R 2 and f is a locally integrable function on U . If there exists a locally integrable function Dk f defined on U such that ) ) dϕ Dk f (x)ϕ(x) dx = − f (x) dx dxk U U

for all ϕ ∈ C0∞ (U ), then f is said to have a weak derivative Dk f. The Sobolev space H 1 (U ) consists of functions f ∈ L2 (U ) that have weak first derivatives Dk f ∈ L2 (U ). The space H 1 (U ) is a Hilbert space with respect to the inner product ) 2 ) f, gH 1 (U ) := f¯(x)g(x) dx + Dk f (x)Dk g(x) dx. U

k=1 1

U

We also define the fractional Sobolev space H 2 (R). For general open subsets U , 1 the fractional Sobolev space H 2 (U ) is subtle to define, but for U = R we can 1 use the Fourier transform to simplify the discussion. The Sobolev space H 2 (R) consists of f ∈ L2 (R) such that ) 1 |fˆ(p)|2 (1 + p 2 ) 2 dp < ∞, R


183

where the Fourier transform fˆ is defined by ) 1 fˆ(p) = √ f (x)e−ixp dx. 2π 1

The space H 2 (R) is a Hilbert space with respect to the inner product ) 1 f, g 1 := fˆ(p)g(p)(1 ˆ + p 2 ) 2 dp. H 2 (R)

R

The language of Sobolev spaces is convenient for the discussion of certain aspects of the localization idea. However, once past some motivational discussions we 1 require only elementary properties of the space H 2 (R). The reader can find the properties of these spaces developed in [80]. The operator Q− defined above extends to a continuous projection on the 1 Sobolev space H 2 (R), and we define a localization of mI − D to the upper halfplane as follows. Let H−1 (H2 ) denote the subspace of the Sobolev space H 1 (H2 ) 1

consisting of functions that have boundary values in the subspace Q− H 2 (R). The restriction of mI − D to the domain H−1 (H2 ) is a proper localization of mI − D defined on H 1 (R 2 ) in the following sense. Suppose that supp(g) is contained in the open upper half-plane. Then there exists a solution f ∈ H−1 (H2 ) to (mI − D)f = g, and this solution f extends to a solution in H 1 (R 2 ) of the same equation. Localizing in the lower half-plane, one encounters the projection ) f (x) = lim G1 (x + iǫ, y)f1 (y) dy + G2 (x + iǫ, y)f2 (y) d y¯ := Q+ f (x), ǫ↓0

R

(4.67) for x ∈ R. These constructions are of interest to us since Q := Q+ − Q− is identified with the scaling limit of the Ising polarization. We next verify the properties of Q± that are of most importance for us by approaching these operators from a different angle. Remark 4.6.1. The Green function representation for projections is probably more familiar to the reader in the case of the ∂¯ operator. The Cauchy kernel (x−y)−1 ¯ The projections on the Hardy spaces of (for x, y ∈ C) is a Green function for ∂. 1 boundary values of H analytic functions in the upper and lower half-planes are given by the familiar formulas ) 1 f (y) 1 H 2 (R) ∋ f (x) → ± lim dy. ǫ↓0 2πi R x − y ± iǫ

It is instructive to rewrite the Dirac equation (mI − D)f = 0 by solving for the x2 derivative ∂2 f to obtain i∂1 −im ∂2 f = f. (4.68) im −i∂1

184


Let fˆ denote the Fourier transform 1 fˆ(p, x2 ) = √ 2π

)

f (x1 , x2 )e−ix1 p dx1

R

of f in the x variable. Then (4.68) becomes −p −im ˆ ˆ ∂2 f = f := −ω(p)P (p)fˆ, im p ( where ω(p) = m2 + p 2 and p im −1 . P (p) = ω(p) −im −p

(4.69)

It is easy to check that P 2 = I and that the multiplication operators associated with Q± (p) := 21 (I ± P (p))

are self-adjoint projections on the ±1 eigenspaces for the multiplication operator 1 P (p) acting on H 2 (R) (which is just a weighted L2 space in the Fourier transform variables). It is easy to see that the forward initial value problem for (4.68) with data f (x1 , x2 )|x2 =0 = f (x1 , 0) is well posed, provided fˆ(p, 0) = Q+ (p)fˆ(p, 0). In this case the solution f (x) to (mI − D)f = 0 in the upper half-plane with boundary value f (x1 , 0) for x2 = 0 is ) 1 f (x) = √ eix1 p−x2 ω(p) fˆ(p, 0) dp for x2 ≥ 0. 2π R We leave it to the reader to check that f H 1 (H2 ) = f

1

H 2 (R)

.

Relations of this sort make the technical discussion of localization (which we don’t much pursue) simpler in the Sobolev space setting. In a similar fashion, the backward initial value problem for (4.68) is well posed if fˆ(p, 0) = Q− (p)fˆ(p, 0). In this circumstance the solution f (x) to (mI −D)f = 0 defined in the lower half-plane with boundary value f (x1 , 0) for x2 = 0 is ) 1 f (x) = √ eix1 p+x2 ω(p) fˆ(p, 0) dp for x2 ≤ 0. 2π R For use in a later calculation and also because it makes possible a simple verification that the projections Q± defined in two different ways above are indeed the same, we introduce a “spectral transform” based on a relative of the uniformization of the curve (p, ω(p)) described in the contour integral calculations above. Define r :=

ω(p) + p , m

(4.70)


185

so that

ω(p) − p . m Note that for p ∈ R we have r ∈ R + (the positive reals). This leads to a rational parametrization m p(r) = r − r −1 , 2 (4.71) m ω(r) = r + r −1 . 2 Note that we have written ω(r) for ω(p(r)) to unburden the notation; this usage should be clear in context. For the spectral representation we are about to introduce, this real r-parametrization of the appropriate “cycles” on the spectral curve is more useful for us than the earlier u-parametrization that made it easy to keep track of the directions for exponential decay in the factor e(x, u). Substitution of (4.71) in the expression for Q± (p) gives 1 r i Q+ (r) = r + r −1 −i r −1 r −1 =

and Q− (r) =

−1 1 r r + r −1 i

−i . r

Define vectors e± (r) that span the range of Q± as follows: iπ 1 1 m− 2 e 4 r2 , e+ (r) := 1 iπ −1 − r +r e 4 r− 2 iπ 1 1 m− 2 e− 4 r 2 e− (r) := . 1 iπ r + r −1 e 4 r − 2

(4.72)

Then Q± (r)e± (r) = e± (r), and with respect to the usual Hermitian symmetric −1 and e+ (r), e− (r) = 0. inner product on C2 we have |e± (r)|2 = m−1 r + r −1 iπ

The choice of the phase factors e± 4 is dictated by a desire to match the spectral transform that arises from scaling the spectral transform on the lattice that was used in Chapters 2 and 3. An explicit comparison can be found in Appendix B. Observe that Q± (r)f = f± e± (r), where

and

1

iπ

1

iπ

f+ = e− 4 e(mr) 2 f1 + e 4 (m/r) 2 f2 iπ

1

iπ

1

f− = e 4 (m/r) 2 f1 + e− 4 (mr) 2 f2 .

Next rewrite the identity

fˆ(p) = Q+ (p)f (p) + Q− (p)f (p)

186


in the r variables to get fˆ(p(r)) = f+ (r)e+ (r) + f− (r)e− (r), where

iπ 1 1 iπ f+ (r) := e− 4 (mr) 2 fˆ1 (p(r)) + e 4 (m/r) 2 fˆ2 (p(r)), 1 1 iπ iπ f− (r) := e 4 (m/r) 2 fˆ1 (p(r)) + e− 4 (mr) 2 fˆ2 (p(r)).

(4.73)

(4.74)

For calculations involving the Green function for the Dirac operator it is useful to rewrite this spectral transform in terms of m ¯ −1 ) 1 e(x, u) = e− 2 (xu+xu −u as f+ (s) = e

− iπ 4

and f− (s) = e

iπ 4

=

ms 2π

=

)

m 2πs

R

)

e(x, −is −1 )τ f (x) dx

(4.75)

e(x, is)τ f (x) dx,

(4.76)

R

where s was used in the argument instead of r for convenience in a later comparison. In this chapter we refer to (4.75) and (4.76) as the spectral transform of f . Now substitute (4.73) in the Fourier inversion formula ) 1 f (x) = √ eixp fˆ(p) dp 2π R and make the change of variables p←⊣ r to get ) ∞ 1 dr ip(r)x f+ (r) e ω(r)[e+ (r), e− (r)] , f (x) = √ f− (r) 2π 0 r which is more conveniently written = ) $ m iπ ∞ dr # 1 1 f (x) = r 2 f+ (r)e(x, ir −1 ) + ir − 2 f− (r)e(x, −ir) . (4.77) e4 2π 2r 0 In this chapter we refer to (4.77) as the inverse spectral transform. Use (4.71), the norms of the vectors e± (r), and the change of variables p←⊣ r to see that ) ) ∞ dr 2 ˆ |f± (r)|2 . |Q± f (p)| dp = (4.78) 2r 0 R Since |Q+ fˆ(p)|2 + |Q− fˆ(p)|2 = |fˆ(p)|2 and the Fourier transform is a unitary map, we see that f+ (r) f → (4.79) f− (r)


187

is a unitary map from L2 (R, dx) to L2 R + , dr (with values in C2 of course). 2r The operators that appear in our Pfaffian representation of the scaling functions . for the Ising model in Theorem 3.4.1 act on the weighted L2 space L2 R + , dr 2r This is consequently where we want to work and is the reason for the particular normalization (4.74) we chose for the spectral transform; however, because H 1 1 functions in the upper (or lower) half-plane restrict continuously to H 2 functions on the boundary, it is easier to tell the localization story if we operate in the 1 fractional Sobolev space H 2 (R) instead. We shall see that it does not matter in 1 the end. The analogue of (4.78) for H 2 (R) is ) ∞ ) m dr |f± (r)|2 ω(r) |Q± f (p)|2 ω(p) dp = . 2r 0 R 1 Thus H 2 (R) maps unitarily onto the weighted L2 space L2 R + , ω(r) m2rdr , under the spectral transform (4.79). Next we check that the Green function formulas reproduce the projections Q± . Suppose that y is real and ℑx > 0 such that ℑ(x − y) > 0. The contour in the representation (4.30) for the Green function G(x, y) can consequently be deformed to iR + . Do this and then parametrize the resulting integral by u = ir for r ∈ R + . Change variables r←⊣ r −1 in the resulting integral to obtain ) ∞ m dr re(x, ir −1 )e(y, −ir −1 )τ for ℑ(x − y) > 0. (4.80) G(x, y) = 4πr 0 Since dy = d y¯ on the real axis, the contour integral (4.67) we are interested in is ) G(x, y)f (y) dy for ℑx > 0. R

Substitute (4.80) in this integral and recall (4.75) to find that = ) ) m ∞ dr 1 iπ 4 r 2 f+ (r)e(x, ir −1 ). G(x, y)f (y) dy = e 2π 0 2r R Compare this with (4.77) to see that the limit as x2 ↓ 0 is just Q+ f. If ℑ(x − y) < 0 then the contour representation (4.30) for G(x, y) may be deformed to u ∈ −iR + . Parametrize u = −ir for r ∈ R + to obtain ) ∞ m dr −1 G(x, y) = − r e(x, −ir)e(y, ir)τ for ℑ(x − y) < 0. (4.81) 4π 0 Substitute (4.81) in −

)

G(x, y)f (y) dy

R

and make use of (4.75) to obtain = ) ) m ∞ dr − 1 iπ 4 r 2 f− (r)e(x, −ir). − G(x, y)f (y) dy = e i 2π 0 2r R

188


Consult (4.77) to see that in the limit x2 ↑ 0 this is Q− f . The localization that is important for the Ising model is a localization of the twisted Dirac operator (mI − D) with a domain that reflects Ising-type branch points at aj ∈ R 2 for j = 1, 2, . . . , n. The localization takes place in the complement of the union of disjoint horizontal strips. The horizontal strips Sj are chosen to contain the branch points aj and associated branch cuts, the horizontal rays emerging to the right from each point aj for the “functions” in the domain of (mI − D). We first consider the case of just one branch point a ∈ R 2 . Let = (L , U ) denote a pair of positive real numbers and write S (a) for the horizontal strip S (a) := {x : a2 − L < x2 < a2 + U }. When the point a and parameter don’t have an important role or are just apparent we unburden the notation to write S = S (a). The natural orientation of the boundary ∂S (more precisely the boundary of the closure of S) associated with Stokes’s theorem is reflected by the positive parametrizations (−∞, ∞) ∋ t → (−t, a2 + U ),

(−∞, ∞) ∋ t → (t, a2 − L )

(i.e., the upper boundary of S is “negatively oriented”). Write L = {x : x2 = a2 − L } and U = {x : x2 = a2 + U }

for the lower and upper boundaries of S. Suppose f is a function defined on R 2 ; we write f L and f U for the restrictions of f to L and U respectively. Also we write f± := Q± f for the function obtained from f by the action of Q± on the function of x1 obtained by fixing the second variable x2 . For the next result it is useful to introduce a direct sum decomposition of the fractional Sobolev space 1 H 2 (∂S). Write  L  L  U  f+ f f+ f = → ⊕  f−L f−U fU 1

for the isometry that takes H 2 (∂S) to the direct sum of the subspaces 1

1

X(∂S) := Q+ H 2 (L) ⊕ Q− H 2 (U ), 1 1 Y (∂S) := Q+ H 2 (U ) ⊕ Q− H 2 (L).

(4.82) 1

This splitting is of interest to us because two of the subspaces of H 2 (∂S) we are concerned with are both graphs over the subspace X(∂S). A subspace W ⊂ 1 H 2 (∂S) is a graph over X(∂S) if there exists a continuous linear map Z : X(∂S) → Y (∂S)


189

such that W = {x + Zx : x ∈ X(∂S)}.

The image W of the projection

Pf (x) = lim ′

S∋x →x

)

G(x ′ , y)f (y) dy

(4.83)

∂S

is one of these two subspaces. The limit x ′ → x is taken nontangentially from the interior of S. We leave it to the reader to check that this subspace is a graph over X(∂S) with L U e−( + )ω 0 . (4.84) Z= L U 0 e−( + )ω We can safely leave this to the reader, since we exhibit in detail the calculations needed to deal with the second of the subspaces of interest. These more elaborate calculations specialize in a simple way to confirm (4.84). The following theorem connects the projection onto the localization subspace for a single branch point with the scaling limit calculations of Chapter 2. At this point we have transformed the rotational characterization of the Green function into an appropriate action on the strips that enter into the “transfer formalism.” 1

Theorem 4.6.1 Suppose that f ∈ H 2 (∂S). Define ) G(x ′ , y; a)f (y) dy. Pr(a)f (x) = lim ′ S∋x →x

(4.85)

∂S

The limit is taken nontangentially as x ′ → x from the interior of S. Then the action of Pr(a) in the direct sum decomposition (4.82) is given by  L  L f+ f+ (4.86) Pr(a)f =   + Z(a)   , f−U f−U where

Z(a) =

α γ

β δ

1

1

is the matrix of a map from X(∂S) to Y (∂S). Thus, α : Q+ H 2 (L) → Q+ H 2 (U ), 1 1 β : Q− H 2 (U ) → Q+ H 2 (U ), etc. In the spectral representation (4.75) and (4.76) 1 1 for the subspaces Q± H 2 (L) and Q± H 2 (U ), which identifies all these subspaces ), the maps α, β, γ , and δ are given by integral operators with L2 (R + , ω(v) dv v ) ∞ r + s U,L ds αf+L (r) = ea, (r, s)f+L (s) , (4.87) r −s 2π is 0 ) ∞ 1 − rs U,U ds βf−U (r) = − ea, (r, s)f−U (s) , (4.88) 1 + rs 2π is )0 ∞ ds 1 − rs L,L γf+L (r) = − ea, (r, s)f+L (s) , (4.89) 1 + rs 2π is 0

190


and δf−U (r) = −

)

∞

0

The exponential factor ea, (r, s) is

r + s L,U ds ea, (r, s)f−U (s) . r −s 2π is

X,Y ea, (r, s) := eix(p(r)−p(s))−

X ω(r)−Y ω(s)

(4.90)

,

with a = (a1 , a2 ),

U = y(U ) − a2 ,

and L = a2 − y(L).

The singular kernels are understood in the sense of the principal value 1 r +s r +s r +s = + . r −s 2 r − s − i0 r − s + i0 Proof. Obvious translation invariance shows that it is enough consider a = 0. To evaluate (4.85) we use the formulas (4.62) and (4.63). We rewrite these formulas for the Green function G(x, y; 0) depending on the disposition of the arguments x and y. There are four cases to consider: (1) ℑy < ℑx < 0, (2) ℑy < 0 < ℑx, (3) 0 < ℑx < ℑy, and (4) ℑx < 0 < ℑy. We concentrate on the details for cases (2) and (3) and leave the details of the other two cases to the reader. Case (2). Suppose that ℑy < 0 < ℑx. In the formula (4.62) for G0 (x, y; 0) one can rotate the contours C(x) and Cǫ (x) to C(i) and Cǫ (i) (since ℑx > 0) and one can rotate the contours C(y) and Cǫ (y) to C(−i) and Cǫ (−i) (since ℑy < 0). Then it is possible to collapse the contours Cǫ (±i) to C(±i) without encountering any poles. The situation is pictured in Figure 4.3 and the reader should keep in mind that the contours C(±i) are shorthand for the limiting contours described in (4.58) above. One finds that for both |x| < |y| and |y| < |x| the Green function G0 (x, y; 0) for ℑx > 0 > ℑy is given by im G (x, y; 0) = 4 0

)

C(−i)

dv 2πv

1

)

C(i)

1

du u 2 v − 2 e(x, u)e(y, v)τ . 2πu u + v

Consulting (4.75) we see that e(y, is)τ or e(y, −is −1 )τ occurs in the spectral transform formula. Because v lives on the contour C(−i), the second parametrization will give a match for the spectral transform. Thus we parametrize the incoming branch of C(−i) by R + ∋ s → −is −1 . On the incoming branch of C(−i) the , so on this branch, argument of v is 3π 2 1

v − 2 = e−i

3π 4

1

s2. 1

The orientation of the outgoing branch and the sign flip for v − 2 cancel, and we find that the integration over v is just twice the result for the incoming branch, or −

im −i 3π e 4 2

)

0

∞

ds 2πs

)

C(i)

1

1

du u 2 s 2 e(x, u)e(y, −is −1 )τ , 2πu u − is −1

(4.91)


191

−1

since dss −1 = − dss . Now consult the formula (4.77) for the inverse spectral transform to see that either e(x, ir −1 ) or e(x, −ir) occurs. Since u ∈ C(i) we see that it will be simplest to parametrize the incoming branch of C(i) by u = ir −1 − 0. On the incoming branch of C(i) the argument of u is π2 , so that on this branch, 1

π

1

u 2 = ei 4 r − 2 . Thus the contribution to the Green function from the incoming branch of C(i) in (4.91) is just 1 1 ) ) ∞ dr r− 2 s 2 m ∞ ds e(x, ir −1 )e(y, −is −1 )τ . − 2 0 2πs 0 2πr ir −1 − is −1 − 0 1

The sign flip for u 2 and the change of orientation on the outgoing branch of C(i) again cancel, but u = ir −1 +0 produces a different boundary value in the outgoing contribution. Introducing the principal value 1 1 1 1 := + , r −s 2 r − s − i0 r − s + i0 we find that for ℑx > 0 > ℑy, 1 3 ) ∞ ) ∞ dr r 2 s 2 ds e(x, ir −1 )e(y, −is −1 )τ . G0 (x, y; 0) = −im 2πs 2πr r − s 0 0

The calculation for G1 (x, y; 0) is much the same. The only significant change 1 1 occurs in an intermediate calculation and arises from the factor u− 2 v 2 , which is 1 1 the inverse of u 2 v − 2 . The phase of this term changes by a sign. We obtain 3 1 ) ∞ ) ∞ dr r 2 s 2 ds e(x, ir −1 )e(y, −is −1 )τ . G1 (x, y; 0) = −im 2πs 0 2πr r − s 0 Recall that G(x, y; 0) = 12 G0 (x, y; 0) + G1 (x, y; 0) , so that ) ) ∞ dr 1 r + s 1 m ∞ ds r2 s 2 e(x, ir −1 )e(y, −is −1 )τ , G(x, y; 0) = −i 2 0 2πs 0 2πr r − s (4.92) for ℑy < 0 < ℑx. Of special interest for us is x = x1 + iU and y = y1 − iL . It is easy to check that e(x1 + iU , ir −1 ) = e−

e(y1 − iL , −is −1 ) = e−

U ω(r) L ω(s)

e(x1 , ir −1 ),

e(y1 , −is −1 ).

Substituting this in (4.92) and consulting the spectral transform (4.75) and its inverse (4.77), one finds that the contribution made by (4.92) to Pr(a)f (x) is given by ) G(x, y; 0)f (y) dy for x ∈ U, L

which is (4.87) above.

192


Case 3. We suppose 0 < ℑx < ℑy and consider G0 (x, y) in the case |x| < |y|, so that by (4.62), im G (x, y; 0) = 4 0

)

C(y)

dv 2πv

1

)

Cǫ (x)

1

du u 2 v − 2 e(x, u)e(y, v)τ . 2πu u + v

(4.93)

We want to collapse the contour Cǫ (x) to C(x) to pick up a contribution from the pole at u = −v. This can always be done, but if it is not done carefully the contribution from the pole term may not be integrable in v (this is less of an issue here than it is for the case |x| > |y|). Note that the pole term has an exponential contribution arising from e(x, −v)e(y, v)τ . This is small as v tends to 0 or ∞, provided (x − y) · v < 0. Since ℑ(x − y) < 0, the inner product (x − y) · v is negative for v ∈ iR + . Because ℑy > 0, we can take advantage of this by rotating the contour C(y) in (4.93) to C(i). Do this and then use ℑx > 0 to rotate the contour 1 1 Cǫ (x) to Cǫ (i). Collapse the contour Cǫ (i) to C(i) and note that (−v) 2 v 2 = −i for v on the outgoing branch of C(i). One obtains the pole contribution ) ) ∞ m dv −1 m ds −1 − v e(x, −v)e(y, v)τ = − s e(x, −is)e(y, is)τ . + 4π v 4π s iR 0 (4.94) Deform the ray (x −y)R + in (4.30) to −iR + and parametrize the resulting integral by u = −is to see that the pole contribution (4.94) is just G(x, y) (the “free” Green function). Thus for |x| < |y| and 0 < ℑx < ℑy we have )

1

)

1

du u 2 v − 2 e(x, u)e(y, v)τ . C(i) C(i) 2π u u + v (4.95) Consult (4.75) again to see that ∓is ∓1 occurs as an argument in the spectral transform. This suggests the parametrization v = is for the outgoing branch of C(i) in (4.95). The inverse spectral transform (4.77) suggests the parametrization u = ir −1 for the incoming branch of C(i) in (4.95). Adopt these parametrizations in (4.95) to see that 1 1 ) ∞ ) ∞ ds dr r 2 s − 2 G0 (x, y; 0) = G(x, y) + m e(x, ir −1 )e(y, is)τ 2πs 2πr 1 + rs 0 0 (4.96) for |x| < |y| and 0 < ℑx < ℑy. It is not surprising that the same representation is valid for G0 (x, y; 0) when |y| < |x| and 0 < ℑx < ℑy, but there is a variation in the calculation that proceeds from (4.62), which we now describe. For |y| < |x| and 0 < ℑx < ℑy, G0 (x, y; 0) = G(x, y) +

G0 (x, y; 0) =

im 4

)

C(x)

im 4

du 2πu

dv 2πv

)

Cǫ (y)

1

1

dv u 2 v − 2 e(x, u)e(y, v)τ . 2πv u + v

(4.97)

As above, we want to collapse the Cǫ (y) contour to C(y), picking up a pole at v = −u. The exponential terms in e(x, u)e(y, −u)τ will be small for u close to 0


193

and ∞, provided (x − y) · u > 0. Since x − y is in the lower half-plane, it is not suitable to take u ∈ iR + . Instead, since x · y > 0, it is permissible to rotate the contour Cǫ (y) in (4.97) to Cǫ (x). Since |x| > |y|, it follows that (x − y) · x > 0 and hence (x − y) · u > 0 for u ∈ C(x). The pole contribution that now arises from the collapse of Cǫ (x) to C(x) is a contour integral over C(x). Since x − y is in the lower half-plane, this contour may be rotated to C(−i) and the result identified as G(x, y), as was done above for |x| < |y|. The C(x) contours in the double integral that remains can both be rotated to C(i), and the calculation of this term is then exactly the same as (4.95). The upshot is that (4.96) is valid for |y| < |x| as well. The same techniques work for G1 (x, y; 0) and we obtain )

∞

ds 2πs

)

∞

3

1

dr r 2 s 2 e(x, ir −1 )e(y, is)τ . G (x, y; 0) = G(x, y) − m 2πr 1 + rs 0 0 Since G(x, y; 0) = 21 G0 (x, y; 0) + G1 (x, y; 0) , we see that 1

G(x, y; 0)

)

∞

)

∞

dr 1 1 − rs − 1 r2 s 2 e(x, ir −1 )e(y, is)τ 2πr 1 + rs 0 0 (4.98) for 0 < ℑx < ℑy. The contribution this makes to Pr(a)f (x) is the boundary value for x ↑ U of ) − G(x, y; 0)f (y) dy. = G(x, y) +

m 2

ds 2πs

U

Comparing this with the spectral transform and its inverse, we see that the double integral in (4.98) makes the contribution (4.89) to Pr(a)f . The “free” Green function G(x, y) makes the contribution [0, f−U ]τ to Pr(a)f . We leave it to the reader to verify that G(x, y; 0) is given by ) ) ∞ m ∞ ds dr − 1 1 − rs 1 G(x, y) + r 2 s 2 e(x, −ir)e(y, −is −1 )τ 2 0 2πs 0 2πr 1 + rs for ℑy < ℑx < 0 and m G(x, y; 0) = −i 2

)

0

∞

ds 2πs

)

0

∞

dr − 1 r + s − 1 r 2 s 2 e(x, −ir)e(y, is)τ 2πr r −s

for ℑx < 0 < ℑy. These representations simply confirm the remaining assertions of the theorem. There is another way to think about the graph representation of the subspace Wint (a), that is the image of the projection Pr(a) defined above, that makes a connection between the operators D −1 , BD −1 , D −1 C, and D −τ in Theorems 3.4.1 and 3.4.2 and the maps α, β, γ , and δ that appear in the preceding Theorem 4.6.1. This is only motivational for us, so we will proceed somewhat informally. Let a = (x, y) ∈ R 2 and define L = {x ∈ R 2 : x2 = y − L } and U = {x ∈ 2 R : x2 = y + eU }. A pair of functions (FL , FU ) is in Wint (a) provided that FU is

194


obtained from FL in the following manner. First transfer FL from x2 = y − U to x2 = y via the free Dirac propagator. Multiply the result by −sgn(x1 − x). Then transfer the result from x2 = y to x2 = y +U again via the free Dirac propagator. The result is FU if (FL , FU ) is in Wint (a). Let Q± denote the projection on the boundary values of well-behaved solutions to the Dirac equation from the upper (+) and lower (−) half-planes as above. Write L U and FU = f−U + g+ , FL = f+L + g− To denote the splittings of FL and FU according to the polarization Q. The somewhat odd choice of f and g for this splitting is done because we want to describe the subspace by solving for g in terms of f . It will be convenient to write vectors as column vectors according to the polarization W+ ⊕ W− . Let A B C D

denote the matrix of multiplication by −sgn(x1 − x) relative to the splitting W+ ⊕ W− (for simplicity we write A, B, C, D for what we earlier denoted by A(x), B(x), etc.). Then for (FL , FU ) to be in Wint (a) we want U −U ω L e 0 A B e− ω g+ 0 f+L Uω Lω U = L . f− C D g− 0 e 0 e L We can use the equation for f−U to solve for g− . The result is L

L

L

L g− = −e− ω D −1 Ce− ω f+L + e− ω D −1 e−

Uω

f−U .

(4.99)

U L Substituting this result for g− yields in the equation for g+ U g+ = e−

Uω

L

D −τ e− ω f+L + e−

Uω

BD −1 e−

Uω

f−U ,

(4.100)

where we used A−BD −1 C = D −τ , which follows from the fact that multiplication by −sgn(x1 − x) is complex orthogonal. Note that although Dirac propagation involves the action of the unbounded operator eǫω on W− , the graph representation of the subspace by (4.99) and (4.100) involves only bounded operators. Comparing this with Theorem 4.6.1, we expect that α = e−

β = e− γ = −e

Uω

D −τ e− ω ,

L

Uω

BD −1 e−

−L ω L

D −1 Ce

δ = e− ω D −1 e−

Uω

,

−L ω

Uω

(4.101) ,

.

These identities are central to establishing the connection between the scaling functions for the Ising model and the transfer-matrix calculation of tau functions


195

that we turn to in the next chapter. Rather than trying to make the argument that we have given rigorous, we confirm (4.101) by simply comparing the integral representations for D −1 , D −τ , BD −1 , and D −1 C that one finds in Remark 3.4.1 with the integral representations for α, β, γ , and δ that one finds in Theorem 4.6.1. The identities (4.101) are immediate.

5 Scaling Functions as Tau Functions

In this chapter we introduce tau functions associated with Dirac operators acting on the sections of a vector bundle over the punctured plane C\a, where we write a := {a1 , a2 , . . . , an }. In order to make comparisons with the formulas for the scaling functions of the Ising model found in Chapter 3, we develop a “transfer formalism” to study these Dirac operators. The points ai are isolated in disjoint strips Sj . The boundary conditions at the points ai for i = 1, . . . , n that determine the domains of the [±] (a) Dirac operators relevant to the Ising model are reflected in subspaces Wint 1

of H 2 (∂S). Here S = ∪i Si , and ∂S = ∪i ∂Si is the union of the boundaries 1 ∂Si ; H 2 (∂S) is the Sobolev space of order 21 , and [±] indexes the sub ([−]) and [±] super ([+]) critical boundary conditions. The subspaces Wint (a) are the boundary conditions for the localization of the Dirac operators to the exterior of the union of the strips Si for i = 1, . . . , n. These subspaces belong to a Grassmannian that $ is # [±] (a) for the base for a holomorphic line bundle det ∗ , and the tau functions τ Wint

the Dirac operators are defined by trivializations of the det ∗ bundle over the family [±] (a). The situation is quite analogous to the Segal–Wilson [142] of subspaces Wint interpretation of the τ functions introduced by Sato Miwa, Jimbo, and collaborators for KdV and related equations. The principal results of this chapter are (Theorems 5.1.1 and 5.2.1) $ τ [−] (ma) 2 # [−] [−] Wint (a) = τ τ [−] (ma0 )

198

5. Scaling Functions as Tau Functions

and

[+] 2 # $ τ (ma) [+] n Wint (a) = i τ . τ [−] (ma0 ) Here τ [±] (ma) are the sub- and supercritical scaling functions for the Ising model and a0 is a reference point used to define the Grassmannian. Note that in the second formula it is τ [−] (ma0 ) that appears rather than τ [+] (ma). The connection between the mass m in the Dirac operator m − D and the scaling behavior of the functions τ [±] can be seen in these formulas. The vector bundle of interest is a generalization to n points of the vector bundle introduced in Section 4.4, but we do not introduce it explicitly until the next chapter. We get away with this because the localization by strips allows us to fix a trivialization for the bundle that isolates the branch cuts for the relevant sections inside the strips (this is the O0 , trivialization of the next chapter). Such sections restrict to smooth functions on ∂S, and there is no need to introduce the bundle E. However, in the following chapter, which is devoted to the deformation analysis of these τ functions, we localize in the complement of the union of a family of disjoint disks Di containing the points ai ; for this localization there is no single trivialization that reflects the appropriate smoothness everywhere. [+]

5.1 The Subcritical Tau Function In this section we localize the Dirac operators of interest in the complement of horizontal strips about each of the branch points aj for j = 1, . . . , n. These lo[−] (a) calizations are defined by boundary conditions associated with subspaces Wint that are elements of an infinite-dimensional Grassmannian. Remark 5.1.1. Because we have systematically used ± to index polarizations and now we want to use ± to index limits from above and below the critical temperature Tc , we again introduce the modified convention [−] ≃ from T < Tc , [+] ≃ from T > Tc .

Living over the Grassmannian of boundary conditions for localizations of the Dirac operator is a line bundle det ∗ , the dual of the determinant bundle, first defined by Segal and Wilson [142]. We define a τ function for the localization by trivializing [−] the det ∗ bundle over the family of subspaces Wint (a). As in Segal and Wilson’s work on τ functions for the KdV hierarchy, we define a τ function as the ratio [−] # $ σ (Wint (a)) [−] , τ Wint (a) := [−] δ(Wint (a))

where a → σ (a) is the canonical section and a → δ(a) is the trivialization. The principal result is # $ τ [−] (ma) 2 [−] τ Wint (a) = , (5.1) τ [−] (ma0 )

5.1 The Subcritical Tau Function

199

where τ [−] (ma) is the scaling function for the Ising correlations from below Tc , and a0 is a reference point used to define the Grassmannian. This result is the key ingredient in the deformation analysis of τ [−] (ma), which is worked out in Chapter 6. Working in the det ∗ bundle produces a square in the relation (5.1). There is “real” Grassmannian and a Pfaffian bundle over it that eliminates the square in this relation, but it is more complicated to describe, so we don’t take it up here (but see [111]). Since the deformation analysis applies to the logarithmic derivative # $ [−] da log τ Wint (a) = 2da log τ [−] (ma),

not much, apart from a more satisfying theoretical match, is gained for the additional effort. In order to concentrate on some computational details in this section a number of technical details here are deferred until Chapter 6, where they fit more naturally into the functional-analytic framework developed there. Suppose that aj0 = (xj0 , yj0 ) ∈ R 2

and

for j = 1, . . . , n,

y10 < y20 < · · · < yn0 .

For ǫ > 0 let Sj denote the horizontal strip

Sj := {(x, y) : |y − yj0 | < ǫ},

and write

Uj = {(x, yj0 + ǫ) : x ∈ R},

Lj = {(x, yj0 − ǫ) : x ∈ R},

for the upper and lower components of the boundary of Sj . Choose ǫ > 0 so small that the closed strips S¯j are disjoint. That is, S¯j ∩ S¯k = ∅ for j = k. The Grassmannian we are interested in is a family of closed subspaces in the Hilbert space 1

H 2 (∂S) := 1

n *

1

H 2 (∂Sj ),

j =1

where H 2 (∂Sj ) is the fractional Sobolev space of order 21 for the boundary ∂Sj of the j th strip Sj and we write S = ∪j Sj . Choose aj ∈ Bǫ (aj0 ), the open ball of radius ǫ about aj0 . Then aj ∈ Sj for each j and and where

Uj = {(x, yj + Uj ) : x ∈ R} Lj = {(x, yj − Lj ) : x ∈ R}, yj + Uj = yj0 + ǫ, yj − Lj = yj0 − ǫ.

200


y = y(Uj)

aj

∆Uj

Sj y = y(Lj)

L y =yj ∆ j

∆j y = y(Uj –1)

∆j

aj–1

∆Uj

Sj–1 L

∆ j– 1

y = y(Lj –1)

Figure 5.1: The strips that occur in the localization of the Dirac operator. The coordinates of the branch point aj = (xj , yj ) and the average y¯j = (yj + yj −1 )/2. ¯ j and j arise in the formulas for the scaled correlations and The increments ¯ U,L in the formulas for the tau functions. The increments and U,L arise in the j j formulas that connect the scaling limit and tau function formalisms. The situation is pictured in Figure 5.1. 1 [−] For each j = 1, . . . , n there is a closed subspace Wint (aj ) ⊂ H 2 (∂Sj ) that is the image of the projection Pr(aj ) defined by the nontangential limit from the interior, ) Pr(aj )f (x) = lim G(x ′ , y; aj )f (y)dy. ′ Sj ∋x →x

∂Sj

That is, 1

[−] Wint (aj ) := Pr(aj )H 2 (∂Sj ).

The subscript “int” is short for “interior,” and refers to the fact that the functions in this subspace are boundary values of functions that are branched solutions to the [−] Dirac equation in the interior of Sj . The functions in Wint (aj ) extend to functions in Sj that are well behaved enough at aj to be in the domain of the twisted Dirac operator that arises for the subcritial scaling limit of the Ising model. Of principal interest for us is the family of subspaces [−] Wint (a) =

n * j =1

1

[−] Wint (aj ) ⊂ H 2 (∂S). 1

[−] 0 Write W0 for Wint (a ) and define a subspace Wext ⊂ H 2 (∂S) that consists of the 1

H 2 boundary values of solutions to the Dirac equation (mI − D)f (x) = 0 for


201

x ∈ R 2 \S with f ∈ H 1 (R 2 \S) (the Sobolev space of order 1). In Chapter 6 we show that 1 [−] H 2 (∂S) = Wint (a) + Wext is a continuous direct sum decomposition. This allows us to introduce a variant of the Segal–Wilson Grassmannian. Let Gr(W0 ) denote the space of closed subpaces 1 of H 2 (∂S) that are “close to W0 ” in the following sense. Write P0 for the projection 1 from H 2 (∂S) onto W0 along Wext . A closed subspace W is in Gr(W0 ) if P0 : W → W0 is Fredholm with index 0 and 1 − P0 : W → Wext is compact. In Segal and Wilson’s paper the map P0 : W → W0 is required only to be Fredholm. The restriction we make to Fredholm maps with index 0 keeps us in the connected component of the Grassmannian containing the subspace W0 and is sufficient for our purpose (at least for the scaling limit from below Tc ). In order to describe the line bundle det ∗ we first introduce the notion of an admissible frame. An isomorphism F : W0 → W is said to be an admissible frame for the subspace W if P0 F : W0 → W0 is a trace class perturbation of the indentity. The fiber of the line bundle, det∗ → Gr(W0 ), over the subspace W ∈ Gr(W0 ) is an equivalence class of ordered pairs (F, λ), where F : W0 → W is an admissible frame, λ ∈ C, and (F1 , λ1 ) ≃ (F2 , λ2 ) if and only if λ1 = λ2 det(F2−1 F1 ). It is easy to check that since both F1 and F2 are admissible frames, the map F2−1 F1 : W0 → W0 is a trace class perturbation of the identity and hence has a determinant [147]. The linear structure on the fiber is (F, λ1 ) + (F, λ2 ) = (F, λ1 + λ2 ), which the reader may verify is independent of the frame F . The technical elaboration of these matters is considered in more detail in [126] and [142] but will not much concern us. We are principally interested in the calculation of a τ function that emerges from the det ∗ construction and that will directly relate to the scaling function τ [−] (ma) defined in Chapter 3.

202


The canonical section of the det ∗ bundle vanishes over those subspaces W ∈ Gr(W0 ) for which the projection P0 : W → W0 fails to be invertible. On the other hand, if the restriction of P0 to W is invertible, then the inverse P−1 0 : W0 → W is an admissible frame that defines the canonical section over such W ∈ Gr(W0 ). It [−] is convenient at this point to note that Wint (a) ∈ Gr(W0 ); a simple proof of this fact [−] is given in Chapter 6. For the family of subspaces Wint (a) there is an alternative representation of the inverse of the projection P0 that is useful. Let P[−] int (a) denote 1

[−] [−] [−] the projection of H 2 (∂S) onto Wint (a) (a) : W0 → Wint (a) along Wext . Then Pint [−] [−] inverts the projection P0 : Wint (a) → W0 . If f ∈ Wint (a) is the sum f0 + fext , [−] (a)f0 = f , since both where f0 ∈ W0 and fext ∈ Wext , then P0 f = f0 and Pint projections are along the same complementary subspace Wext . The map [−] [−] (a), 1) (a)) = (Pint σ (Wint

is thus a representation of the canonical section of det ∗ over the family of subspaces [−] a → Wint (a). The trivialization of det ∗ over this family of subspaces in which we are interested is obtained from the direct sum of the projections Pr(a) := Pr(an ) ⊕ · · · ⊕ Pr(a1 ). Note the right-to-left ordering; this is the ordering that we will use for all matrix representations. The restriction of Pr(a) to W0 is an admissible frame (confirmed in Chapter 6), and we define the trivialization [−] (a)) := (Pr(a), 1). δ(Wint [−] [−] [−] (a) and Pr(a) (a) to W0 with range Wint (a) for the restriction of Pint Writing Pint [−] for the restriction of Pr(a) to W0 with range Wint (a), we see that

# # $$ [−] [−] (Pint (a), 1) ≃ Pr(a), det Pint (a)−1 Pr(a) .

Thus if we define

[−] τ (Wint (a)) :=

[−] (a)) σ (Wint [−] (a)) δ(Wint

,

we obtain $ # [−] [−] τ (Wint (a)−1 Pr(a) . (a)) = det Pint

The principal result of this section is the following theorem.

(5.2)


203

Theorem 5.1.1 For any set of points {a1 , a2 , . . . , an } with aj in the j th strip Sj , we have [−] 2 τ (ma) [−] τ (Wint (a)) = . τ [−] (ma0 ) We turn to the calculation of (5.2) that makes it possible to prove this theorem. 1 [−] (a) Suppose that F ∈ H 2 (∂S), and we want to split F = f + g with f ∈ Wint [−] Uj and g ∈ Wext so f = Pint (a)F . For a function F defined on ∂S we let F denote the restriction of F to the upper boundary of ∂Sj and F Lj the restriction of F to the lower boundary of ∂Sj . Let αj βj Z(aj ) = γj δj

denote the graph operator for the representation in Theorem 4.6.1 of the subspace Wint (aj ). The condition that the restriction fj of f to ∂Sj lie in Wint (aj ) is fj = Pr(aj )fj , and Theorem 4.6.1 allows us to write this as U

L

L

L

U

f Uj = f− j + αj f+ j + βj f− j , U

f Lj = f+ j + γj f+ j + δj f− j . We will use these relations in the form, U

L

U

f+ j = αj f+ j + βj f− j , L L U f− j = γj f+ j + δ j f− j , U

(5.3)

L

L

U

to eliminate f+ j and f− j in favor of the graph coordinates f+ j and f− j . The condition that the pair g Uj and g Lj +1 constitute the boundary value of a solution to the Dirac equation (mI − D)g = 0 in the strip that lies between Uj and Lj +1 is L

U

g+j +1 = e−j +1 ω g+j , L

U

g−j = e−j +1 ω g−j +1 ,

where

j +1 = y(Lj +1 ) − y(Uj ) = yj +1 − Lj+1 − (yj + Uj ) = yj0+1 − yj0 − 2ǫ.

The conditions at the ends are a little different. Namely, L

g+1 = 0,

Un g− = 0.

We turn these into equations for (note the right-to-left ordering) L

L

f+L = [f+Ln , f+ n−1 , . . . , f+ 1 ]τ

and

U

U

f−U = [f−Un , f− n−1 , . . . , f− 1 ]τ

204


by substituting g = F −f into the equations for g and then using (5.3) to eliminate f+U and f−L in favor of f+L and f−U . One obtains L

L

f+ 1 = F+ 1 , and for j = 1, . . . , n − 1, L

L

L

U

U

f+ j +1 − e−j +1 ω αj f+ j − e−j +1 ω βj f− j = F+ j +1 − e−j +1 ω F+ j , L

U

U

L

U

f− j − e−j +1 ω γj +1 f+ j +1 − e−j +1 ω δj +1 f− j +1 = F− j − e−j +1 ω F− j +1 , (5.4) and f−Un = F−Un . These equations can be rewritten in matrix form as L 1 − α() −β() f+ F+ () = , −γ () 1 − δ() f−U F− () where α() and β() are the superdiagonal matrices,  0 e−n ω αn−1  0 ···  0 e−3 ω α2 α() :=    0 

0 e−n ω βn−1  0  β() :=   

··· 0

e

−3 ω

where γ () and δ() are subdiagonal matrices,  0 e−n ω γn 0  ··· γ () :=    e−3 ω γ3 0 

β2

0



  ,  −2 ω  e α1 0 

0 ···

e−3 ω δ3

(5.6)

  .  −2 ω  e β1 0 

e−2 ω γ2

0

  ,  



0

e−n ω δn  δ() :=   

(5.5)

0 e−2 ω δ2

0

  ,  

(5.7)


and

  U F+Ln − e−n ω F+ n−1   ..   . ,  F+ () =    F+L2 − e−2 ω F+U1  L F+ 1   F−Un   Un−1 − e−n ω F−Ln  F− . F− () =  ..     . U

Note that the invertibility of

205

(5.8)

(5.9)

L

F− 1 − e−2 ω F− 2 −β() , 1 − δ()

1 − α() −γ ()

which is needed to solve (5.5) for [f+L , f−U ]τ , follows from the transversality of the subspaces Wint (a) and Wext established in Chapter 6. [−] (a)−1 Pr(a) as a map on W0 . Suppose that Now we use this to calculate Pint g ∈ W0 . Then gj , the restriction of g to ∂Sj , is represented in the direct sum 1 decomposition X(∂Sj ) ⊕ Y (∂Sj ) of H 2 (∂Sj ) as  L   L  g+j g+j    0  gj ≃   ⊕ Z(aj )   . (5.10) U U g−j g−j It follows that



L

g+j





L

g+j



    Pr(aj )gj ≃   ⊕ Z(aj )   . U U g−j g−j

(5.11)

[−] Pint (a)F = Pr(a)g.

(5.12)

The right-hand side of (5.11) is evidently in Wint (aj ), and it differs from the right-hand side of (5.10) by an element in Y (∂Sj ). The subspace Y (∂Sj ) is the complementary subspace (i.e., null space) for both of the projections Pr(aj0 ) and Pr(aj ), and so we have (5.11). Now suppose that F ∈ W0 and

We want to “solve” this for the graph coordinates [F+L , F−U ]τ of F using (5.5). Because F ∈ W0 we can use  U   L  F+ j F+ j    0    = Z(aj )   Lj Uj F− F−

206


to eliminate [F+U , F−L ] in favor of [F+L , F−U ] on the right-hand side of (5.8) and (5.9). Doing this, we find that the equation (5.12) becomes   L  L  −1  g+ 1 − α 0 () −β 0 () F+ 1 − α() −β()   =  ,    U g− F−U −γ 0 () 1 − δ 0 () −γ () 1 − δ() (5.13) where α 0 () is defined in the same way as α() but with αj replaced by αj0 and [−] so on. Thus in graph coordinates we find that the map Pint (a)−1 Pr(a) acting on W0 is −1 1 − α 0 () −β 0 () 1 − α() −β() [−] . (a)−1 Pr(a) = Pint −γ 0 () 1 − δ 0 () −γ () 1 − δ() (5.14) −1 [−] (a) Pr(a) can be factored. Observe that since α() − The determinant of Pint α 0 () and δ() − δ 0 () are both in the trace class, the determinant −1 0 1 − α() 0 1 − α 0 () det (5.15) 0 1 − δ 0 () 0 1 − δ() is absolutely convergent. In fact, since (1 − α 0 ())(1 − α())−1 , a trace class perturbation of the identity by a strictly upper triangular matrix, it has determinant equal to 1. The other entry in (5.15) is a trace class perturbation of the identity by a strictly lower triangular operator and so has determinant 1 as well. It follows that (5.15) 1. $ # also has determinant [−] −1 Rewrite det Pint (a) Pr(a) by reversing the order of the factors in (5.14) (justified by the similarity invariance of the determinant) and multiply the result on the left by (5.15). Use the multiplicative property of determinants and similarity invariance one more time to see that # $ [−] det Pint (a)−1 Pr(a) = det P(a)P(a0 )−1 , (5.16) where

P(a) =

1 −(1 − α())−1 β() . −(1 − δ())−1 γ () 1

(5.17)

The multiplicative property of determinants suggests that det (P(a)) . det P(a)P(a0 )−1 = det P(a0 )

(5.18)

We now verify that this is correct and that

det (P(a)) = τ [−] (ma)2 , which completes our discussion of (5.1).

(5.19)


207

Since β() and γ () are Schmidt class operators but not obviously trace class, it is clear that P(a) is a Schmidt class perturbation of the identity but perhaps not a trace class perturbation of the identity. This makes the application of the product rule in (5.18) a little suspect. However, there is a continuous multiplicative extension of the determinant that justifies this calculation. Suppose that a and d are trace class operators and b and c are Schmidt class operators on a Hilbert space H . Write a b S= c d

for the matrix of an operator on H ⊕ H . Operators of the form I + S are closed under multiplication and we define det(I + S) := det (I + S)e−S exp (Tr(a) + Tr(d)) . (5.20)

The “regularized” determinant

det2 (I + S) = det (I + S)e−S

is well defined for S in the Schmidt class and is continuous in the Schmidt norm for S [147]. Since we suppose a and d are trace class, (5.20) is well defined. If S is trace class, then it is known that [147] det (I + S)e−S = det(I + S) exp(−Tr(S)),

and since Tr(S) = Tr(a)+Tr(d) it follows that (5.20) reproduces the multiplicative determinant det(I + S) when S is in the trace class. Since the family of trace class S is dense in the family of maps with diagonal entries in the trace class and offdiagonal entries in the Schmidt class, it follows that (5.20) is multiplicative on this extended class, and this justifies the factorization (5.18). In order to rewrite (5.17) and (5.18) so that the connection with the scaling function formula (3.68) is manifest, observe that since α() is strictly upper triangular, the geometric series for the inverse (1 − α())−1 truncates: (1 − α())−1 =

n−1

α()k .

k=0

For the typical case n = 5, one obtains the strictly upper triangular   0 β4 α4 β3 α4,3 β2 α4,2 β1   4 0 β3 α3 β2 α3,2 β1   k , α() β() =  0 β α β 2 2 1    k=0 0 β1  0 where

βi := e−i+1 ω βi , αi := e−i+1 αi ,

(5.21)

208


and αi,j := e−i+1 ω αi e−i ω · · · e−j +1 ω αj

for i > j.

In a similar fashion one sees that (again for the typical n = 5)  0  γ5 4 0  γ4 0 δ()k γ () =   δ4 γ5 δ3,4 γ5 δ3 γ4 γ3 0 k=0 δ2,4 γ5 δ2,3 γ4 δ2 γ3 γ2

This matrix is strictly lower triangular, with

(5.22) 

  .   0

γi := e−i ω γi , δi := e−i ω δi ,

and δi,j := e−i ω δi e−i−1 ω · · · e−j ω δj

for j > i.

(5.23)

To relate this to the expression for τ [−] (ma) in Theorem 3.4.2 it is useful to make j suitable identifications between the subspaces W± that appear as the domain and U L range of the operators in Theorem 3.4.2 and the subspaces W± j and W± j that arise in the present calculation. The following diagrams summarize the domain and range information for the maps Dj−1 , Bj Dj−1 , Dj−1 Cj , and Dj−τ and also for αj , βj , γj , and δj (they are not commutative diagrams): Bj Dj−1

j

j +1

W+ −−−−→ W+ B  D−τ  Dj−1 A  j j −1

W−

Dj−1 Cj

j

←−−−− W+

and U

βj

U

L

γj

L

W− j −−−−→ W+ j B  αj  δj A  W− j ←−−−− W+ j

This suggests the natural identifications U W+ j −1

and L W− j +1

e

e

¯ Uω − j

−→ ¯L ω − j +1

−→

j W+

j W−

¯ Lω −

e j Lj −→ W+ e

¯U ω − j +1

−→

U

W− j ,

(5.24)

(5.25)


209

where ¯ Uj : = |y¯j − y Uj −1 | = y¯j − y Uj −1 , ¯ Lj : = |y¯j − y Lj | = y Lj − y¯j ,

(5.26)

and y¯j =

yj + yj −1 . 2

Remark 5.1.2. A possible source of confusion about these domain and range identifications concerns the matrix representations for L, U, B, and C. The matrix representation for B is diagonal, but the entries of B, namely Bj Dj−1 , are not maps j

j

from W− to W+ . This is a consequence of the fact that the transformation that led to the product UB is not a similarity transformation on the individual factors U and B. However, the product UB is a map from W−⊕n to W+⊕n , and this can be used to determine the proper domain and range for Bj Dj−1 and Dj−τ . Similar considerations apply to LC, Dj−1 Cj , and Dj−1 . U

U

At first blush, βj : W− j → W+ j makes it appear that βj acts diagonally. Keep U in mind, however, that W+ j is not a component of the base space W+L ⊕ W−U . The superdiagonal matrix for β() is accounted for by the composition U W− j

βj e−j +1 ω L Uj −→ W+ −→ W+ j +1 .

It is simple to sort out the action of α(), γ (), and δ() in the same way. The relations that are expected from these identifications are given in the following theorem. Theorem 5.1.2 . Subject to the identifications (5.24) and (5.25), Dj−τ = e Dj−1 = e Bj Dj−1 = e

¯U ω − j +1 ¯ Lω − j

δj e

¯U ω − j +1

Dj−1 Cj = −e

αj e

¯U ω − j +1

βj e

¯ Lω − j

¯ Lω − j

,

,

¯U ω − j +1

γj e

¯ Lω − j

,

.

Proof. These relations are just a transcription of (4.101) from the preceding chapter. Substituting the relations of the last theorem into the formula (3.68) for the scaled Ising correlation and making the obvious connections with the formulas (5.21) through (5.24), we see that 1 −UB (5.27) = E −1 P(a)E, LC 1

210


where E is the matrix

¯ Lω

with e−

¯U

¯L e − ω E= 0

the n × n diagonal matrix  ¯L e−n ω  ¯L  e−n−1 ω   

and e−+1 ω the diagonal matrix  ¯U e−n+1 ω     

0 ¯U

e−+1 ω

..

,

 . ¯L

e−1 ω



¯U

e − n ω ..

    

. ¯U

e−2 ω

  .  

¯ L1 and ¯ Un+1 are not so far defined. We choose them positive The increments but otherwise arbitrary; the reader may check that they do not make an explicit appearance on the right-hand side of (5.27). Based on (5.27) we assert that 1 −UB det = det P(a). (5.28) LC 1 Of course, the determinant is a similarity invariant, but we must take some care since E −1 is an unbounded operator. However, there is a well-known identity that sorts this out for us and also allows us to deal with an issue that so far we have swept under the rug: the Sobolev space topology for W+L ⊕ W−U . Because of the exponential factors e−ω , > 0, the map E : W+⊕n ⊕ W−⊕n → W+L ⊕ W−U 1

is bounded from the L2 space topology of W+⊕n ⊕ W−⊕n to the H 2 Sobolev space topology for W+L ⊕ W−U . Now define X := E −1 P(a).

(5.29)

Then it is easy to check that the unbounded factor E −1 is more than compensated for by factors e−j ω that appear in the second factor on the right-hand side of (5.29). In fact, one sees that X : W+L ⊕ W−U → W+⊕n ⊕ W −⊕n is a Schmidt class map. Then we can use det(I + XE) = det(I + EX).

5.2 The supercritical Tau Function

211

As usually stated, this result requires that X be trace class and E be bounded. A density argument shows that it is true for the extension of the determinant we have defined with X trace class on the diagonal and Schmidt class on the off-diagonal. Since det(I + EX) = det P(a), this does finish the proof of (5.28) and hence via (5.18) and (3.68) it proves Theorem 5.1.1.

5.2 The supercritical Tau Function [+] In this section we introduce the subspace Wint (a), which is appropriate for the [+] analysis of the scaling function τ (ma) defined in Chapter 3. We will show that [+] (a) ∈ Gr(W0 ), and for an appropriately defined τ function, Wint [+] τ (Wint (a)) = i n

τ [+] (ma) τ [−] (ma0 )

2

.

(5.30)

Note that the denominator involves τ [−] (ma0 ) and not τ [+] (ma0 ). To define [+] [+] (a) (a)) we introduce a trivialization δ of the det ∗ bundle over a → Wint τ (Wint and write [+] (a) σ Wint [+] τ (Wint (a)) := [+] δ Wint (a)

as before. It is useful to begin by describing the subspace of interest for n = 1 with a1 = a. [−] [+] Roughly speaking, the space Wint (a) by the boundary con(a) differs from Wint [−] ∗ dition at a. The symmetric condition c0 (w) = c0 (w) that determines Wint (a) [+] becomes the antisymmetric condition c0 (w) = −c0∗ (w) for Wint (a). We give a precise definition for this subspace below in (5.34), but it is useful to proceed intuitively for just a bit. An important role is played by the function (the normalization by 2i is not important but is convenient) ˆ − a) = w(x

i i (w0 (x − a) − w0∗ (x − a)) = wˆ 0 (x − a). 2 2

We have seen above that this function is globally in L2 (R 2 ). Thus it is in [+] Wint (a) ∩ Wext ,

and hence in the kernel of the projection [−] [+] Pr(a) : Wint (a). (a) → Wint

(5.31)

212


The cokernel of this projection is also important. To understand it suppose that 1 f ∈ H 2 (∂S). Then (4.59) implies that ) G(x, y; a)f (y) dy Pr(a)f (x) = ∂S

has a local expansion near x = a with 0th-order expansion coefficients ) im c0 (Pr(a)f ) = c0∗ (Pr(a)f ) = − wˆ 0 (y − a)τ f (y)dy. 8 ∂S

(5.32)

1

Now choose f ∈ H 2 (∂S) such that ) i im wˆ 0 (y − a)τ f (y)dy = . − 8 ∂S 2 Again the choice of the factor

i 2

is not important but is convenient. Define

w(x, a) = Pr(a)f

for x ∈ S.

Of course, w(x, a) is not uniquely defined, but any choice for such an f will produce a suitable representative w for the cokernel of the projection (5.31). Intuitively [+] ˆ − a) and an this is because each vector in Wint (a) is the sum of a multiple of w(· [+] [−] ˆ element of the intersection Wint (a) ∩ Wint (a). The projection Pr(a) kills the w component and is the identity on the intersection. However, the linear functionals c0 and c0∗ clearly vanish on the intersection, since c0 −c0∗ and c0 +c0∗ both vanish on [−] this subspace. Since w is constructed to be in Wint (a) with c0 (w) = 0, it is a suit[−] [+] (a). (a) → Wint able representative for the cokernel of the projection Pr(a) : Wint [−] For g ∈ Wint (a) we now define rank-1 maps ˆ − a) − w(x, a) (5.33) F ± (a)g(x) = i c0 (g) ± c0∗ (g) w(x

[+] and define the subspace Wint (a) by

[−] [+] (a). (a) := (1 − F + (a))Wint Wint [−] It is easy to see that for g ∈ Wint (a), g(x) + i c0 (g) + c0∗ (g) w(x, a)

(5.34)

is in the null space of both c0 and c0∗ . Thus the map 1 − F + (a) annihilates the component of g in the w(x, a) direction and replaces it with a component in the ˆ − a) direction. It is easy to check that w(x [−] [+] 1 + F − (a) : Wint (a) (a) → Wint

is the inverse of 1 − F + (a).


213

Now recall the notation of the previous section. A base point a0 ∈ R 2n is fixed with aj0 = (xj0 , yj0 ) and y10 < y20 < · · · < yn0 . The points aj0 are isolated in disjoint horizontal strips Sj . Each point aj belongs the strip Sj . We define n * [+] [+] Wint (a) := (aj ) (5.35) Wint j =1

and we recall [−] 0 W0 = Wint (a ) (note the [−]). [+] (a) ∈ Gr(W0 ) and we define an admissible frame Then Wint [+] (a) Pr [+] (a) : W0 → Wint [+] (a) as the direct sum for Wint

Pr [+] (a) :=

n * j =1

(1 − F + (aj ))Pr(aj ).

(5.36)

[+] This determines a trivialization of det ∗ over the family a → Wint (a) given by [+] (a)) ≃ (Pr [+] (a), 1). δ(Wint

(5.37)

As above, we define a τ function [+] (a)) = τ (Wint

[+] σ (Wint (a)) [+] (a)) δ(Wint

.

A difference between this case and the τ function defined in the last section is that [+] (a) when n is odd. This the canonical section σ vanishes over the subspaces Wint is, of course, a reflection of the fact that the odd correlations vanish above Tc . Because of this we are not much interested in n odd (except, of course, for n = 1), and henceforth in this section we suppose that n is even. [+] (a) that defines the canonical Next we introduce an admissible frame for Wint section and that will allow us to prove (5.30). To do this in a simple way we introduce the L2 Ising-type solution to the Dirac equation Wj , which has 0thorder coefficients at ai given by c0i Wj = 12 Tij + iδij , (5.38) c0∗i Wj = 12 Tij − iδij .

214


As is shown in Chapter 6, an L2 (R 2 ) Ising-type solution to the Dirac equation satisfying c0i (Wj ) − c0∗i (Wj ) = iδij exists and is unique. It is also shown in Chapter 6 that {W1 , . . . , Wn } is a basis for the L2 Ising-type solutions to the Dirac equation (mI − D)ψ = 0. This is the only property of {W1 , . . . , Wn } that we use in an essential way, but this particular basis will considerably facilitate the connection with the scaling result Theorem 3.4.2. The coefficients Tij are the matrix elements of an n × n skew-symmetric matrix T . When n is even we will see in Chapter 6 that T is invertible. Suppose now that n is even and write Tij−1 := T −1 ij [−] (a) define for the i, j matrix element of the inverse matrix T −1 . For g ∈ Wint $ # ∗j j (5.39) Tij−1 c0 (g) + c0 (g) Wi . F(a)g = i,j

We first verify that [+] [−] (a) (a) → Wint 1 − F(a) : Wint

is an isomorphism. Since c0i (Wj ) + c0∗i (Wj ) = Tij , [−] (a) and f = (1 − F(a))g, then we see that if g ∈ Wint ∗j

j

c0 (f ) + c0 (f ) = 0. [+] (a). A straightforward calculation shows that the map Thus f ∈ Wint [+] [−] (a) ∋ g → g + i (c0i (g) − c0∗i (g))Wi ∈ Wint (a) Wint

(5.40)

i

[−] inverts the restriction of 1 − F(a) to Wint (a). Next we check that [+] [−] [+] (a) (a) : W0 → Wint (a) := (1 − F(a))Pint Pint

(5.41)

inverts the projection [+] (a) → W0 . P0 : Wint

Since the restriction of Wj to ∂S is in Wext , it follows that P0 kills the range of F(a), and so [−] [−] (a). (a) = P0 Pint P0 (1 − F(a))Pint


215

[−] But we have already seen that the restriction of P0 Pint (a) to W0 is the identity. Thus the canonical section is [+] [+] (a), 1). (a)) ≃ (Pint σ (Wint

(5.42)

Now we compare the canonical section to the trivialization (5.36) to define a τ function [+]

τ (Wint (a)) =

[+] (a)) σ (Wint [+] (a)) δ(Wint

# $ [+] = det Pint (a)−1 Pr [+] (a) .

(5.43)

We have arrived at the principal theorem of this section: Theorem 5.2.1 Suppose that each aj in {a1 , a2 , . . . , an } is in the strip Sj . Then [+]

τ (Wint (a)) = i

n

τ [+] (ma) τ [−] (ma0 )

2

.

The rest of this section is devoted to a proof of this result. Suppose that Hj for j = 1, 2 are Hilbert spaces, A : H2 → H1 and B : H1 → H2 are bounded invertible maps such that AB is a trace class pertubation of the identity, and F is a finite-rank map on H2 . Then using the mulitiplicative property of determinants and similarity invariance we obtain det(A(1 + F )B) = det(AB + AFB) = det(AB) det(1 + (AB)−1 AFB) = det(AB) det(1 + B −1 F B) = det(AB) det(1 + F ).

Consulting (5.36) and (5.41), we see that $ # [−] [+] (a)−1 Pr(a)) det(1 + F ), (a)−1 Pr [+] (a) = det(Pint det Pint

where

1 + F = (1 − F(a))−1

n * j =1

(1 − F + (aj )),

(5.44)

[−] defined as a map on Wint (a). Thus Theorem 5.1.1 implies

[+] (a) = τ Wint

τ [−] (ma) τ [−] (ma0 )

2

det(1 + F ).

We turn to the evaluation of det(1 + F ). Let wj denote the restriction of w(x, aj ) 1 [−] to ∂Sj in the Hilbert space H 2 (∂S). Then Wint (a) is the direct sum of the span of the wj for j = 1, . . . , n and the intersection [−] [+] (a) ∩ Wint (a). Wint

(5.45)

216


On this intersection the linear functionals c0i and c0∗i vanish for i = 1, . . . , n. As a conseqence we see from (5.36) and (5.40) that 1 + F acts as the identity on the intersection. On the other hand, ˆ j, (1 + F )wj = (1 − F(a))−1 w ˆ − aj ) to ∂Sj . Now use (5.40) to see that ˆ j is the restriction of w(x where w ˆ j − Wj . ˆj = w (1 − F(a))−1 w This can be expressed as a linear combination of the wi and a vector in the intersection (5.45). The coordinate functional for wi is −i(c0i + c0∗i ), and so (1 + F )wj + i

ˆ j − Wj )wi = (1 + F )wj − (c0i + c0∗i )(w iTij wi i

i

is in the subspace (5.45). Thus the matrix of 1 + F relative to the direct sum decomposition [+] [−] span{w1 , . . . , wn } ⊕ Wint (a) (a) ∩ Wint

is

iT 1+F ≃ ∗

0 . 1

It follows that det(1 + F ) = det(iT ) = i n det T . In order to verify Theorem 5.2.1 we need only show that the n × n matrix T that appears in (5.38) is the same as the n × n matrix T that appears in Theorem 3.4.2. To see this, it is useful to formulate a “transfer-matrix” characterization of Wj . (5.46) We claim that [−] ˆ j. (a)w Wj |∂S = Pext

(5.47)

[−] ˆ j ∈ Wext , this function is the boundary value of an H 1 solution Since Pext (a)w to the Dirac equation defined in the complement of the union of the strips ∪j Sj . [−] ˆ j extends to an Ising-type To see that (5.47) is true we will show that Pext (a)w solution to the Dirac equation in the interior of the strips Sj with local expansion coefficients that characterize the solution Wj . We begin with the identity [−] [−] ˆ j = Pint ˆ j +Pext ˆ j. w (a)w (a)w


217

Restricting this identity to ∂Sk for k = j , one sees that [−] [−] ˆ j. ˆ j = −Pint Pext (a)w (a)w [−] ˆ j extends to the interior of Sk as an L2 Ising-type (a)w Thus the function Pext solution to the Dirac equation with [−] ∗k ˆ j = 0 for k = j. (a)w (5.48) c0 − c0k Pext

On ∂Sj we have

[−] [−] ˆj = w ˆ j − Pint ˆ j. Pext (a)w (a)w [−] ˆ j extends to an Ising-type solution to the Dirac equation This shows that Pext (a)w in Sj with # $ j ∗j [−] ˆ j = 1. −i c0 − c0 Pext (a)w (5.49)

But Wj is uniquely characterized as the L2 Ising-type solution to the Dirac equation with local expansions (5.48) and (5.49). We have confirmed (5.47). We next employ the contour integral representation for the local expansion coefficients to see that for ǫ > 0 sufficiently small, ) i im i Tij = c0i∗ + c0i Wj = − wˆ 0,1 dz, wˆ 0,2 d z¯ Wj , (5.50) 4 Ci (ǫ) i where Ci (ǫ) is the circle of radius ǫ about ai and wˆ 0,j is the j th component of the function wˆ 0 (z − ai ). Since the integrand in (5.50) is a closed form and both wˆ 0i and Wj decay exponentially at ∞, the contour Ci (ǫ) can be deformed to ∂Si . Because of (5.47) we see that ) i [−] im i ˆ j. wˆ dz, wˆ 0,2 d z¯ Pext (a)w Tij = − 4 ∂Si 0,1 Since c0i∗ + c0i wˆ 0i = 0, this is ) i [−] im i ˆ j. Tij = wˆ 0,1 dz, wˆ 0,2 (a)w d z¯ Pint 4 ∂Si j

ˆ j = 2i wˆ 0 , the expression Now on ∂Si we have dz = d(x + iy) = dx and since w for Tij becomes ) ) m [−] [−] ˆ j (z) dx , (5.51) ˆ j (z) dx − ˆ i (z)τ Pint ˆ i (z)τ Pint Tij = (a)w w (a)w w 2 Ui Li [−] ˆj where Li and Ui are the lower and upper pieces of ∂Si . The restriction of Pint (a)w to ∂Si is in $ $ # # 1 U L L U H 2 (∂Si ) = W+ i ⊕ W− i ⊕ W+ i ⊕ W− i , (5.52)

218


and since

U

ˆ i|Ui ∈ W+ i , w L

U

L

ˆ i|Li ∈ W− i , w

and the subspaces W+ i and W− i are isotropic with respect to the complex bilinear forms ) ) f (z)τ g(z) dx, f (z)τ g(z) dx and Li

Ui

L

U

[−] ˆ j in W+ i ⊕W− i makes a contribution it follows that only the component of Pint (a)w to the integrals that appear in (5.51). Now write $ # U U [−] ˆj = fj −i + fj +i (a)w Pint Ui

and

#

[−] ˆj Pint (a)w

$

L

Li

L

= fj +i + fj −i

[−] ˆ j according to the polarization (5.52). Then (5.51) is for the splitting of Pint (a)w ) ) m U L ˆ i (z)τ fj −i (z) dx , ˆ i (z)τ fj +i (z) dx − Tij = w (5.53) w 2 Ui Li

and according to (5.5), L fj + 1 − α() = −γ () fjU−

−β() 1 − δ()

−1

Fj + () , Fj − ()

(5.54)

where   0 Un−1  Ln ˆ j,+ ˆ j,+ w − e−n ω w ..       . ..    Uj  . − ω    Fj + () =  L = −e j +1 w ˆj   ←⊣ j + 1, 2 −2 ω U1    ˆ ˆ wj,+   wj,+ − e .   .. L1 ˆ j,+ w 0 

and



Un ˆ j,− w −n ω





0 .. .



     Un−1 Ln  ˆ j,− ˆ j,− − e w    w −j ω Lj  ←   = Fj − () =  ˆ j  ⊣ j − 1. w ..  −e      . ..   U1 L2 . ˆ j,− ˆ j,− w − e−2 ω w 0

X ˆ j,± Here we have written w for the components of splitting of the restriction 1

ˆ j to X = Ui , Li relative to the polarization H 2 (X) = W+X ⊕ W−X , and the of w


219

notation ←⊣ j ± 1 denotes the location of the nonvanishing entry of the adjacent vector measured from the bottom. In order to make use of (5.27) it is useful to employ the factorization 1 − α() −β() 1 − α() 0 = P(a) −γ () 1 − δ() 0 1 − δ() in (5.54). One computes  U  ˆjj αn−1,j +1 e−j +1 ω w  U  αn−2,j +1 e−j +1 ω w ˆj j     ..   .   −j +1 ω Uj  (1 − α())−1 Fj + () = −  ˆj  w  αj +1 e   U   ˆjj e−j +1 ω w     ..   . 0 and



(5.55)



0 .. .

      Lj  e−j ω w  ˆj     −j ω Lj  ˆ w δ e (1 − δ())−1 Fj − () = −  j .  j −1   ..   .   L   j ˆj  δ3,j −1 e−j ω w

(5.56)

L

−j ω ˆjj w δ2,j −1 e

Now define (column vectors)

j

and

¯U U − ˆ j j , 0, . . . , 0]τ vj+ := −[0, . . . , 0, e j +1 w

(5.57)

j

vj− := −[0, . . . , 0, e

¯L − j

L ˆ j j , 0, . . . , 0]τ . w

(5.58)

In both instances the nonzero entry occurs in the j th slot up from the bottom. Next use the identities ¯ Lω ¯L = e−i+1 ω Di−τ · · · Dj−τ e j αi,j and ¯U

= e−i ω Di−1 · · · Dj−1 e δi,j

¯U ω j +1

220


in (5.55) and (5.56) and recall (3.70) and (3.71) to see that ¯L

(1 − α())−1 Fj+ = e− ω Uvj+ and ¯U

(1 − δ())−1 Fj− = e−+1 ω Lvj− . Combine this with (5.54) and (5.27) to obtain L fj + 1 = E LC fjU−

−UB 1

−1 + Uvj . Lvj−

(5.59)

To make a comparison with (3.82) we want to rewrite the formula (5.53) for Tij as an inner product. The natural bilinear form ) F, G → F (x)τ G(x) dx ∂S

is not quite right for us, since the upper boundaries Uj are negatively oriented. However, since e−sω is symmetric with respect to the bilinear pairing ) f (x)τ g(x) dx, R

we can use (5.59) to rewrite (5.53) in terms of the bilinear form (from (3.82)) ) ) f+ g τ τ f−,j (x)g+,j (x) dx f+,j (x)g−,j (x) dx + = , + g− f− R R j

j

on W+⊕n ⊕ W−⊕n to obtain m Tij = 2

7

1 vi+ , −vi− LC

−UB 1

−1

Uvj+ Lvj−

8

.

(5.60)

To finish the proof of Theorem 5.2.1 we need only calculate the spectral representations for vj± and note that they are appropriately related to δj± from (3.82). Theorem 5.2.2 The spectral representation (4.77) for wˆ 0 (z) for ℑz = y > 0 is ) ∞ # $ dr iπ 1 − my r+r −1 , wˆ 0 (z) = −2e− 4 r2e 2 e(x, ir −1 ) 2π r 0 and for ℑz = y < 0, wˆ 0 (z) = 2ie

− iπ 4

)

0

∞

r

− 21

e

#

my −1 2 r+r

$

e(x, −ir)

dr . 2π r


Proof. We start with the representation of wˆ 0 (z) (4.39): ) du 1 wˆ 0 (z) = u− 2 e(z, u) . 2π u C(z)

221

(5.61)

For ℑz = y > 0 we deform the contour to the “positive” imaginary axis and introduce the parametrization u = ir −1 . On the incoming branch of the contour C(i) the argument of i is π/2. Thus on this branch, −1 − 21 π 1 ir = e−i 4 r 2 .

Combining this with the contribution from the outgoing branch, one obtains ) ∞ # $ dr 1 − my r+r −1 − iπ 2 4 2 wˆ 0 (z) = −2e r e e(x, ir −1 ) . 2π r 0 Now suppose that ℑz < 0. Deform the contour C(z) to the “negative” imaginary axis and parametrize (5.61) by u = −ir. On the outgoing branch the argument of −i is 3π/2, so that on this branch, 1

(−ir)− 2 = e−i

3π 4

1

r− 2 .

Combining this with the contribution from the incoming branch, one obtains ) ∞ dr 1 my π −1 wˆ 0 (z) = −2ei 4 r − 2 e 2 (r+r ) e(x, −ir) . 2π r 0 ˆ From this theorem and (4.77) we can read off the spectral transform for w(z) (in the x = ℜz variable) restricted to the horizontal lines ℑz = y. For y > 0 the ˆ spectral transform of w(z) restricted to ℑz = y is = # $ 2 − my2 r+r −1 + ˆ (r, y) = w . e mπ ˆ restricted to ℑz = y is For y < 0 the spectral transform of w = # $ 2 my2 r+r −1 − ˆ (r, y) = − w e . mπ These representations allow one to verify that (see Remark 3.4.1) = 2 ± ± vj = ±i δ . m j Substituting these representation in (5.60), we obtain 7 8 −1 Uδj+ 1 −UB δi+ Tij = − , . δi− LC 1 −Lδj− Comparing this with (3.4.2), we have finished the proof of Theorem 5.2.1.

6 Deformation Analysis of Tau Functions

This chapter is devoted to the analysis of the tau functions that were introduced in the preceding chapter. The central observation is that the Green function for the twisted Dirac operator has a finite-rank derivative with respect to the branch points. The finite-dimensional subspace spanned by the “response functions” that arise in this derivative are flat sections for a “Dirac-compatible” connection. The logarithmic derivative of the tau functions with respect to the branch points are shown to be low-order Fourier coefficients in the local expansions for the response functions. The zero-curvature condition for the connection becomes a system of nonlinear deformation equations. Following Sato, Miwa, and Jimbo we find that the low-order Fourier coefficients of response functions can be expressed in terms of the solutions to the deformation equations. For the scaling limit of the Ising two-point function this translates into an expression for the logarithmic derivative that can be written in terms of Painlevé functions of the third kind, a surprising result that was first obtained by Wu, McCoy, Tracy, and Barouch in [159]. We start this chapter with the existence theory for the appropriate Green function. Next we prove that the derivative of the Green function is of finite rank and characterize the response functions. Then we use the formulas for the tau functions given in the preceding chapter to show that the logarithmic derivative of the tau functions is expressed as the level-1 Fourier coefficient for a response function. We then do the zero-curvature local expansion analysis (à la Sato, Miwa, and Jimbo) to see that these Fourier coefficients have a characterization in terms of solutions of the deformation equations. For the two-point scaling functions we reproduce the WMTB formulas. We also describe how this characterization has been used to verify rotational invariance for the n-point functions and the Luther–Peschel formulas for the “short distance” asymptotics of the n-point Ising scaling functions.

224

6. Deformation Analysis of Tau Functions

6.1 Boundary Conditions for m − D on E Suppose that a = {a1 , . . . , an } with ai ∈ C for i = 1, . . . , n and ai = aj for i = j . We introduce a line bundle E over C\a associated with “Ising transition functions” that was already described in Section 3.4 for the case of just one point a = a1 . The bundle E is constructed so that the smooth sections of E correspond to smooth “multivalued functions” on C\a that change by a multiplier −1 when followed around a complete circuit that encloses a single point ai . To be more precise, note that there are only a finite number of vectors ai − aj for i, j = 1, . . . , n. Thus it is possible to choose a vector r ∈ C\{0} that is not a multiple of any of these vectors. Then the rays defined by ri = {z : z = ai + tr, t > 0} do not intersect. Choose an argument θr for r so that r = |r|eiθr with |θr | ≤ π and let θ (z) denote the polar angle defined by z = |z|eiθ (z)

with θr − π < θ (z) < θr + π.

This angle is branched along the ray −r. For ǫ > 0 define a tubular neighborhood ti of ri by π ti (ǫ) := z ∈ C\a : dist(z, ri ) < ǫ ∩ z : |θ (z − ai ) − θr | < , 4 where dist(z, ri ) is the distance from the point z to the ray ri . Now choose such an ǫ > 0 that the tubular neighborhoods ti are mutually disjoint and (for later convenience) such that the disks Di (2ǫ) = {z : |z − ai | < 2ǫ} are also mutually disjoint. The situation is pictured in Figure 6.1. Now we introduce a covering of C\a over each element of which the bundle E is trival. Let O0 := {z ∈ C\a : z ∈ / ri , i = 1, . . . , n} and Oi = ti (ǫ) for i = 1, . . . , n. To define the bundle E we glue together the trivial bundles Oi × C → Oi by giving the transition functions si that relate the trivializations over O0 and Oi . Define −1 for θ (z − ai ) < 0, si (z) = 1 for θ (z − ai ) > 0.

6.1 Boundary Conditions for m − D on E

225

2ε a1 t 1(ε) D1 2ε z= a1+tr a2 t 2(ε) D2 z= a2+tr

Figure 6.1: The geometry associated with the trivializations of E. Then the bundle E is determined by the transition map between the vector (z, v)0 ∈ O0 × C and the vector (z, u)i ∈ Oi × C defined such that (z, v)0 = (z, u)i if and only if u = si (z)v. Note that si (z) is a smooth function since it is constant on each connected component of Oi ∩ O0 . It is useful to introduce a slightly different trivialization in a neighborhood of ∞. Choose R > 0 such that R > |ai | for i = 1, . . . , n. Suppose that n is even and define O∞ := {z ∈ C : |z| > R}. The intersection O0 ∩ O∞ splits into n components: one “large” component (that contains −tr for t > 0 sufficiently large) and “smaller” components that lie between branch cuts. Let φ0 denote the section of E over O0 ∩ O∞ that is identically equal to 1 in the large component of O0 ∩ O∞ and that is alternately either −1 or +1 in the other components as one moves around the circle of radius R. Since n is even, it is easy to see that φ 0 extends to a smooth section φ of E over O∞ (the trivialization φi of φ over Oi is again either the function +1 or −1). The situation is pictured in Figure 6.2. Any smooth section of E over O∞ is the product of a smooth function f : O∞ → C and the section φ. We will refer to this as the O∞ trivialization of E. For n odd we modify this construction a bit. Choose a branch cut Bri := {z = ai + tr, for t > 0}

226

6. Deformation Analysis of Tau Functions φ0 =1 φ0 =1

a2

φ0 = –1

φ2 = 1

φ0 = –1 φ0 = –1

a1

φ0 = 1

φ1= –1

φ0 =1 φ0 =1 φ0 =1

φ 3= 1 φ0 = –1 φ0 = –1 φ0 = –1

a3

a2

φ0 = –1 φ0 = –1 φ0 = –1 φ1= –1 φ0 =1

a1

φ0 = 1

Figure 6.2: The trivialization of E at infinity. There are two cases depending on whether n is even (top) or n is odd (bottom). In the odd case a residual branch cut is depicted at a2 in the figure.

and define φ0 on O0 ∩ O∞ by making φ0 = 1 on the “large” component, then propagating alternating ±1 values in both directions around the circle until they meet at the branch cut Bri . Because n is odd, the values for φ0 agree on both sides of Bri , and hence φ0 does not extend across Bri as a smooth section of E. Let φ∞ denote the smooth section of E defined by φ0 on O∞ \Bri . The nonvanishing smooth section φ∞ defines a trivialization of E over O∞ \Bri . Of principal interest for us is that L2 solutions w to (m−D)w = 0 in C ∞ (E) have uniformly absolutely convergent expansions for |z| > R in the φ∞ trivialization given by w(z) = cˆℓ wˆ ℓ (z) ℓ∈Z+ 21

for ℓ even, or w(z) =

ℓ∈Z

cˆℓ wˆ ℓ (z − ai )

when ℓ# is odd. $ In either case, Theorem 4.4.2 implies that w and all its derivatives −m|z| are O em|z| as z → ∞.


Since

m 2∂¯

2∂ m

m −2∂¯

2 ¯ −2∂ m − 4∂∂ = m 0

0 ¯ , m2 − 4∂∂

227

(6.1)

it follows that smooth solutions to the Dirac equation (m − D)f = 0 have components that are solutions to the Helmholtz equation ¯ (m2 − 4∂∂)f j = 0 for j = 1, 2. It is technically convenient to begin our discussion of the Green function for the Dirac operator acting on sections of the vector bundle E by first considering the action of the Helmholtz operator ¯ = m2 − m2 − 4∂∂ on sections of the line bundle E. Here is the usual Laplacian, the sum of the squares of the coordinate derivatives. We start with a discussion of the subspace of L2 (E) that consists of smooth sections of the line bundle E that are solutions to the elliptic Helmholtz equation ¯ (m2 − 4∂∂)v = (m2 − )v = 0. Following SMJ, we show that this space is 2n-dimensional and has a “canonical” basis that is useful for sorting out the boundary conditions for the twisted Dirac operator m − D that are relevant for the Ising model scaling functions. In order to use the standard existence theory of functional analysis, we introduce appropriate Hilbert spaces. Let L2 (E) denote the closure of the space of smooth sections C ∞ (E) with respect to the norm determined by the inner product ) d z¯ dz f, g = f (z)g(z) . 2i C We write f 2 = f, f . Here f and g are two smooth sections of E, and the integrand in the inner product is well defined, since the selection of a trivialization f (z) and g(z) for f and g introduces an ambiguity of at worst the same ± sign in f (z) and g(z), and this ensures that the function z → f (z)g(z) is unambiguous. More succinctly, since the transition maps are unitary valued on C, there is a natural unitary structure in the fibers of E. In the same fashion we introduce the (Sobolev) Hilbert space H1 (E) defined as the closure of C ∞ (E) with respect to the norm defined by the inner product ) # $ ¯ ∂g ¯ + m2 f¯g d z¯ dz , f, g1 := 2∂f ∂g + 2∂f 2i C

228


and as usual we write ∂ :=

∂ ∂z

and

∂ ∂¯ := . ∂ z¯

Suppose now that f ∈ L2 (E). Then the linear functional H1 (E) ∋ g → f, g is clearly a bounded linear functional, since f, g ≤ f · g ≤ m−1 f · g1 . Thus by the Riesz representation theorem there exists F ∈ H1 (E) such that f, g = F, g1 . Now suppose that g ∈ C0∞ (E). If F is also smooth, then integration by parts shows that F, g1 = F, (m2 − )g. This remains true in the limit if F is approximated in H1 (E) by smooth sections, and we see that for g ∈ C0∞ (E), f, g = F, (m2 − )g. Thus F ∈ H1 (E) is a distribution solution to (m2 − )F = f. The “boundary conditions” implicit for sections in H1 (E) allow us to make use of such solutions to construct what SMJ refer to as the canonical basis for the space of solutions to (m2 − )v = 0

such that v ∈ L2 (E). This basis is distinguished by the behavior of local expansions in eigenfunctions for infinitesimal rotations at the branch points ai , and we turn next to a consideration of such expansions. Define a function vk(0) (z) on C\R + by 1

vk(0) (z) := ei(k− 2 ) Ik− 1 (mr), 2

where z = ei r

with 0 < < 2π and r > 0.

Define a function vk(π) (z) on C\R − by 1

vk(π) (z) := ei(k− 2 )θ Ik− 1 (mr), 2


229

where z = eiθ r

with |θ| < π and r > 0.

We write vk (z − ai ) for the local section of E defined in a small disk about ai (one that does not contain any other points aj for j = i) by vk(0) (z − ai ) in the O0 trivialization and by vk(π) (z − ai ) in the Oi trivialization. The reader may check that the Bessel function recursion relations ℓ Iℓ′ (r) + Iℓ (r) = Iℓ−1 (r), r ℓ Iℓ′ (r) − Iℓ (r) = Iℓ+1 (r), r and the polar representations i ∂ e−iθ ∂ i ∂ eiθ ∂ ¯ ∂= − + and ∂ = 2 ∂r r ∂θ 2 ∂r r ∂θ imply that m vk−1 (z − ai ), 2 ¯ k (z − ai ) = m vk+1 (z − ai ). ∂v 2 From this it is easy to see that the local sections vk (z − ai ) are solutions to the Helmholtz equation ∂vk (z − ai ) =

2 ¯ (m2 − 4∂∂)v k (z − ai ) = (m − )vk (z − ai ) = 0.

A second family of solutions to the Helmholtz equation is obtained by just taking the complex conjugate vk → v¯k . The ladder relations for derivatives become m ∂¯ v¯k (z − ai ) = v¯k−1 (z − ai ), 2 m ∂ v¯k (z − ai ) = v¯k+1 (z − ai ). 2 As in the local expansion results for Chapter 4, the Fourier expansion of a local section v of E that is a solution to the Helmholtz equation (m2 − )v = 0 in a neighborhood of ai can be written v(z) =

∞

k=−∞

ak vk (z − ai ) + bk v¯k (z − ai ).

We are particularly interested in sections that are locally in L2 (E). A section v(z) is locally in L2 (E) near ai if and only if the local expansion at ai has a truncated expansion ∞ v(z) = ak vk (z − ai ) + bk v¯k (z − ai ). k=0

230


We formulate our first theorem (a result of SMJ [135]): Theorem 6.1.1 The space of solutions v to (m2 − )v = 0 that are elements of L2 (E) is 2n-dimensional. Furthermore, there exists a basis {vi , v¯ i } (i = 1, 2, . . . , n) for this space that is uniquely characterized by the local expansion near aj : vi (z) = δij v0 (z − aj ) + · · · and v¯ i (z) = δij v¯0 (z − aj ) + · · · , where + · · · refers to terms involving vk (z − aj ) and v¯k (z − aj ) for k ≥ 1. Proof. First we consider the existence and then uniqueness. Let ϕi denote a smooth function of compact support on C that is identically 1 in a small disk about ai and that vanishes outside a larger disk about ai that is, however, small enough not to contain any of the points aj for j = i. Define a smooth section fi of E by fi (z) = ϕi (z)v0 (z − ai ). Let gi denote the solution to (m2 − )gi = −fi ¯ i are locally in L2 (E) about each such that gi ∈ H1 (E). Then since ∂gi and ∂g of the branch points, none of the local expansions for gi near aj with j = i can contain terms that involve v0 (z − aj ) or v¯0 (z − aj ) because ∂v0 = v−1 , ∂¯ v¯0 = v¯−1 and both v−1 (z − aj ) and v¯−1 (z − aj ) are both too singular at z = aj to be square integrable in a neighborhood of aj . To verify this conclusion in a more precise fashion compute the local L2 norm in polar coordinates and use the orthogonality of the Fourier series to isolate the contribution that v−1 and v¯−1 make to this norm. The asymptotics Iℓ (r) ∼ r ℓ as r → 0 then show that no terms involving v−1 or v¯−1 are possible if the L2 norm is to be finite. The section vi := gi + fi is clearly in L2 (E) with (m2 − )vi = 0. The difference vi (z) − v0 (z − ai ) is equal ¯ that are in L2 (E) to fi for z near ai and hence has a complex derivative ∂fi and ∂f near ai . This implies that the local expansion is given by vi (z) = v0 (z − ai ) + · · · , where + · · · are terms involving vk (z − ai ) and v¯k (z − ai ) for k ≥ 1. This finishes the existence proof for vi . Note that elliptic regularity [26] implies that vi is actually a smooth section of E. To prove uniqueness suppose that v is a smooth section of E such that (m2 − )v = 0. Let ¯ d z¯ − i v¯ ∂v dz. ω := i v¯ ∂v


231

Then one can check that ¯ ∂v ¯ + m2 vv) dω = (2∂v ∂v + 2∂v ¯

d z¯ dz . 2i

Excise small disjoint disks Di about each point ai inside a large disk D about 0 that contains all the Di . Then Stokes’s theorem implies )

d z¯ dz ¯ ∂v ¯ + m2 vv) = ¯ (2∂v ∂v + 2∂v 2i D\∪i Di

)

∂D

ω−

n ) i=1

ω.

∂Di

If v ∈ H1 (E), then we let the radius of the disk D tend to infinity in this last equation and use the exponential decay of v and its derivatives at ∞ to conclude that the contour integral on the right-hand side over ∂D vanishes in this limit. Thus for such a v we have ) ) d z¯ dz ¯ ∂v ¯ + m2 vv) (2∂v ∂v + 2∂v ¯ =− ω. 2i C\∪i Di ∂Di i Now we use the local expansions v(z) = ai v1 (z − ai ) + bi v¯1 (z − ai ) + · · · to see that if we now take the limit in which all the radii of the disks Di tend to 0, then the contour integrals ) ¯ d z¯ − v¯ ∂v dz v¯ ∂v ∂Di

all tend to 0. Hence )

C

¯ ∂v ¯ + m2 vv) ¯ (2∂v ∂v + 2∂v

d z¯ dz = 0. 2i

Thus an L2 (E) solution to (m2 −)v = 0 that is also in H1 (E) must be identically 0. Now suppose that v is an L2 (E) solution of (m2 −)v = 0 that has local expansions v = ai v0 (z − ai ) + bi v¯0 (z − ai ) + · · · . Then v−

i

ai vi −

i

bi v¯ i

is in H1 (E) and hence must be identically 0. This shows that {vi , v¯ i } is a basis for the space of L2 (E) solutions to the Helmholtz equation and finishes the proof of the theorem. The vector bundle on which we want to study the action of the Dirac operator m − D is E ⊗ C2 . To avoid extra notation, we continue to denote this bundle by E

232


and refer to it as the vector bundle E if there is a possibility of confusion with the line bundle E. Recall that 1 vk (z) = ei(k− 2 )θ I

k−

1 (mr), 2

where z = reiθ . The local eigenfunctions for the infinitesimal rotation that commutes with the Dirac operator are  k k  m −z − 1 k− vk (z) + )  + O(|z|k+ ), wk (z) = (6.2) =  2mk+Ŵ(k k± := k ± k+ z vk+1 (z) 2 k+ 2

Ŵ(k+ +1)

and

 k k  m + z¯ + v ¯ (z) k+1 ∗ 2k+ Ŵ(k+ +1)   wk (z) = = + O(|z|k+ ), mk− z¯ k− v¯k (z) k

2 − Ŵ(k+ )

1 k± := k ± . 2

(6.3)

We have included the local asymptotics as z → 0 as a convenient reference for some “residue” calculations to follow. It is not important for those calculations that (6.2) and (6.3) do not explicitly identify the next-to-leading-order asymptotics; only the leading order makes a contribution. The following theorem (again due to SMJ) is important for us in sorting out the boundary conditions for m − D of interest for the Ising model. Theorem 6.1.2 Let N denote the space of smooth sections w of the vector bundle E that are solutions of the Dirac equation (m − D)w = 0 and that are in L2 (E). We refer to N as the space of L2 Ising-type solutions to the Dirac equation. The space N is n-dimensional and has a basis of sections vi wi = 2 ¯ ∂vi m that are characterized by the local expansions wi (z) = δij w0 (z − aj ) + cij w0∗ (z − aj ) + · · · ,

(6.4)

where + · · · are terms involving wk (z − aj ) and wk∗ (z − aj ) for k ≥ 1. Proof. It is easy to check that wi is in L2 (E), since vi ∈ H1 (E). Also, (m − D) wi = 0 follows immediately from (m2 − )vi = 0 and the fact that vi is a smooth ¯ i are section. Since the lowest-order terms in the local expansions of vi and m2 ∂v vi (z) = δij v0 (z − aj ) + cij v¯1 (z − aj ) + · · ·


233

and

2¯ ∂vi (z) = δij v1 (z − aj ) + cij v¯0 (z − aj ) + · · · , m it follows that the local expansion of type (3.50) for wi is (6.4). In order to see that {wi } is a basis, we localize the calculation for the L2 inner product of two solutions to the Dirac equation (m2 − D)w = 0. Suppose that u, v ∈ C ∞ (E) are solutions to the Dirac equation. Then it is easy to check that d(i u¯ 1 v2 d z¯ ) = m(u¯ 1 v1 + u¯ 2 v2 )

d z¯ dz 2i

(6.5)

and

d z¯ dz . (6.6) 2i As we did in the last theorem, let Di denote small disjoint disks centered at ai and let D denote a disk centered at 0 that is large enough to contain all the Di . Then Stokes’s theorem and (6.5) imply ) ) ) d z¯ dz i u¯ 1 v2 d z¯ − i u¯ 1 v2 d z¯ . = u¯ · v m 2i ∂Di ∂D D\∪i Di i d(−i u¯ 2 v1 dz) = m(u¯ 1 v1 + u¯ 2 v2 )

Provided that u, v ∈ L2 (E), we can let the radius of D tend to ∞ and the radius of each Di tend to 0. The exponential decay of w at infinity eliminates the contribution from ∂D in this limit. Substituting the local expansions w(z) =

∞ k=0

cki (w)wk (z − ai ) + ck∗i (w)wk∗ (z − ai )

into the contour integrals over ∂Di and passing to the limit, one obtains ) d z¯ dz 4 i u¯ · v c0 (u)c0∗i (v), =− 2 2i m C i

(6.7)

where we used the limit lim ǫI− 1 (ǫ)2 =

ǫ→0

2

2 . π

Using (6.6) instead of (6.5), one also obtains ) d z¯ dz 4 ∗i u¯ · v =− 2 c (u)c0i (v). 2i m i 0 C These formulas show that the maps N ∋ w → c0 (w) := (c01 (w), c02 (w), . . . , c0n (w))τ ∈ Cn and N ∋ w → c0∗ (w) := (c0∗1 (w), c0∗2 (w), . . . , c0∗n (w))τ ∈ Cn

(6.8)

234


are isomorphisms. Since c0i (wj ) = δij , it is clear that {wi } is a basis for N, each element of which is characterized by its c0 local expansion coefficient. The Dirac operators we are interested in are localizations of m − D away from the branch points ai . Suppose that ǫ > 0 is small enough that the open disks Di := Dǫ (ai ) of radius ǫ about ai have mutually disjoint closures. Let Eǫ denote the restriction of E to the base C\ ∪i D¯ i . For each subspace V ⊂ Cn ⊕ Cn we define a domain H1 (V) for m − D acting in H1 (Eǫ ) in the following manner. Let Ei denote the restriction of the bundle E to Di \{ai }. Write Ki for the subspace of w ∈ L2 (Ei ) such that (m − D)w = 0. Let K(V) denote the subspace of the direct sum ⊕i Ki such that w = (w1 , w2 , . . . , wn ) is in K(V) provided (c0 (w), c0∗ (w)) ∈ V ⊂ Cn ⊕ Cn . We write (m − D)V for the restriction of m − D to the domain H1 (V) defined by H1 (V) = {w ∈ H1 (Eǫ ) : w|∂D ∈ K(V)}. What we mean by w|∂D ∈ K(V ) is that w|∂D := (w|∂D1 , w|∂D2 , . . . , w|∂Dn ) is the boundary value of an element in K(V). Said another way, the domain H1 (V) consists of H1 sections of Eǫ that extend to L2 solutions of the Dirac equation inside each disk Di with a restriction on the local expansion coefficients c0 and c0∗ determined by the subspace V. Of special interest are the boundary conditions that make (m − D)V invertible. We can describe these subspaces as follows. Let N denote the image of N in Cn ⊕ Cn under the map N ∋ w → (c0 (w), c0∗ (w)). Then we have the following theorem. Theorem 6.1.3 If V is transverse to N in Cn ⊕ Cn , then for each f ∈ L2 (Eǫ ) there exists a unique F ∈ H1 (V) such that (m − D)F = f. Proof. Extend f to all of C by setting it equal to 0 on ∪i Di . Let gj ∈ H1 (E) be the solution to (m2 − )gj = fj and define G=

mg1 + 2∂g2 ¯ 1 + mg2 . 2∂g


235

Then G ∈ L2 (E) and (m − D)G = f. Thus G extends into ∪i Di as a solution to the homogeneous equation (m − D)G = 0. Since we have assumed that V and N are transverse, we can adjust the boundary value G|∂D on ∪i ∂Di by an element of N so that the result is in V. In other words, there exists a w ∈ N such that G|∂D − w|∂D ∈ V. But then F = G − w is a solution to (m − D)F = f with F |∂D ∈ V.

A priori one might worry that w is only in L2 (E), but Theorem 4.4.2 shows that w ∈ H1 (Eǫ ). This finishes the existence proof. Now suppose that F is a solution to (m − D)F (z) = 0 for z ∈ / ∪i Di and F |∂D ∈ V.

This implies that F extends to an L2 section of E that is in the null space of m − D. Thus F |∂D ∈ N ∩ V = {0}. This in turn implies that c0 (F ) = 0 and hence by (6.7) that F = 0.

One subspace that is transverse to N is 0 ⊕ Cn . Equation (6.7) implies that N ∩ 0 ⊕ Cn = {0}. On the other hand, N is isomorphic to N and so also has dimension n. These two observations imply that the n-dimensional space 0 ⊕ Cn and N must be transverse. For similar reasons Cn ⊕0 is transverse to N . The subspaces that are most relevant for the Ising model are determined by the “boundary conditions” c0 (w) = c0∗ (w) (the subcritical limit) and c0 (w) = −c0∗ (w) (the supercritical limit). We write I[−] for the “diagonal” subspace of Cn ⊕Cn that consists of all vectors v ⊕v for v ∈ Cn and I[+] for the subspace of Cn ⊕ Cn that consists of all vectors v ⊕ (−v) for v ∈ Cn . Note that the [±] notation is again tied to the subcritical–supercritical distinction. The formula (6.7) shows that any section w ∈ N that has boundary

236


values in I[−] will have a negative L2 norm if any coefficient c0i (w) is nonzero. This contradiction shows that N ∩ I[−] = {0}. Since the dimension of I[−] is n, the subspace I[−] is transverse to N . We refer to I[−] as the subcritical Ising boundary conditions. For n odd the subspace I[+] is not transverse to N (for n = 1 I[+] = N ); for n even I[+] is transverse to N . This is proved below. We refer to I[+] as the supercritical Ising boundary conditions.

6.2 Existence Theory for the Subcritical Green Function In this section we show how to construct the Green function for the Dirac operators (m − D)I[−] . Suppose that f ∈ C0∞ (Eǫ ) is a smooth section of compact support. We know that there exists a section F ∈ H1 (I[−]) such that (m − D)F = f. We want to show that there exists a Green function G[−] (x, y; a) such that ) d y¯ dy F (x) = G[−] (x, y; a)Jf (y) . 2i C In this formula G[−] (x, y; a) is a 2 × 2 matrix whose entries x → G[−] ij (x, y; a) [−] and y → Gij (x, y; a) are smooth sections of the line bundle E away from the diagonal x = y. We start by proving existence for a pair of sections of the vector bundle E that turn out to be the columns of the Green function. Theorem 6.2.1 Suppose that x, y ∈ C\ ∪i D¯ i (the base of Eǫ ). Let Gj (x) denote the j th column of the Green function (4.36) for the Dirac operator m − D acting on C ∞ (C). For j = 1, 2 there exist unique smooth sections x → gj (x, y) of Eǫ defined for x = y such that gj ∈ H1 (I[−]), (m − D)x gj (x, y) = 0 for x = y,

(6.9)

gj (x, y) − Gj (x − y)

(6.10)

and is smooth for x in a neighborhood of y. If y is not on any branch cut Bri , then x → Gj (x − y) is identified with a local section of Eǫ in the O0 trivialization. If y ∈ Bri then we identify x → Gj (x − y) with a local section of Eǫ in the Oi trivialization. Proof. Let ϕ denote a C ∞ function of compact support on C that is identically 1 in a neighborhood of 0. Fix a choice of y ∈ C\ ∪i Di . If y ∈ O0 choose ϕ such

6.2 Existence Theory for the Subcritical Green Function

237

that the support of x → ϕ(x − y) is inside O0 . If y ∈ Bri choose ϕ such that the support of x → ϕ(x − y) is contained in Oi . We wish to solve (m − D)x fj (x, y) = −(m − D)x ϕ(x − y)Gj (x − y)

(6.11)

for fj (·, y) ∈ H1 (I[−]). Before we do this, we need to be a little more explicit about the meaning of the right-hand side. Since (6.12)

(m − D)x G(x − y) = δ(x − y)I,

we want to exclude the singularity at x = y from the right-hand side of (6.11). The Leibniz rule for the derivative of a product and (6.12) imply −(m−D)x ϕ(x −y)Gj (x −y) = Dx (ϕ(x − y)I ) Gj (x −y)

for x = y. (6.13)

The right-hand side of (6.13) can now be regarded as a smooth section of compact support in Eǫ . Thus by Theorem 6.1.3 and the transversality of I[−] and N there exists a section fj ∈ H1 (I[−]) such that (m − D)x fj (x, y) = Dx (ϕ(x − y)I ) Gj (x − y).

(6.14)

In fact, since the right-hand side is a smooth section of Eǫ , local elliptic regularity implies that x → fj (x, y) is a smooth section of Eǫ . Now define gj (x, y) := fj (x, y) + ϕ(x − y)Gj (x − y).

(6.15)

From (6.14), and (6.13) it follows that (m − D)x gj (x, y) = 0 for x = y, and from (6.15) we see that gj (x, y) − G(x − y) is smooth for x in a neighborhood of y. This proves existence. To show uniqueness, let gj denote the difference of two solutions to (6.9) and (6.10). Then (6.10) implies that gj (x, y) has a smooth extension to x = y and (m − D)x gj (x, y) = 0 for x ∈ C\ ∪i Di . Since gj (·, y) ∈ H1 (I[−]), the transversality of I[−] and N implies that gj = 0. This finishes the proof of uniqueness.

Remark 6.2.1. It is not really necessary to use the Helmholtz solution in the existence proof, though it is natural. Suppose that one wants to solve (m − D)f = F,

238


where F has support outside the union of strips Sj , with aj ∈ Sj , and [−] f|∂S ∈ Wint (a).

Define f0 = (m − D)−1 F, where (m − D)−1 is the inverse of the Dirac operator on L2 (R 2 ). Then write f0 = fint + fext for the splitting of the boundary values of f0 on ∂S. Define f = f0 − fext in the exterior of the union of the strips S. Then (m − D)f = F, and the restriction of f to ∂S is [−] (a). f = f0 − fext = fint ∈ Wint

One advantage of this construction for the Green function is that it is easy to see that if F is a smooth function of a, then f depends smoothly on a, since the formula for [−] (a) found in Chapter 5 is smooth in a. Thus we see that gi,j (x, y) constructed Pint above depends smoothly on a. Since we are about to confirm that gi,j (x, y) is the Green function, G[−] i,j (x, y; a), this implies that the Green function of interest to us is also smooth in a. The following two technical lemmas are useful in establishing the connection between gj (x, y) and the Green function we are interested in. Lemma 6.2.1 Let Cǫ (ai ) denote the counterclockwise-oriented circle of radius ǫ about ai ∈ C. Suppose that f (z) and g(z) are smooth local sections of E that are L2 solutions to the Dirac equation in a punctured neighborhood of ai . Then ) 4i f1 (z)g1 (z) dz + f2 (z)g2 (z) d z¯ = (c0i (f )c0i (g) − c0∗i (f )c0∗i (g)). lim ǫ→0 Cǫ (a ) m i Proof. This is a simple consequence of the leading local asymptotic behavior of f given by   θ c0i (f )e−i 2 I− 1 (mr) 2  as r = |z − ai | → 0, f (z) ∼  ∗i θ c0 (f )ei 2 I− 1 (mr) 2

and the analogous result for g(z), both of which follow from the existence of local expansions (4.42).

6.2 Existence Theory for the Subcritical Green Function

239

A short calculation shows that f1 g1 dz + f2 g2 d z¯ is a closed one-form whenever f and g are solutions to the Dirac equation, so that it is not necessary to take the limit ǫ → 0 in this lemma. That is, for all small enough ǫ we have ) 4i f1 (z)g1 (z) dz + f2 (z)g2 (z) d z¯ = (c0i (f )c0i (g) − c0∗i (f )c0∗i (g)). (6.16) m Cǫ (ai ) Lemma 6.2.2 Let gij (x, y) denote the ith component of the section gj (x, y) for i = 1, 2. Suppose that f is a local section of E that is smooth in a neighborhood of y ∈ C\a. Then ) g1j (x, y)f1 (x) dx + g2j (x, y)f2 (x) d x¯ = −fj (y). lim ǫ→0 Cǫ (y)

Proof. This follows from the fact that, modulo a smooth function of x, the matrix g = [g1 , g2 ] looks like G(x − y) near the diagonal x = y, where x¯ K0′ (m|x|) K0 (m|x|) im − |x| , G(x) = x K ′ (m|x|) −K0 (m|x|) 2π |x| 0 and the r → 0 asymptotics of K0 and K0′ are K0 (r) ∼ − log r and K0′ (r) ∼ − 1r . As a first step in showing that g is the Green function we are interested in, we prove the following. Theorem 6.2.2 For all x, y ∈ C\ ∪i Di with x = y we have gj k (x, y) + gkj (y, x) = 0. Proof. Choose ǫ > 0 such that the disks Dǫ (ai ) (i = 1, . . . , n), Dǫ (x), and Dǫ (y) are all disjoint. Choose R > 0 large enough that the union of all these disks is contained in the disk DR of radius R about 0. Since gij (z, x) and gij (z, y) are solutions to the homogeneous Dirac equation for z ∈ D, where D := DR \ ∪i Dǫ (ai ) ∪ Dǫ (x) ∪ Dǫ (y), it follows from (4.25) that ω := g1j (z, x)g1k (z, y) dz + g2j (z, x)g2k (z, y) d z¯ is a closed form in D. Thus Stokes’s theorem implies ) ) ) ) ) ω− ω− ω− ω= ∂D

∂DR

i

∂Dǫ (ai )

∂Dǫ (x)

∂Dǫ (y)

ω = 0.

Theorem 4.4.2 implies that the contribution from ∂DR vanishes in the limit R → ∞, and Lemma 6.2.1 implies that the contribution from the integrals over ∂Dǫ (ai ) vanishes as well, since c0i (gj (·, x)) = c0∗i (gj (·, x)) and

c0i (gj (·, y)) = c0∗i (gj (·, y)).

240


Thus we see that )

∂Dǫ (x)

ω+

But Lemma 6.2.2 implies that ) ω = −gj k (x, y) ∂Dǫ (x)

)

∂Dǫ (y)

and

ω = 0.

)

∂Dǫ (y)

This finishes the proof.

ω = −gkj (y, x).

Now we are ready to confirm that the matrix g = [g1 , g2 ] with columns g1 and g2 is the Green function G[−] (x, y; a). Suppose that f is a smooth section of compact support in Eǫ . Then Theorem 6.2.2 implies that y → [gi1 (x, y), gi2 (x, y)]τ is a solution to the Dirac equation, and we see from (4.19) that ) d y¯ dy [gj 1 (x, y), gj 2 (x, y)]J (m − D)f (y) 2i C\Dǫ (x) ) gj 1 (x, y)f1 (y) dy + gj 2 (x, y)f2 (y) d y¯ = ∂Dǫ (x) ) g1j (y, x)f1 (y) dy + g2j (y, x)f2 (y) d y. ¯ =− ∂Dǫ (x)

Lemma 6.2.2 now implies that the right-hand side of this last equation tends to fj (x) as ǫ → 0. Thus we have ) d y¯ dy g(x, y)J (m − D)f (y) = f (x). 2i C

6.3 Existence for the Basis {Wj } and the Derivative of G(x, y; a) In this section we show that there exists a basis {Wj } for the L2 sections of E that are solutions to the homogeneous Dirac equation that is uniquely characterized by the linear conditions c0i (Wj ) − c0∗i (Wj ) = iδij on the lowest-order expansion coefficients. This basis already played a role defining the canonical section for the supercritical subspaces in the last chapter. It is useful to begin with a description of the relation between the “canonical basis” {wj } and the “dual canonical basis” {wj∗ } obtained from the first by the conjugation ∗ a b¯ = . b a¯

6.3 Existence for the Basis {Wj } and the Derivative of G(x, y; a)

241

The leading-order local expansion of the canonical basis for z − ai small is wj (z) = δij w0 (z − ai ) + c0∗i (wj )w0∗ (z − ai ) + · · · . Applying the conjugation, we obtain wj∗ (z) = δij w0∗ (z − ai ) + c0∗i (wj )w0 (z − ai ) + · · · . Proposition 6.3.1 The canonical and dual canonical bases are related by c0∗i (wj )wi∗ . (6.17) wj = i

The n × n matrix

c0∗

whose i, j matrix element is (c0∗ )ij = c0∗i (wj )

is Hermitian symmetric, negative definite, and satisfies c0∗ c0∗ = I . Also, the level-one coefficients are related by j

c1∗i (wj∗ ) = c1 (wi ).

(6.18)

Proof. Using (6.8) and the local expansion for wj recorded above, we see that ) d z¯ dz 4 4 ¯ i · wj w =− 2 δik c0∗k (wj ) = − 2 c0∗i (wj ). 2i m m C k

This shows that c0∗ is Hermitian symmetric and negative definite. Computing local expansion coefficients, one easily sees that all the c0∗i linear functionals vanish on c0∗i (wj )wi∗ . wj − i

Consulting (6.8), we see that the L2 norm of this section is 0 and hence the section is 0. This proves (6.17). Taking conjugates of (6.17), we see that c0∗i (wj )wi . wj∗ = i

Combining this with (6.17), we see that c0∗ c0∗ = I . Now let u = wi and v = ∂¯z wj∗ . Since both u and v are Ising-type solutions to the Dirac equation, we have d(i u¯ 1 v2 d z¯ ) = −d(i u¯ 2 v1 dz). Integrating both sides of this last equality over the complement of the union of disks of radius ǫ about ak for k = 1, . . . , n and making use of the fact that u and v are exponentially small at ∞, we see that Stokes’s theorem implies ) ) i u¯ 2 v1 dz. (6.19) i u¯ 1 v2 d z¯ = − k

∂Dǫ (ak )

k

∂Dǫ (ak )

242


Using the limits lim ǫI− 1 (ǫ)I− 1 (ǫ) = − lim ǫI− 3 (ǫ)I 1 (ǫ) =

ǫ→0

2

2

and the integrals )

ǫ→0

2

2

2 π

2π

0

eiℓθ dθ = 2πδ0 (ℓ),

we can use local expansions to evaluate the left-hand side of (6.19) in the limit ǫ → 0. We find that this limit is j

2(c1∗i (wj∗ ) − c1 (wi )). Doing the same for the right-hand side, we get 0. This finishes the proof.

Now suppose that α is an n × n matrix and define Wj = αij wi . i

The conditions c0i (Wj ) − c0∗i (Wj ) = iδij translate to the matrix equation α − c0∗ α = i, which has the unique solution α = i(1 − c0∗ )−1 , where c0∗ − 1 is invertible, since c0∗ is negative definite. Note that since α is invertible, it follows that {Wj } is a basis for N, the L2 Ising-type solutions to the Dirac equation. It is convenient to reformulate this just slightly. Define an n × n matrix T by c0∗i (Wj ) = 12 (Tij − iδij ).

(6.20)

c0i (Wj ) = 21 (Tij + iδij ),

(6.21)

Then we see that

and solving (6.20) for T , we obtain T = 2c0∗ α + i = i(1 + c0∗ )(1 − c0∗ )−1 .

(6.22)

Since c0∗ is Hermitian symmetric, T is skew Hermitian. Since c0 ∗ = c0∗−1 , it follows that T =T.


243

Thus T is real and being skew Hermitian must be a real skew-symmetric matrix. In particular, Tii = 0. Consulting (6.20) and (6.21), we can rewrite the leading-order local expansions for Wj as Wj (z) =

iδij Tij ∗ (w0 (z − ai ) − w0∗ (z − ai )) + (w0 (z − ai ) + w0 (z − ai )) + · · · . 2 2

This suggests that Wj is something like the “canonical basis” relative to the local wave functions i (wk (z − ai ) − wk∗ (z − ai )) 2

and

1 (wk (z − ai ) + wk∗ (z − ai )). 2

These wave functions are just a natural “real” basis for the conjugation ∗. Now we turn to the calculation of the derivative of the Green function G[−] (x, y; a) with respect to the branch points aj . We begin with a calculation of the 0th- and 1st-order coefficients in the local expansion of the Green function. We write these local expansions in the following way: G[−] (x, y; a) = (w0 (xi ) + w0∗ (xi ))g0i (y)τ + w1 (xi )g1i (y)τ + w1∗ (xi )g1∗i (y)τ + · · · .

(6.23)

In this formula xi := x − ai and gji (y) is a 2 × 1 column vector, so that the transpose gji (y)τ is a 1 × 2 row vector. Both gji (y) and gj∗i (y) depend on a, but for brevity we don’t explicitly exhibit this dependence. We compute gj(∗)i (y) in terms of the wave function Wi . To simplify this calculation it is useful to introduce the following wave function: 4i ∂Wj im j Fj = c1 (Wj )Wj . − m ∂aj 2 Our interest in this has to do with its local expansion, which we characterize as follows: j w−1 (xj ) + c0 (Fj )(w0 (xj ) + w0∗ (xj )) + · · · for x ∼ aj , Fj (x) = i c0 (Fj )(w0 (xi ) + w0∗ (xi )) + · · · for x ∼ ai , i = j. (6.24) The ladder relations (4.40) and (4.41), the derivative identity ∂ ∂ w(x − a) = − w(x − a), ∂a ∂x and the observation that differentiation by aj will alter the “shape” of local expansions only at x = aj are enough to confirm (6.24). Note that Fj has only one nonvanishing term at level −1 and all the terms in its local expansion at level 0 are symmetric (c0 (Fj ) = c0∗ (Fj )).

244


Theorem 6.3.2 The expansion coefficients gℓi (y) and gℓ∗i (y) in (6.23) for ℓ = 0, 1 are given by m g0i = − Wi (6.25) 4 and ∂Wi im i i g1 = − − c (Wi )Wi (6.26) ∂ai 2 1 and

im i ∂Wi + c (Wi )Wi . =− ∂ a¯ i 2 1

g1∗i

(6.27)

Proof. The calculations are quite analogous to those in Theorem 6.2.2. Choose ǫ > 0 such that the disks Dǫ (ai ) (i = 1, . . . , n) and Dǫ (y) are all disjoint. Choose R > 0 large enough that the union of all these disks is contained in the disk DR of radius R about 0. Since Wi (x) and gj (x, y) are solutions to the homogeneous Dirac equation for x ∈ D, where D := DR \ ∪i Dǫ (ai ) ∪ Dǫ (y), it follows from (4.19) that ω := g1,j (x, y)Wi,1 (x) dx + g2,j (x, y)Wi,2 (x) d x¯ is a closed form in D. Thus Stokes’s theorem implies ) ) ) ) ω− ω− ω= ∂D

∂DR

∂Dǫ (ak )

k

∂Dǫ (y)

ω = 0.

Theorem 4.4.2 implies that the contribution from ∂DR vanishes in the limit R → ∞, and Lemma 6.2.1 implies that the contribution from the integrals over ∂Dǫ (ak ) for k = i vanishes as well, since c0k (gj (·, x)) = c0∗k (gj (·, x)) Thus we see that

)

Cǫ (ai )

ω+

)

and

Cǫ (y)

c0k (Wi ) = c0∗k (Wi ).

ω = 0.

But Lemma 6.2.2 implies that lim

)

ǫ→0 Cǫ (y)

ω = −Wi,j (y),

and Lemma 6.2.1 implies that ) i i 4 i 4i i i (y). g0,j (y) − g0,j (y) − = − g0,j ω= m 2 2 m Cǫ (ai )


245

Thus

m Wi (y). 4 To do the next calculation recall Fi defined by (6.24) and define g0i (y) = −

= g1,j (x, y)Fi,1 (x) dx + g2,j (x, y)Fi,2 (x) d x. ¯ Then as above, Stokes’s theorem implies ) ) )

−

= ∂D

∂DR

∂Dǫ (ak )

k

−

)

∂Dǫ (y)

= 0.

Theorem 4.4.2 implies that the contribution from the integral over ∂DR vanishes in the R → ∞ limit. Lemma 6.2.1 implies that the integrals over all the circles ∂Dǫ (ak ) for k = i vanish. Thus we have ) )

= 0.

+ Cǫ (y)

Cǫ (ai )

Lemma 6.2.2 implies that lim

)

ǫ→0 Cǫ (y)

= −Fi,j (y).

The only term that survives in the integral of over Cǫ (ai ) is the term that pairs w−1 (x) in the expansion of Fi with the w1 (x) expansion term for gj (x, y). Using lim ǫI− 3 (mǫ)I 1 (mǫ) = − 2

ǫ→0

2

2 , mπ

we obtain )

2

= 2πi − mπ Cǫ (ai )

i g1j (y) = −

4i i g (y). m 1j

Thus im Fi (y). 4 Recalling the definition of Fi , this is just (6.26). A similar calculation in which Fi is replaced by Fi∗ suffices to prove (6.27). g1i (y) =

We come now to a central result. Theorem 6.3.3 The derivative of the Green function G[−] (x, y; a) with respect to the branch point ai is the kernel of a finite-rank operator given by im ∂Wi (x) ∂Wi (y)τ ∂ [−] τ G (x, y; a) = − − Wi (y) (6.28) Wi (x) ∂ai 2 ∂ai ∂ai and

∂ [−] im ∂Wi (x) ∂Wi (y)τ Wi (x) G (x, y; a) = − Wi (y)τ . ∂ a¯ i 2 ∂ a¯ i ∂ a¯ i

(6.29)

246


Proof. Differentiation with respect to ai kills off the singularity in the Green function at x = y. The function x →

∂ [−] G (x, y; a) ∂ai

is a section of E that satisfies the Dirac equation and is in L2 near infinity. Our strategy in determining this function is to subtract from it solutions to the Dirac equation that are in L2 near infinity and chosen to make the difference an L2 Isingtype solution with symmetric coefficients (c0 = c0∗ ). Since we know that such L2 Ising-type solutions must be 0, we succeed in identifying the derivative in this way. Differentiating the local expansion (6.23) with respect to ai , we obtain ∂ [−] m m G (x, y; a) = − w−1 (xi )g0i (y)τ − w0 (xi )g1i (y)τ + sym0 + · · · ∂ai 2 2 for x near ai and ∂ [−] G (x, y; a) = sym0 + · · · ∂ai

for x near aj , j = i,

where sym0 stands for terms at level 0 that are symmetric. It is easy to see from this that ∂ [−] ∂Wi (x) im i G (x, y; a) + 2i − c1 (Wi )Wi (x) g0i (y)τ ∂ai ∂ai 2

has a local expansion with no terms below level 0 and with the single nonsymmetric contribution m − w0 (xi )g1i (y)τ 2 at level 0 in the expansion about ai . This coefficient can be symmetrized by the addition of im − Wi (x)g1i (y)τ . 2 Thus the columns of ∂ [−] ∂Wi (x) im i im G (x, y; a) + 2i − c1 (Wi )Wi (x) g0i (y)τ − Wi (x)g1i (y)τ ∂ai ∂ai 2 2 are L2 sections of E that are solutions to the Dirac equation in the x variable with purely symmetric local expansion coefficients at level 0 and so must vanish identically. If we couple this observation with the expressions in Theorem 6.3.2 for g0i (y) and g1i (y) and make an obvious simplification, we arrive at (6.28). An obvious adaptation of this argument proves (6.29). Remark 6.3.1. The fact that the Green function has a finite-rank derivative im[−] plies that the same is true for the projections Pint (a). Without difficulty one sees [−] [−] 0 [−] that this implies that Pint (a) − Pint (a ) is trace class and hence that Wint (a) is in [−] 0 the Grassmannian of subspaces close to Wint (a ) in the sense of Segal–Wilson.

6.4 The Derivative of log τ

247

We can supply the result needed in the preceding chapter for the supercritical case with the following observation. Theorem 6.3.4 The intersection is N ∩ I[+] equal to {0} if and only if c0∗ does not have −1 as an eigenvalue. This in turn happens if and only if T is invertible. Proof. Suppose that −1 is an eigenvalue for c0∗ , so that for some nonzero element α ∈ Cn we have c0i (wj )αj = −αi . j

Then it is easy to check that     c0∗  αj wj  , αj wj  = −c0  j

j

which implies that N ∩ I[+] = {0}. The steps are reversible, so the intersection is nonzero iff c0∗ has eigenvalue −1. Since T = i(c0∗ + 1)(1 − c0∗ )−1 , it is clear that T is invertible iff c0∗ + 1 is invertible. This finishes the proof.

One consequence of this theorem is that N ∩ I [+] = {0} if n is odd. This follows from the fact that T is a real skew-symmetric n × n matrix. For odd n, such a matrix always has 0 as an eigenvalue and hence fails to be invertible. On the other hand, the application of the GKS inequalities at the end of Chapter 2 showed that the scaling functions τ [+] (a1 , . . . , a2n ) are positive. This implies that in these circumstances Pf(−T ) = 0 and hence that T is invertible. This is what was needed for the tau function analysis for the supercritical case to proceed in the previous chapter.

6.4 The Derivative of log τ [−] (a) in terms of the Green We begin with the representation of the projection Pint function G[−] (x, y; a). Suppose, as in Chapter 3, that a base point a0 ∈ Cn consisting of n points ai0 with distinct second coordinates is fixed. Choose open horizontal strips Si whose closures are disjoint such that ai0 ∈ Si . Let S = ∪i Si and ∂S = ∪i ∂Si . The orientation of ∂Si is the standard positive orientation t → t + p for p on the lower component of ∂Si and the negative orientation t → −t + p for p on the upper component of ∂Si (it might be better to write ∂ S¯i instead of ∂Si , but we are not so fussy).

248


Theorem 6.4.1 Suppose that f ∈ C0∞ (∂S). Then ) [−] ′ [−] ′ G[−] ¯ Pint (a)f (x) = lim 1 (x , y; a)f1 (y) dy + G2 (x , y; a)f2 (y) d y. ′ x →x

∂S

Here G[−] j (x, y; a) is the j th column of the Green function in the O0 ′

trivialization, and the limit is a nontangential limit as x approaches a point x on the boundary ∂S from the interior of S. The foundation of the deformation analysis of the subcritical scaling function is given by the following theorem. Theorem 6.4.2 Suppose that a ∈ Cn \HC, the site configurations without horizontal coindences. The logarithmic derivative of the subcritical scaling function τ [−] (ma) is given by $ 1 # d log τ [−] (ma) = − Tr Pr(a)dP[−] (a) , (6.30) int 2 where we have written d = da,a¯ , since no other variables are present and there is little chance of confusion. Of course, since we have proved the convergence of this scaling function only for noncoincident second coordinates in a, we must restrict a to this subset. Proof. Theorem 5.1.1 implies that # $ 1 (a) , d log τ [−] (ma) = − d log det Pr(a)−1 P[−] int 2

[−] [−] 0 where both P[−] int (a) and Pr(a) are regarded as maps from Wint (a ) to Wint (a) and −1 −1 we used (det X) = det(X ). The formula for the derivative of the logarithm of the determinant [147],

d log det (X) = Tr(dX · X −1 ), implies that # $ # # $ $ [−] −1 [−] −1 d log det Pr(a)−1 P[−] (a) = Tr d Pr(a) P (a) P (a) Pr(a) . (6.31) int int int

Recall that

*

[−] 0 Wext (ai )

i

is the null space for both the projection Pr(a) and Pr(a0 ). This implies that if [−] 0 (a ) x ∈ Wint

and Pr(a)x = y, then x = Pr(a0 )y. Thus Pr(a)−1 = Pr(a0 )


249

[−] [−] 0 if both sides are regarded as maps from Wint (a) to Wint (a ). In a similar fashion, [−] 0 [−] 0 since Wext (a ) is the null space for both the projection P[−] int (a) and Pint (a ), it follows that −1 0 P[−] = P[−] int (a) int (a ) [−] 0 [−] if both sides are regarded as maps from Wint (a ). Substituting these (a) to Wint results in the trace on the right-hand side of (6.31), we obtain # # $ $ [−] 0 Tr Pr(a0 ) dP[−] (a) P (a )Pr(a) . (6.32) int int

Since

# $ [−] 0 P[−] (a) I − P (a ) = 0, int int

0 we can eliminate the factor P[−] int (a ) and use the central character of the trace to rewrite (6.32) as $ $ ## 0 Tr dP[−] (a) Pr(a)Pr(a ) . int

[−] Note that the trace is taken for the restriction of the relevant map to Wint (a). However,

Pr(a)(I − Pr(a0 )) = 0 shows that we can eliminate the factor Pr(a 0 ) in this last trace. Eliminating this factor in the resulting trace, we see that it is not necessary to restrict the map to [−] Wint (a), since Pr(a) vanishes identically on the complement * [−] Wext (ai ). i

In the unrestricted trace that results we need only use the central character of the trace once more to obtain (6.30). Next we use Theorems 6.4.1 and 6.3.3 to calculate the trace in the last theorem. The result provides the connection between the scaling function τ [−] (ma) and the deformation analysis that follows. Theorem 6.4.3 Suppose that a ∈ Cn \HC, the site configurations without a horizontal coincidence. Then the derivative of log τ [−] (ma) is given by d log τ [−] (ma) = −

im j ∗j c Wj daj − c1 Wj d a¯ j . 2 j 1

Proof. If a is a vector in a Hilbert space and b is a linear functional on the Hilbert space, then Tr(a ⊗ b) = b(a). (6.33)

250


For brevity we temporarily introduce the notation ) f1 (x)g1 (x) dx + f2 (x)g2 (x) d x. ¯ (f, g) =

(6.34)

∂S

Of course d x¯ = dx on ∂S, but in the case that f and g are solutions to the Dirac equation, the integrand in (6.34) is the closed form that allows a contour deformation. Employing the representation of Theorem 6.4.1 in Theorem 5.1.1 and the formula for the derivative of the Green function Theorem 6.3.3 in the trace calculation associated with (6.33), we see that ∂ log τ [−] (a) im =− ∂aj 4

Pr(a)

∂Wj , Wj ∂aj

∂Wj − Pr(a)Wj , ∂aj

(6.35)

and im ∂ log τ [−] (ma) =− ∂ a¯ j 4

∂Wj ∂Wj . , Wj − Pr(a)Wj , Pr(a) ∂ a¯ j ∂ a¯ j

(6.36)

We can evaluate these bilinear pairings by noting that all the functions that occur in (6.35) and (6.36) are solutions to the Dirac equation that are small at infinity. Thus the contour integrals in (6.34) over ∂Si can all be collapsed to integrals over small circles Cǫ (ai ) surrounding the branch cuts aj . We can then use Lemma 6.2.1 to evaluate the resulting integrals. We first observe that Pr(ai )Wj = Wj |∂Sj

for i = j,

since in this case the restriction of Wj to ∂Si extends to a solution of the Dirac equation in Si that is in L2 and has symmetric coefficients at level 0. Write j j wˆ 0 (z) =wˆ 0 (z − aj ). Since Wj − 2i wˆ 0 has level-0 expansion coefficients at aj that are 0, the restriction of this function to ∂Sj is in the range of Pr(aj ). Since j Pr(aj )wˆ 0 = 0, we have i j i j Pr(aj )Wj = Pr(aj ) Wj − wˆ 0 = Wj − wˆ 0 , 2 2 restricted to ∂Sj . Thus the level-0 expansion coefficients of Pr(aj )Wj vanish. As a consequence it is not hard to see by “residue” calculation that ∂Wj j = −c1 (Wj ) Pr(a)Wj , ∂aj and ∂Wj ∗j Pr(a)Wj , = −c1 (Wj ). ∂ a¯ j


251

The only contribution comes from the level-1 coefficents for Pr(a)Wj and the level −1 coefficients of ∂Wj /∂aj and ∂Wj /∂ a¯ j , making use of lim ǫI− 3 (mǫ)I 1 (mǫ) = − ǫ↓0

2

2

2 . mπ

Turning to the other pairings, we note that Pr(aj )

∂Wj ∂Wj = ∂aj ∂aj

restricted to ∂Si for i = j

and Pr(ai )

∂Wj ∂Wj = ∂ a¯ j ∂ a¯ j

restricted to ∂Si for i = j.

Since the level-0 expansion coefficients of ∂Wj /∂aj , ∂Wj /∂ a¯ j , and Wj are all symmetric at ai for i = j , none of the integrals on ∂Si for i = j makes a contribution to ∂Wj ∂Wj Pr(a) , Wj , Wj . or Pr(a) ∂aj ∂ a¯ j It remains to compute the contribution from ∂Sj . A short calculation shows that the expansion coefficients for j

∂Wj i ∂ wˆ 0 m j j − + c1 (Wj )wˆ 0 ∂aj 2 ∂aj 4 at aj below level 0 all vanish and the coefficients at level 0 are symmetric. Since j the restrictions of wˆ 0 and its aj derivative to ∂Sj are in the kernel of Pr(aj ), it follows that j ∂Wj ∂Wj i ∂ wˆ 0 m j j Pr(aj ) = − + c1 (Wj )wˆ 0 ∂aj ∂aj 2 ∂aj 4 restricted to ∂Sj , and one sees that ∂Wj ∂Wj m j ∗j j = c0 Pr(aj ) = − c1 (Wj ). c0 Pr(aj ) ∂aj ∂aj 4 In a precisely similar fashion one finds that ∂Wj ∂Wj m ∗j ∗j j = c0 Pr(aj ) = − c1 (Wj ). c0 Pr(aj ) ∂ a¯ j ∂ a¯ j 4 Combining this with Lemma 6.2.1 we see that ∂Wj j Pr(a) , Wj = c1 (Wj ) ∂aj

252


and ∂Wj ∗j Pr(a) , Wj = c1 (Wj ). ∂ a¯ j Together with the results for the other significant pairings this finishes the proof.

6.5 Holonomic Systems and the Deformation Equations In the previous section, the tau functions that we identified as scaling functions for the Ising model were seen to have logarithmic derivatives in the branch points that are expressed as level-1 coefficients in the Fourier expansion of the appropriate response functions. In this section we reproduce the SMJ analysis of these Fourier coefficients. We show that they can be expressed in terms of the solution to nonlinear deformation equations. In the subsequent section we show that for the two-point tau functions the deformation equations boil down to a Painlevé equation of the third kind. This is the SMJ account of the original result of WMTB [159]. The ladder relations for the derivatives of the local wave functions are m m ∂z wℓ (zi ) = wℓ−1 (zi ), ∂¯z wℓ (zi ) = wℓ+1 (zi ), 2 2 m ∗ m ∗ ∗ ∗ ¯ ∂z wℓ (zi ) = wℓ+1 (zi ), ∂z wℓ (zi ) = wℓ−1 (zi ). 2 2 As above, we write zi = z − a i .

These relations and the corresponding results for derivatives with respect to the branch points aj are the principal tools in the analysis. It is useful to introduce some notation. Suppose that uj for j = 1, . . . , n are n Ising-type solutions to the Dirac equation. Each of these solutions has a local expansion at z = ai given by uj = cℓi (uj )wℓi + cℓ∗i (uj )wℓ∗i , ℓ

where wℓi (z) = wℓ (z − ai )

and

wℓ∗i (z) = wℓ (z − ai ).

We write u = [u1 , . . . , un ] for the row vector of solutions and define n × n matrices cℓ (u) and cℓ∗ (u) with i, j entries i,j

cℓ (u) := cℓi (uj )

and

∗i,j

cℓ (u) := cℓ∗i (uj ).

6.5 Holonomic Systems and the Deformation Equations

253

Recall the local expansions for the canonical basis wj = δij w0i + c0∗i (wj )w0∗i + c1i (wj )w1i + c1∗i (wj )w1∗i + · · · . In the notation we just introduced, the row vector w := [w1 , . . . , wn ] is then characterized by the local expansion coefficients cℓ (w) = cℓ∗ (w) = 0 for ℓ < 0 and c0 (w) = I. Our first goal is to use the infinitesimal rotational symmetry for the Dirac operator to extend the Dirac equation for w to a Pfaffian differential system for w (every derivative of every function is specified). Recall that 1 1 0 ¯ Rw = z∂z w − z¯ ∂z w + w 2 0 −1 is a differential operator that commutes with the Dirac operator m − D. Thus if we apply R to a C ∞ (E) solution w to the Dirac equation, we find another C ∞ (E) solution to the Dirac equation. We can use this observation to prove the following theorem. Theorem 6.5.1 The row vector w for the canonical basis satisfies the differential equation m Rw − ∂z wa + ∂¯z w∗ a¯ c0∗ = w[a, c1 ], (6.37) 2 where a is the diagonal matrix   a1   .. a :=  , . an

we write c¯ for the usual entry-by-entry complex conjugate of a matrix, we introduce the abreviated notation cℓ := cℓ (w), and cℓ∗ := cℓ∗ (w),

and we write [a, c1 ] = ac1 − c1 a. Proof. We prove this by first showing that the left-hand side of (6.37) has local expansion coefficients that are 0 for all levels ℓ < 0. We then compare the c0 coefficients of both sides to see that these coefficients are the same, and hence by Theorem 6.1.3, both sides must also be equal. First write 1 1 0 ¯ , Ri = (z − ai )∂z − (¯z − a¯ i )∂z + 2 0 −1

254


and note that Writing

Ri wℓi = ℓwℓi

and

Ri wℓ∗i = −ℓwℓ∗i .

(6.38)

R = Ri + ai ∂z − a¯ i ∂¯z , we can use (6.38) to calculate the action of R on the local expansion coefficients of w. We find that cℓ (Rw) and cℓ∗ (Rw) are 0 for ℓ < −1 and m m ac−1 (∂z w) = a, 2 2 m m ∗ ¯ ∗ c−1 (Rw) = − a¯ c−1 (∂z w) = − a¯ c0∗ . 2 2 c−1 (Rw) =

But # m $ c−1 ∂z w a = 2 $ # ∗ ∗m ¯ c−1 ∂z w a¯ c0∗ = 2

m a, 2 m ∗ a¯ c . 2 0

∗ Thus it follows that the c−1 and c−1 coefficients of the left-hand side of (6.37) are both 0. To prove (6.37) in light of Theorem 6.1.3 we need only compute the c0 coefficients of both sides. Using (6.38) and the ladder relations for derivatives of the local wave functions, we see that m cℓ (Rw) = ℓcℓ + (acℓ+1 − a¯ cℓ−1 ) , 2 (6.39) m ∗ ∗ acℓ−1 − a¯ cℓ+1 . cℓ∗ (Rw) = −ℓcℓ∗ + 2

Also, it is clear that

m cℓ+1 , 2 m ∗ cℓ∗ (∂z w) = cℓ−1 . 2 cℓ (∂z w) =

(6.40)

Applying the conjugation ∗ to both sides of the local expansion for w and ∂z w, we find that cℓ (w∗ ) = cℓ∗ ,

(6.41)

cℓ (∂¯z w∗ ) = cℓ∗ (∂z w), cℓ∗ (∂¯z w∗ ) = cℓ (∂z w).

(6.42)

cℓ∗ (w∗ ) = cℓ , and

We henceforth use (6.41) and (6.42) to eliminate the local expansion coefficients of w∗ and ∂¯z w∗ in favor of the local expansion coefficients for w.


255

It is a simple matter to use these results to calculate c0 (Rw − ∂z wa + ∂¯z w∗ a¯ c0∗ ) =

m [a, c1 ]. 2

Since the c0 coefficient of the right-hand side of (6.37) is also m2 [a, c1 ], this finishes the proof of the theorem. Note that in (6.37) we can use the relation w = w∗ c0∗ to eliminate w∗ in favor of w. All the coefficients in the resulting differential equation can be expressed in terms of c0∗ and [a, c1 ]. Following SMJ [135] we introduce G := −c0∗ , F := m2 [c1 , a].

(6.43)

The change of sign for G is dictated by the desire that it be positive definite rather than negative definite. We aim to show that F and G satisfy a coupled system of nonlinear differential equations in the ai variables. Because of the connection with monodromy-preserving deformations of the Dirac equation, SMJ refer to these equations as the deformation equations. Before we turn to this matter it is instructive to recognize that (6.37) and the Dirac equation can be reformulated as a Pfaffian system for w. Write u wj = j vj for the two components of wj and u := [u1 , . . . , un ], v := [v1 , . . . , vn ].

Then the combination of the Dirac equation for w, the identity w∗ G = −w, and (6.37) yield mu − 2∂z v = 0, mv − 2∂¯z u = 0, and ∂z u(z − a) − ∂¯z uG−1 (¯z − a¯ )G + u(F + 21 ) = 0, ∂z v(z − a) − ∂¯z vG−1 (¯z − a¯ )G + v(F − 21 ) = 0. Using the first two equations to eliminate the ∂z v and ∂¯z u in the second pair of equations, we obtain the Pfaffian system d(u, v) = (u, v)P dz + (u, v)Q d z¯ ,

(6.44)

256


where

−F − 12 P = m (¯z − a∗ ) 2 0 Q= m (¯ z − a∗ ) 2

− a) (z − a)−1 , 0 m (z − a) 2 (¯z − a∗ )−1 , F − 12

m (z 2

and a∗ := G−1 a¯ G. Remark 6.5.1. Note that if A is the matrix of a linear transformation with respect to the basis {wi }, then the adjoint of A for the L2 inner product on the space spanned by {wi } has a matrix in the basis {wi } given by A∗ = G−1 A¯ τ G,

(6.45)

where A¯ τ is the usual conjugate transpose. We write a∗ for G−1 a¯ G for brevity. The usual conjugate transpose of a is a¯ , and we will always use a¯ for this matrix to avoid confusion. In fact, in this section it is illuminating to use the notation A∗ for (6.45). We write A¯ τ for the usual conjugate transpose. Note that ck∗ and w∗ will retain their usual meaning. It is useful in what follows to separate the ai and a¯ i derivatives, and so we write ∂a w :=

n ∂w dai ∂ai i=1

and

∂¯a w :=

n ∂w d a¯ i . ∂ a¯ i i=1

As a first step toward obtaining the deformation equations we prove the following result. Theorem 6.5.2 The row vectors w and w∗ satisfy the differential equations ∂a w + ∂z wda = w

(6.46)

¯ ∂¯a w∗ + ∂¯z w∗ d a¯ = w∗ ,

(6.47)

and

where :=

m [c1 , da]. 2

The matrix-valued one-form is determined by F : i,j = Fi,j

dai − daj . ai − aj


257

Proof. The second relation is obtained from the first by applying the conjugation ∗. Thus we can concentrate on (6.46). In the following calculation the appearance of many of the equations will be simpler if we introduce Ai :=

m ai 2

and

A=

m a. 2

Since i ∂a wℓi = −wℓ−1 dAi

and

∗i dAi ∂a wℓ∗i = −wℓ+1

and differentiation with respect to ai commutes with the Dirac operator (and so maps solutions of the Dirac equation into new solutions of the Dirac equation), we easily see that ∗ c−1 (∂a w + ∂z wda) = c−1 (∂a w + ∂z wda) = 0.

Thus the local expansion coefficients of the left-hand side of (6.46) vanish at all negative levels ℓ. But c0 (∂a w) = −dAc1 and c0 (∂z wda) = c1 dA, and so it follows that the c0 coefficients of both sides of (6.46) are equal. The uniqueness result Theorem 6.1.3 then implies that both sides of (6.46) must be equal. Our strategy to obtain the deformation equations for F and G and also to find the expression for the coefficients c1i (Wi ) and c¯1i (Wi ) that appear in the derivative formula for the tau functions in terms of F and G is to simply explore the identities at low levels for the local expansions of the equations (6.37) and (6.46). The following identities are easy to check and facilitate these calculations: c−1 (∂a w) = −dA, ∗ c−1 (∂a w) = 0, c0 (∂a w) = −dAc1 , c0∗ (∂a w) = ∂a c0∗ , c1 (∂a w) = ∂a c1 − dAc2 , c1∗ (∂a w) = ∂a c1∗ − dAc0∗ .

(6.48)

New information is extracted from (6.37) by computing the c0∗ coefficients of both sides. One obtains ¯ 1∗ + c1 Ac ¯ 0∗ = c0∗ [A, c1 ]. −Ac

Multiplying both sides of this equation by c0∗−1 and using the identity c1∗ c0∗−1 = c1∗ (w∗ ) = c¯1

258


yields ¯ = c0∗ [A, c1 ]c∗−1 , [c¯1 , A] 0 or F¯ = −GF G−1 .

(6.49)

Equating the c1 coefficients on both sides of (6.37), we find that ¯ 0∗ = −c1 [c1 , A]. ¯ + c¯0∗ Ac c1 − [c2 , A] − A This is most useful for us as the identity ¯ − A∗ − c1 [c1 , A]. −[c2 , A] = −c1 + A

(6.50)

Now we turn to the identities that arise in the low-order expansion for (6.46). Using (6.48) we see that the c0∗ coefficient of (6.46) is ∂a c0∗ = c0∗ [c1 , dA]. Thus where as above,

∂a G = G,

(6.51)

:= [c1 , dA]. Equation (6.51) is half the deformation equation for G. To find the other half we just take the complex conjugate of both sides of (6.51) and use the fact that ¯ = G−1 (see Proposition 6.3.1). We obtain G ¯ ∂¯a G = −G,

(6.52)

¯ is just the complex conjugate where ¯ ij = F¯i,j

d a¯ i − d a¯ j . a¯ i − a¯ j

Thus ¯ da,a¯ G = G − G.

(6.53)

This is the deformation equation for G. Next consider the c1 coefficient of (6.46); we obtain ∂a c1 − dAc2 = −c2 dA + c1 [c1 , dA], or ∂a c1 = −[c2 , dA] + c1 [c1 , dA].

(6.54)


259

We want to find an equation for the derivative ∂a F that can be written in terms of G and F . First observe that ∂a F = [∂a c1 , A] + [c1 , dA]. Since [c1 , dA] = is determined by F , we need only deal with the first term in this last equation. Taking the commutator of (6.54) with A and making use of the derivation rule for iterated commutators (the Jacobi identity), we see that [∂a c1 , A] = −[[c2 , A], dA] + [c1 , A][c1 , dA] + c1 [[c1 , A], dA]. Substituting (6.50) in the preceeding equation, we find that [∂a c1 , A] = −c1 + A − A∗ − c1 [c1 , A] , dA + [c1 , A][c1 , dA] + c1 [[c1 , A], dA]. Add this to [c1 , dA] to obtain ∂a F = [F, ] − [A∗ , dA].

(6.55)

This is the first half of the deformation equation for F , and to find the ∂¯a derivative of F one might take the complex conjugate of both sides of (6.55) to get ¯ ∗ , d A]. ¯ ¯ − [A ∂¯a F¯ = [F¯ , ] In principle one can substitute (6.49) in this last result to obtain an equation for ∂¯a F . However, it is a little simpler to proceed differently. Substitute w∗ = −wG−1 in (6.47) and use the deformation equation for G to simplify the result. One obtains ¯ = ∂¯a w + ∂¯z w(dA)∗ = 0. ∂¯a w + ∂¯z wG−1 d AG Equating the c1 coefficient of the left-hand side of (6.56) to 0, one obtains ¯ − (dA)∗ . ∂¯a c1 = d A Then ∂¯a F = ∂¯a [c1 , A] = [∂¯a c1 , A] = [A, (dA)∗ ]. The deformation equation for F can thus be written dF = [F, ] − [A∗ , dA] + [A, (dA)∗ ]. We formulate these results in a theorem (see [135]). Theorem 6.5.3 The n × n matrices F and G are defined by F = [c1 , A], and G = −c0∗ ,

(6.56)

260


where A = m2 a. The matrix of one-forms = [c1 , dA] is zero on the diagonal and has matrix elements dai − daj for i = j. i,j = Fi,j ai − aj The matrices F and G satisfy the deformation equations

¯ = G + ¯ τ G, dG = G − G dF = [F, ] − [A∗ , dA] + [A, (dA)∗ ].

(6.57)

The matrices F and G satisfy the symmetry conditions ¯ τ = G, G F ∗ = G−1 F¯ τ G = F.

(6.58)

Furthermore, the deformation equations respect these symmetry conditions in the sense that if the symmetry conditions are valid at one point a0 , then the symmetry conditions will be valid on any path-connected domain containing a0 in which the deformation equations are satisfied. Proof. We have already established all but the second version of the first deformation equation (6.57) and the first of the symmetry relations. To prove the second condition in (6.58), recall (6.18): c¯1τ = c1∗ (w∗ ) = c¯1 . Thus c1τ = c1 , and it follows that F τ = −F . Combined with (6.49) we see that G−1 F¯ τ G = −G−1 F¯ G = F.

It is easy to see that c1τ = c1 also implies τ = −, and this establishes the second version of the first deformation equation. To prove the last part of the theorem we use the deformation equations to compute τ ¯ τ − G) + ¯ τ − G). ¯ τ − G) = G + ¯ τ G = (G ¯ τ (G ¯ τ G − G − d(G

¯ τ − G satisfies a linear differential equation along smooth paths in domains Thus G ¯ τ − G vanishes at a point where the deformation equations are satisfied. Thus if G in such a path-connected domain, it must vanish at all points in the domain. Using the deformation equations and X ∗∗ = X one easily calculates dF ∗ = d(G−1 F¯ τ G) = [F ∗ , ] − [A∗ , dA] + [A, (dA)∗ ].

Thus d(F ∗ − F ) = [(F ∗ − F ), ],

and this implies that if F ∗ −F vanishes at one point, it is zero on any path-connected domain on which the deformation equations are satisfied.


261

As this theorem demonstrates, the symmetries (6.58) are naturally associated with the deformation equations. There are a number of other symmetries that it is useful to keep in mind, so we summarize them here: ¯ = 1, GG F τ = −F, −1 ¯ G F G = −F, τ = −, T¯ = −T , T τ = −T .

(6.59)

Mostly these follow from (6.58) and the additional symmetries c¯0∗ c0∗ = 1 and c1τ = c1 .

6.5.1 A Deformation-Theoretic Expression for d log τ We turn now to the expression for d log τ [−] (ma) in terms of solutions to the deformation equations. We assume throughout this discussion that a ∈ Cn \HC, the site configurations without a horizontal coincidence. Our starting point is the representation of Theorem 6.4.3, d log τ [−] (ma) = −

im j ∗j c (Wj ) daj − c1 (Wj ) d a¯ j . 2 j 1

Comparing the level-0 expansion coefficients, we see that Wj∗ = Wj , from which it follows that ∗j

j

c1 (Wj ) = c1 (Wj ). Thus we can rewrite the expression for d log τ as d log τ [−] (ma) = −

im j j c (Wj ) daj − c1 (Wj )d a¯ j . 2 j 1

(6.60)

Let W = [W1 , . . . , Wn ] denote the row vector of Ising response functions. We have already seen that W = iw(1 − c0∗ )−1 = iw(G + 1)−1 . Since we see that

T = i(1 + c0∗ )(1 − c0∗ )−1 = i(1 − G)(1 + G)−1 , G = (1 + iT )(1 − iT )−1 ,

262


from which it follows that (G + 1)−1 = 12 (1 − iT ). Thus W = 12 w(i + T ),

(6.61)

and hence c1 (W) = 21 c1 (i + T ). Therefore the diagonal entries of c1 (W) can be written j,k j,j j c1 (Wj ) = 12 (ic1 + k c1 Tk,j ).

(6.62)

Since Tj,j = 0, the second term on the right-hand side of (6.62) does not involve any diagonal entries for c1 . The off-diagonal entries of c1 have the deformationtheoretic expression 2Fj,k j,k c1 = − . m(aj − ak ) j

j,j

Thus to write c1 (Wj ) in deformation-theoretic terms it is enough to do so for c1 . Return to (6.50) and make note that the commutator [c2 , A] has diagonal entries that are zero. Hence j,k j,j ∗ (6.63) c1 = A¯ j − Aj,j − c1 Fk,j . k =j

Since Fj,j = 0, the sum on the right-hand side does not involve the diagonal entry j,j c1 , and this translates to the deformation-theoretic expression j,j ∗ c1 = A¯ j − Aj,j +

2Fj,k Fk,j . m(aj − ak ) k=j

We take a closer look at the contribution to d log τ [−] (ma) made by the second term in (6.62). Note that j,k c1 Tk,j daj = Tr(c1 T da) = Tr(T dac1 ). (6.64) j,k

Recall that c1τ = c1 and T τ = −T and use Tr(X τ ) = Tr(X) to see that Tr(c1 T da) = −Tr(daT c1 ) = −Tr(T c1 da).

(6.65)

Averaging the results (6.64) and (6.65), we see that the contribution to d log τ [±] (a) from the second term in (6.62) is −

im j,k i i c Tk,j daj = Tr(T [c1 , dA]) = Tr(T ). 4 j,k 1 4 4

(6.66)


263

Taking complex conjugates of this result and noting the T¯ = T , we see that i im j,k ¯ ¯ c¯1 Tk,j d a¯ j = − Tr(T ). 4 j,k 4 Thus (6.60) becomes d log τ [−] (ma) =

m j,j i j,j ¯ (c daj + c¯1 d a¯ j ) + Tr(T − T ). 4 j 1 4

Now we use the deformation equations = G−1 ∂a G,

¯ = −∂a¯ G · G−1 ,

and the fact that T = i(1 − G)(1 + G)−1 and G commute to see that ¯ = Tr(T G−1 dG) = idTr(log(G(G + 1)−2 )). Tr(T − T ) Since G is Hermitian symmetric and positive definite, there exists a Hermitian symmetric matrix H such that

¯ = 1, it follows that Since GG

G = e2H . H¯ τ = H, H¯ = −H.

(6.67)

(6.68)

Clearly, G(G + 1)−2 =

1 −2 ch H, 4

so ¯ = −2idTr(log(ch H )) = −2id log det(ch H ). Tr(T − T ) Thus we have the following result. Theorem 6.5.4 For a ∈ Cn \HC, the logarithmic derivative of the tau function, im j j c (Wj ) daj − c¯1 (Wj ) d a¯ j , d log τ [−] (ma) = − 2 j 1 can be rewritten in deformation-theoretic terms as 1 m j,j j,j (c1 daj + c¯1 d a¯ j ) + d log det(ch H ), d log τ [−] (ma) = 4 j 2

(6.69)

j,j

where the diagonal terms c1 are given by j,j

∗ + c1 = A¯ j − Aj,j

2Fj,k Fk,j . m(aj − ak ) k=j

(6.70)

264


Note that this result implies that τ [+] (ma) is also computable in terms of solutions to the deformation equations, since we know that τ [+] (ma) = τ [−] (ma)Pf(−T ) and T = i(1 − G)(G + 1)−1 .

6.5.2 Rotational Invariance for τ [±] The deformation equations have both translational and rotational symmetries. Using the deformation equations, we see that 7 8 7 8 ∂ ∂ dG = dG = 0, ∂ak ∂ a¯ k k k since 8 7 8 7 ∂ ∂ = = 0. ∂ak ∂ a¯ k k k These equalities coupled with the deformation equations also imply that 8 7 > ∂ m? = 0. = − A∗ , dF ∂ak 2 k A similar calculation shows that 8 7 > m? ∂ = 0. = A, dF ∂ a¯ k 2 k This implies that solutions to the deformation equations are invariant under simultaneous translation ai → ai +b of all the coordinates ai . An elementary calculation shows that 7 8 ai − aj ∂ ∂ − a¯ k = ak Fi,j = 0, = Fi,j ∂ak ∂ a¯ k ai − aj k i =j i=j ¯ implies since F is skew-symmetric, F τ = −F . A precisely analogous result for that 8 7 ∂ ∂ = 0. dG ak − a¯ k ∂ak ∂ a¯ k k Since [A∗ , dA]

7 k

∂ ∂ − a¯ k ak ∂ak ∂ a¯ k

8

= [A∗ , A]


and ∗

[A, (dA) ]

7

k

∂ ∂ ak − a¯ k ∂ak ∂ a¯ k

8

265

= −[A, A∗ ],

the deformation equations for F imply that 8 7 ∂ ∂ − a¯ k = −[A∗ , A] − [A, A∗ ] = 0. dF ak ∂a ∂ a ¯ k k k This implies that both F and G are invariant under the simultaneous rotation ai → eiθ ai of all the ai . This rotational invariance has some interesting consequences for the rotational invariance of the scaling limits for the Ising model. We can’t quite prove complete rotational invariance for these scaling limits because of the limitations of our transfer-matrix proof for the convergence of the scaling limit. For example, the site configurations {ai } in which there is a coincidence of second coordinates for some pair of sites ai and aj with i = j are excluded in our proof of convergence. However, we can do a little better than this. On a finite square lattice, the spin correlations for plus boundary conditions and for open boundary conditions are obviously invariant under rotation by 90 degrees coupled with the interchange of the horizontal and vertical coupling. This remains true in the thermodynamic limit, and since the coupling dependence washes out in the scaling limit, it follows that the scaling functions are invariant under global rotation of all the sites by 90 degrees. The explicit formulas for the scaling functions then show that they are smooth functions of the site coordinates at all site configurations {a1 , a2 , . . . , an } for which there are no simultaneous coincidences of horizontal and vertical coordinates for different pairs {ai1 , ai2 } and {aj1 , aj2 }. We next introduce a rotationally invariant set E that consists of all site configurations {a1 , a2 , . . . , an } for which there exists a distinct pair {ai1 , ai2 } and {aj1 , aj2 } with i1 = i2 and j1 = j2 such that the Euclidean inner product (ai1 − ai2 ) · (aj1 − aj2 ) equals 0. Evidently, E consists of all site configurations that can be rotated into bad position, one for which we are not able to prove the convergence of the scaling limits. However, we can use the deformation theory to prove the following result. Theorem 6.5.5 For all configurations of sites {a1 , a2 , . . . , an } ∈ Cn \E the scaling functions τ [±] (a) are rotationally invariant in the sense that τ [±] (eiθ a1 , eiθ a2 , . . . , eiθ an ) = τ [±] (a1 , a2 , . . . , an ). Furthermore, the scaling functions τ [±] have unique rotationally invariant smooth extensions to all of Cn \C, where C is the subset of Cn consisting of all configurations of sites for which there is a coincident pair ai = aj for i = j . Proof. Write d log τ [−] (ma) = ,

266


where is the right-hand side of (6.70) and Rθ denotes the action of multiplication by eiθ on Cn (global rotation by θ). Then using Rθ∗ G = G and Rθ∗ F = F it is simple to see that Rθ∗ = .

This doesn’t immediately imply that the tau function is rotationally invariant, but because this tau function has a “rotationally invariant boundary condition” at ∞ this does follow. Suppose that a ∈ Cn \E. Then we know that lim τ [−] (ta) = 1,

t→∞

since any real multiple of a ∈ Cn \E remains in Cn \E and our explicit formula for τ [−] (a) has this limit. Let Ŵ(a) denote the path [1, ∞) ∋ t → ta, which joins a to infinity. Then Theorem 6.4.3 and (6.70) imply ) ) ) ∗ [−] iθ

= log τ [−] (ma). Rθ = −

=− log τ (e ma) = − Ŵ(a) Ŵ(a) Ŵ (eiθ a) Thus τ [−] is rotationally invariant on Cn \E. The same is true for τ [+] , since the two functions differ by a function of G alone that is rotationally invariant. A bit more is true. Let E0 denote the subset of E that consists of site configurations in which there is a pair {ai1 , ai2 } with coincident first coordinates and a second pair {aj1 , aj2 } with coincident second coordinates. Then the same proof shows that τ [±] is smooth and locally rotationally invariant on E\E0 . If a ∈ E0 and all the coordinates ai are distinct, then it is easy to see that eiθ a ∈ E\E0 for all sufficiently small values of θ . We would like to extend τ [±] to a ∈ E0 \C by defining τ [±] (a) := τ [±] (eiθ a) for all θ sufficiently small.

However, there is the issue that the value obtained for θ > 0 might not be the same as for θ < 0. To settle this matter suppose that θ is small and θ > 0. Let an ∈ Cn \E be a sequence of points that converges to a ∈ E0 . Then global rotational invariance on Cn \E implies that τ [±] (eiθ an ) = τ [±] (e−iθ an ). However, since τ [±] is smooth on Cn \E0 and eiθ a and e−iθ a will be in Cn \E0 for all sufficiently small values of θ , it follows that lim τ [±] (eiθ an ) = τ [±] (eiθ a) n

and lim τ [±] (e−iθ an ) = τ [±] (e−iθ a). n

Thus for all sufficiently small values of θ we have τ [±] (eiθ a) = τ [±] (e−iθ a). This finishes the proof that there is a unique rotationally invariant extension. Smoothness is obvious.


267

6.5.3 The MTWB Result for the Two-Point Scaling Functions The application of the deformation analysis to obtain the MWTB results for the scaled two-point functions is straightforward. Consider the two-point scaling functions τ [±] (ma1 , ma2 ). We know that the 2 × 2 matrix G is a self-adjoint positive definite matrix. Thus G = e2H

¯ = G−1 , so H¯ = −H . This for a self-adjoint matrix H . We also know that G implies that H is pure imaginary, and since it is also Hermitian symmetric it must have the form 0 −iψ 2H = iψ 0 for a real-valued function ψ. Thus ch ψ G= i sh ψ

−i sh ψ . ch ψ

We know that F τ = −F , so we can write 0 −if F = , if 0

(6.71)

(6.72)

where the i is inserted for convenience. The other algebraic relation G−1 F¯ τ G = F becomes f¯ = f , so f should be real. Introduce the new variables a = a1 − a2 and b = a1 + a2 . Translation invariance implies that there is no interesting dependence on b, and if we write a = reiθ , rotational invariance implies that F and G are independent of θ . The matrix of one-forms is 0 −if da a = , if da 0 a and one easily computes that

∂ ∂r

1 0 = r if

−if . 0

Since ∂r∂ commutes with G, the specialization of the deformation equation to the r derivative becomes ∂ 0 −iψ ∂ ∂ −1 ∂G ¯ = − . = G 0 ∂r ∂r iψ ∂r ∂r Thus the deformation equation for G implies f =

r dψ . 2 dr

268


To compute the deformation equation for F observe that a1 = 12 (a + b),

a2 = 21 (−a + b),

implies that A= Thus

ma 1 4 0

∂ dA ∂r Since

∂ ∂r

mb 1 0 + −1 4 0 meiθ 1 = 0 4

0 . 1

(6.73)

0 . −1

commutes with F , we see that ∂F ¯ ∂ G − G−1 AG, ¯ dA ∂ = A, G−1 d A . ∂r ∂r ∂r

¯ in this last equation and observe that the Now substitute (6.73) for A and G−1 AG term that involves b drops out of both commutators. A short calculation using 1 0 ch 2ψ −i sh 2ψ G−1 G= 0 −1 −i sh 2ψ − ch 2ψ then shows that

dF m2 r sh 2ψ 0 −i = , i 0 dr 4 from which it follows that 1 d dψ r = 21 m2 sh(2ψ). r dr dr

(6.74)

Next we calculate the log derivative of τ with respect to r. First observe that on ¯ − A∗ , the term involving b again disappears, and we substituting (6.73) into A have ma¯ 1 0 ch 2ψ −i sh 2ψ ∗ −1 ¯ ¯ ¯ A − A = A − G AG = − . 0 −1 −i sh 2ψ − ch 2ψ 4 Thus mr mr ∗ A¯ 1 − A1,1 = − e−iθ (ch 2ψ − 1) = − e−iθ sh2 ψ, 4 2 mr mr −iθ −iθ ∗ e (ch 2ψ − 1) = e sh2 ψ. A¯ 2 − A2,2 = 4 2 Also, 2if 2 2 2 (−if ) = − f = − e−iθ f 2 , ma ma mr 2 −iθ 2 e f . = mr

c11,2 F2,1 = − c12,1 F1,2


Then from (6.70) and d(a1 − a2 )

∂ ∂r

269

= eiθ we see that

$ ∂ m # 1,1 1 m2 r 2 2,2 c1 da1 + c1 da2 sh ψ + f 2 . =− 4 ∂r 8 2r

(6.75)

We need to add this to its complex conjugate to figure the contribution it makes to (6.69). Coupled with the observation that ψ 1 d log det ch H = d log ch , 2 2 we find that (6.69) becomes m2 r 2 r ∂ log τ [−] (ma) =− sh ψ + ∂r 4 4

dψ dr

2

ψ + d log ch 2

.

(6.76)

To obtain the MTWB result, we want to integrate this r = r to r = ∞. To do this it is useful to know the behavior of ψ(r) as r → ∞. Observe that ψ τ [−] (ma). (6.77) τ [+] (ma) = τ [−] (ma)Pf(−T ) = tanh 2 Since we know that τ [+] (ma) tends to 0 and τ [−] (ma) tends to 1 as r → ∞, it follows that ψ(r) tends to 0 as r → ∞. Thus we can integrate (6.76) from r to infinity to obtain 7 8 ) dψ 2 1 ∞ ψ 2 2 [−] , (6.78) r dr m sh ψ − exp τ (ma) = ch 2 4 r dr and using (6.77) we see that τ

[+]

ψ (ma) = sh 2

1 exp 4

)

r

∞

7

2

2

r dr m sh ψ −

dψ dr

2 8

.

(6.79)

Equation (6.74) and (6.78) and (6.79) are originally due to MWTB [159]. The analysis we have presented here follows SMJ [135] quite closely. The problem of characterizing the solution to (6.74) that is appropriate for the Ising model is a subtle one. This problem is considered in [87], where the solution is characterized by a careful asymptotic matching at r = ∞. The large- r asymptotics of the two-point scaling function are easy to find by an application of the Laplace method to the formulas found at the end of Chapter 2. This gives rise to the question of how this particular solution behaves in a neighborhood of r = 0. This in turn is a special case of a more general problem referred to as the “connection problem” for solutions to Painlevé equations. The substitution η(r) = e−ψ(r) transforms (6.74) (for m = 1) into η′′ = η−1 (η′ )2 − r −1 η′ + η3 − η−1 ,

270


which is a special case of a Painlevé equation of the third kind. Painlevé equations have an interesting history. They are second-order nonlinear ordinary differential equations that depend rationally on the function and its derivatives, and they are characterized by solutions that have the Painlevé property. The solutions to a differential equation have the Painlevé property if the only singularities of solutions to the equation that depend on initial conditions are “pole-type” singularities. In 1888, this property was used by Sonia Kovalevskaya to discover the integrable regime for the rotating top that now bears her name. In 1900, Painlevé did an analysis to find all the second-order equations that have this property (a gap in this analysis was filled by Gambier in 1910). They discovered six families of differential equations, now referred to as Painlevé I–VI, whose integration led to new transcendental functions. This analysis can be found in Ince [60] along with a discussion of what is meant by “new transcendentals.” Fuchs discovered that Painlevé transcendents arise in the solution of certain monodromy-preserving deformations of linear ordinary differential equations. Later, Schlesinger [139] generalized the notion of monodromy-preserving deformations and showed that the solutions to these deformation equations have the Painlevé property. Schlesinger’s analysis was considerably clarified by Malgrange [85]. The connection between Painlevé equations and monodromy-preserving deformations was pursued by Garnier in the period 1912–1920, but it dropped out of sight until Sato, Miwa, and Jimbo revived interest in the subject in 1978–1980. The principal English-language reference on the Painlevé trancendents, Ince [60], doesn’t even mention the connection with monodromy-preserving deformations. Knowledge of the connection between monodromy-preserving deformations and the Painlevé equations appears to have been the inspiration that led Sato, Miwa, and Jimbo to go from the observation of Wu, McCoy, Tracy, and Barouch that Painlevé transcendents arise in the description of the scaled two-point Ising correlations to their theory of holonomic quantum fields. Miwa, Jimbo, and Ueno [67] generalized an example of Flaschka and Newell [53] to give a version of “monodromy-preserving deformations” that included deformations that fix the local Stokes multipliers at irregular singular points. A geometric account of these deformations can be found in Malgrange [85]. Miwa, Jimbo, and Ueno showed that the Painlevé VI equations are naturally associated with the monodromy-preserving deformations of a 2 × 2 matrix system of ordinary differential equations with simple pole-type singularities at three points on the Riemann sphere. They also showed that the other Painlevé transcendents arise in the integration of the deformation equations associated with linear differential equations with irregular singular points that arise from the coalescence of these simple poletype singularities into irregular singular points. This association of linear equations with the Painlevé equations has been exploited to analyze solutions to the Painlevé equations in much the same way that solutions to KdV are analyzed as isospectral deformations of a linear differential operator. A principal example of this is the connection problem. In (6.74) above, for example, it is possible to single out a particular solution by specifying its asymptotics at the singular point r = 0 or at the singular point r = ∞. The connection problem is to determine the asymptotics

6.6 Short-Distance Behavior of the Scaling Functions

271

at r = 0 from the asymptotics at r = ∞ or vice versa. Results on this problem can be found in [87], [64], and [63].

6.6 Short-Distance Behavior of the Scaling Functions The most detailed short-distance asymptotics for the two-point scaling function τ [−] (a1 , a2 ) are due to Tracy [153] with improvements in the analysis due to Tracy and Widom [154]. By analyzing the Fredholm determinant formula for the twopoint function, which is given at the end of Chapter 3, they find that 1

τ [−] (0, a) ∼ c|a|− 4 where c = e3ζ

as a → 0,

′ (−1)− 1 log 2 6

A less-refined result is known for the short-distance behavior of the n-point scaling functions. Using the characterization d log τ [−] (ma) = −

im j j c (Wj ) daj − c1 (Wj ) d a¯ j 2 j 1

given above, it is shown in [115] that lim d log τ [−] (ma) =

m→0

1 d log |ai − aj |2ǫi ǫj 2 |ǫ|=0 i<j

for n even and lim d log τ [−] (ma) =

m→0

1 d log |ai − aj |2ǫi ǫj 2 1 i<j |ǫ|=± 2

for n odd. In each case ǫ is an n-tuple with entries ± 12 and |ǫ| =

ǫj .

j

The first of these formulas is due to Luther and Peschel [82]. The proof in [115] is essentially an analysis of the solutions Wj to the Dirac equation in the limit m → 0. A crucial ingredient in this calculation is the following explicit formula for the mass-zero limit of the Green function G[−] (x, y; a) (when n is even): n 1 j =1 uj (x)vj (y) g(x, y) G0 (x, y; a) = − , n g(x, y) 4πi j =1 uj (x)vj (y)

272


where 1

uj (x) = (x − aj )− 2 g(x, y) = −

c(ǫ)

|ǫ|=0

(x − ak ) 12

1

k =j

,

(aj − ak ) 2

− aj )ǫj (y − aj )−ǫj

j (x

x−y

,

where ǫ = (ǫ1 , ǫ2 , . . . , ǫn ), each ǫj is ± 12 , |ǫ| = and

j

|ai − aj |2ǫi ǫj , 2ǫi ǫj |ǫ|=0 i<j |ai − aj |

ǫj , c(ǫ) = 1

vj (x) = (x − aj )− 2

i<j

|ǫ|=0,ǫj = 21

c(ǫ)

(x − ak )ǫk . (aj − ak )ǫk k =j

There is a similar formula that is valid when n is odd. Although it is not mentioned in [115], the technique developed there is also sufficient to deal with the shortdistance behavior of τ [+] (ma). One needs the analogue of the representation (6.4.3) for d log τ [+] (ma) and the mass-zero limit of the Green function G[+] (x, y; a) (for n even). This mass-zero limit is seen to be the same as the formula we gave above for the limit of the subcritical Green function except for a change of sign on the diagonal. The upshot of these developments is that the limiting behavior of d log τ [+] (ma) as m → 0 is the same as the limiting behavior for d log τ [−] (ma) as m → 0 when n is even.

Appendix A Spin Representations of the Orthogonal Group

In this chapter we prove the Pfaffian formulas for matrix elements in spin representations of the orthogonal group that are used in the text to control the thermodynamic and scaling limits for the Ising correlations.

A.1 Grassmann Algebra We start by recalling the multilinear algebra that is used in defining the Fock representations of Clifford algebras. This is also useful for the “Grassmann calculus” that is developed for spin representations. Suppose that V is an n-dimensional complex vector space. The k-fold tensor product V ⊗k := V ⊗ V ⊗ · · · ⊗ V k factors

is defined so that for any basis ej , j = 1, . . . , n, the nk vectors ej1 ⊗ ej2 ⊗ · · · ⊗ ejk are a basis for V ⊗k . Let Sk denote the permutation group on k letters. Then for v = v1 ⊗ v2 ⊗ · · · ⊗ vk and σ ∈ Sk the map vσ := vσ (1) ⊗ vσ (2) ⊗ · · · ⊗ vσ (k) extends by linearity to a right action of Sk on V ⊗k . The space of antisymmetric k tensors over V , Altk (V ), can be defined either as a subspace of V ⊗k or a quotient of this vector space (by the relations that identify tensor products that have two equal factors with 0). It is probably useful for the reader to be aware of these

274

Appendix A. Spin Representations of the Orthogonal Group

two possibilities, but for simplicity we always think of Altk (V ) as the subspace of vectors v ∈ V ⊗k that transform by the signature character of the permutation group, vσ = sgn(σ )v, where sgn(σ ) is plus or minus 1 depending on whether σ is an even or odd permutation. It is easy to check that V ⊗k ∋ v → alt(v) :=

1 sgn(σ )vσ k! σ ∈S k

defines a projection from V ⊗k onto Altk (V ). It is useful for us to define the wedge product of u ∈ Altk (V ) and v ∈ Altℓ (V ) by √ (k + ℓ)! u ∧ v := √ √ alt(u ⊗ v) ∈ Altk+ℓ (V ). k! ℓ! Choosing the normalization constant with square roots is not necessary (and perhaps not quite standard), but it simplifies some results for duality and inner products that are important for us. Following Spivak [150] we show that this product is associative (Spivak’s normalization constant is (k+ℓ)! , but any constant of the form k!ℓ! φ(k+ℓ) works in the proof of associativity). φ(k)φ(ℓ) Suppose that u ∈ V ⊗k and v ∈ V ⊗ℓ . We first observe that if alt(u) = 0 or alt(v) = 0 then alt(u ⊗ v) = 0. To see this, note that the direct product Sk × Sℓ is an obvious subgroup of Sk+ℓ . Thus Sk+ℓ is a disjoint union of right cosets (Sk ×Sℓ )g, and we can write (k + ℓ)!alt(u ⊗ v) = sgn((σ × τ )g)(uσ ⊗ vτ )g, g

σ ∈Sk τ ∈Sℓ

where the sum on g runs over distinct representatives for the right cosets of Sk × Sℓ in Sk+ℓ . Since sgn((σ × τ )g) = sgn(σ )sgn(τ )sgn(g), it follows that (k + ℓ)!alt(u ⊗ v) = k!ℓ! sgn(g)alt(u) ⊗ alt(v)g. g

From this it follows directly that alt(u ⊗ v) = 0 if either alt(u) = 0 or alt(v) = 0. Thus for u ∈ V ⊗k , v ∈ V ⊗ℓ , and w ∈ V ⊗m , alt ((alt(u ⊗ v) − u ⊗ v) ⊗ w) = 0, or alt(alt(u ⊗ v) ⊗ w) = alt(u ⊗ v ⊗ w). In a similar fashion it follows that alt(u ⊗ alt(v ⊗ w)) = alt(u ⊗ v ⊗ w).

A.1 Grassmann Algebra

275

Inserting the appropriate normalizations in these last two equalities and specializing to u ∈ Altk (V ), v ∈ Altℓ (V ), and w ∈ Altm (V ), we obtain (u ∧ v) ∧ w =

=

(k + ℓ + m)! alt(u ⊗ v ⊗ w) = u ∧ (v ∧ w). k!ℓ!m!

By an obvious induction we see that if vj ∈ V for j = 1, . . . , k, then 1 v1 ∧ v2 ∧ · · · ∧ vk = √ sgn(σ )vσ (1) ⊗ vσ (2) ⊗ · · · ⊗ vσ (k) . k! σ ∈Sk Now let V ∗ denote the dual of V (the space of complex linear functionals on V ). Then V ∗⊗k and V ⊗k are dual to one another via the pairing ϕ1 ⊗ ϕ2 ⊗ · · · ⊗ ϕk (v1 ⊗ v2 ⊗ · · · ⊗ vk ) =

k

ϕj (vj ),

(A.1)

j =1

where ϕj ∈ V ∗ and vj ∈ V for j = 1, . . . , k. Furthermore, n 1 sgn(σ )sgn(τ ) ϕσ (j ) (vτ (j ) ) k! σ,τ j =1 ' ' ϕσ (1) (v1 ) · · · ϕσ (1) (vk ) ' 1 ' .. .. sgn(σ ) ' = . . ' k! σ ' ϕσ (k) (v1 ) · · · ϕσ (k) (vk ) ' ' ' ϕ1 (v1 ) · · · ϕ1 (vk ) ' ' ' ' ' .. .. =' ', . . ' ' ' ϕk (v1 ) · · · ϕk (vk ) '

ϕ1 ∧ · · · ∧ ϕk (v1 ∧ · · · ∧ vk ) =

' ' ' ' ' ' '

where |A| = det A. The first equality follows from the definition of the determinant, and the second is a consequence of the fact that the factor sgn(σ ) is just what is needed to put the permuted rows in the second line into natural order in the third line. The simplicity of this duality relation is one reason for our choice of the square root in the normalization factors for the wedge product. Of course, one could achieve the same effect for any choice of normalization for the wedge product by adjusting the duality relation between Alt(V ) and Alt(V ∗ ). One would then need to be careful not to use the “natural” duality (A.1) induced by restriction from the full tensor algebra. Now suppose that e1 , e2 , . . . , en is a basis for V and ϕ1 , ϕ2 , . . . , ϕn is the corresponding dual basis for V ∗ . It is easy to see that the wedge products ϕj1 ∧ ϕj2 ∧ · · · ∧ ϕjk

for j1 < j2 < · · · < jk

(A.2)

276


span Altk (V ∗ ), and since the duality relation we have proved shows that δjℓ iℓ ϕj1 ∧ ϕj2 ∧ · · · ∧ ϕjk (ei1 ∧ ei2 ∧ · · · ∧ eik ) =

(A.3)

ℓ

whenever both (j1 , j2 , . . . , jk ) and (i1 , i2 , . . . , ik ) are in natural order, it follows that the vectors (A.2) are linearly independent. Hence dim(Altk (V ∗ )) = nk . Conversely, the vectors ej1 ∧ ej2 ∧ · · · ∧ ejk with (j1 , j2 , . . . , jk ) in natural order clearly span Altk (V ), and the same duality relation (A.3) shows that they are linearly independent. Hence dim(Altk (V )) = nk as well. We can now introduce the creation and annihilation operators that are important for the Fock representations of the Clifford relations. Suppose that v ∈ V . Then we define Altk (V ) ∋ w → c(v)w := v ∧ w ∈ Altk+1 (V ). Note that we usually write |0 = 1 ⊕ 0 ⊕ · · · ⊕ 0 for the vacuum vector in Alt(W+ ), and in particular, we define |0 → c(v)|0 := v ∈ Alt1 (V ) = V . This defines an action of the “creation operator” c(v) on the alternating tensor algebra n * Altk (V ). Alt(V ) := k=0

∗

Now suppose that ϕ ∈ V is an element in the dual of V . Then c(ϕ) is a map defined on Alt(V ∗ ). The transpose c(ϕ)τ is thus a map on the dual space Alt(V ∗ )∗ , which we identify with Alt(V ). We now calculate the action of c(ϕ)τ on a vector v = v1 ∧ v2 ∧ · · · ∧ vk in Altk (V ). We pair this vector with c(ϕ) for ∈ Alt(V ∗ ). Only the component of in Altk−1 (V ∗ ) makes a contribution, and so we might as well suppose that = ϕ1 ∧ ϕ2 ∧ · · · ∧ ϕk−1 , since that component of is a sum of such terms. For this choice of , ' ' ' ϕ(v1 ) ··· ϕ(vk ) '' ' ' ϕ1 (v1 ) · · · ϕ1 (vk ) '' ' (c(ϕ)τ v) := vc(ϕ) = ' ' .. .. ' ' . . ' ' 'ϕk−1 (v1 ) · · · ϕk−1 (vk )' =

k (−1)j −1 ϕ(vj )vj , j =1

A.2 Fock Representations of the Clifford Algebra

277

where vj = v1 ∧ v2 ∧ · · · ∧ vj −1 ∧ vj +1 ∧ · · · ∧ vk . It follows that c(ϕ)τ v =

n

(−1)j −1 ϕ(vj )vj .

j =1

We write a(ϕ) := c(ϕ)τ and refer to a(ϕ) as an annihilation operator. Using this last formula it is a simple matter to check that c(v)a(ϕ) + a(ϕ)c(v) = ϕ(v)I

for v ∈ V , ϕ ∈ V ∗ .

These are the canonical anticommutation relations.

A.2 Fock Representations of the Clifford Algebra Next we introduce the Fock representations of the Clifford algebra. Suppose that W is a finite-dimensional complex vector space with a distinguished nondegenerate complex bilinear form (·, ·). We refer to such spaces as orthogonal spaces. The Clifford algebra of the orthogonal space W is the associative algebra with unit e generated by the elements x ∈ W subject to the multiplicative relations xy + yx = (x, y)e

for x, y ∈ W.

(A.4)

Remark A.2.1. It is probably more common to include a factor of 2 in the generator relations, so that xy + yx = 2(x, y)e. (A.5) √ Evidently x → 2x takes generators of the first type to generators of the second √ type. To avoid many vexatious appearances of the factors 2 in the Grassmann calculus that follows, we adopt (A.4) as the generator relations for the Clifford algebra over W rather than (A.5). Note that one can also pass from one to the other by just scaling the distinguished bilinear form on W by a factor of 2. If (wk ) is a basis for W , then e and the monomials wk1 wk2 · · · wkl with 1 ≤ k1 < k2 < · · · < kl ≤ dim W are a basis for Cliff(W ). Thus the dimension of Cliff(W ) is 2dim(W ) , which is the same as the dimension of Alt(W ). The representations of the Clifford algebra we are interested in arise from a splitting of the space W into isotropic subspaces W± : W = W+ ⊕ W− .

(A.6)

A subspace V is isotropic if the bilinear form (x, y) is zero whenever x, y ∈ V . Because the bilinear form (·, ·) is assumed to be nondegenerate, the subspace W+ in an isotropic splitting can be isomorphically identified with the dual space W−∗ through the bilinear form. It follows that W must be even-dimensional if it has an isotropic splitting. We will often identify objects associated with the isotropic splitting W+ ⊕ W− with ± subscripts. However, when the objects themselves have

278


subscripts we will move the ± to the superscript position without making special mention of this alteration. Later we specialize to isotropic splittings that are orthogonal splittings with respect to a Hermitian inner product on W , but for the present it is cleaner to defer the introduction of the inner product. Each isotropic splitting W = W+ ⊕ W− may be parametrized by an operator Q defined by x for x ∈ W+ , Qx = −x for x ∈ W− . Observe that if Q2 = I , then Q± = 21 (I ± Q) is the projection onto the ±1 eigenspace for Q. Defining W± = Q± W , we see that since Q+ + Q− = I and Q+ Q− = 0, the space W is the direct sum W+ + W− . We leave it to the reader to check that the ±1 eigenspaces of an operator Q with Q2 = I are isotropic if and only if Q is skew-symmetric with respect to the complex bilinear form (·, ·), that is, Qτ = −Q. A skew-symmetric operator Q with Q2 = I is called a polarization. Note that this term is also used to refer to the isotropic splitting. We now define the Fock representation of the Clifford algebra associated with polarization Q. Let FQ (x) := c(x+ ) + a(x− )

for x ∈ W,

(A.7)

where x = x+ + x− is the W+ + W− splitting of x, and x− is identified with an element of the dual of W+ by W+ ∋ x → (x, x− ). The canonical anticommutation relations (A.4) and the fact that W± is isotropic are all that are needed to check that FQ (x)FQ (y) + FQ (y)FQ (x) = (x, y)I

for x, y ∈ W.

(A.8)

Thus FQ satisfies the generator relations for the Clifford algebra of W and so extends algebraically to a representation of Cliff(W ) on Alt(W+ ), which we continue to denote by FQ . We refer to FQ as the Q Fock representation of Cliff(W ). Of principal interest for us is a subgroup G of Cliff(W ), called the Clifford Group, which consists of invertible elements g ∈ Cliff(W ) with the property that gxg −1 = T (g)x

for x ∈ W, and T (g) a linear map on W.

(A.9)

Remark A.2.2. There is a mild schism in the mathematics and physics literature concerning these constructions. It is common in the physics literature to concentrate on the canonical anticommutation relations without introducing the Clifford algebra (this isn’t universal; work in supersymmetry and string theory are exceptions). The Clifford group we just introduced is then associated with the group of linear automorphisms of the canonical commutation relations also

A.3 The Grassmann Calculus

279

called Bogoliubov automorphisms. From a Fock representation of the canonical anticommutation relations over V one can recover a Fock representation of the Clifford algebra over W = V ⊕ V ∗ , where V ⊕ V ∗ is an isotropic splitting of W with the natural bilinear form (u + u∗ , v + v ∗ ) = u∗ (v) + v ∗ (u), with u, v ∈ V and u∗ , v ∗ ∈ V ∗ . Of course, this works the other way as well, so the two constructions are just different ways of talking about the same subject. However, there are circumstances that can make the Clifford algebra formalism more illuminating. In [111] the connection between the Pfaffian formalism and the transfer matrix formalism for the Ising model leads naturally to the identification of a distinguished bilinear form via “Stokes’s theorem.” The introduction of the Clifford algebra associated with this bilinear form makes the relation between the two approaches to the Ising model more transparent. By taking anticommutators of both sides of (A.9) it is easy to see that the linear transformation T (g) must be complex orthogonal, that is, (T (g)x, T (g)y) = (x, y)

for x, y ∈ W.

We write O(W ) for the group of complex orthogonals on W . Later we will see that every T ∈ O(W ) is covered by an element g ∈ G. Furthermore, if gk ∈ G for k = 1, 2 and Tg1 = Tg2 , then g1 g2−1 commutes with all of Cliff(W ). The center of Cliff(W ) consists of all multiples of the identity [28], so T

C∗ → G → O(W ) → 0 defines a central extension of O(W ) by C∗ (the multiplicative group of nonzero complex numbers). In a Fock representation of Cliff(W ), the corresponding representation of G gives rise to a projective representation of O(W ) that is called a spin representation. Our goal in this appendix is to develop some useful formulas for matrix elements in spin representations (first obtained by Sato, Miwa, and Jimbo [133]) with infinite-dimensional generalizations that apply to the correlations of the Ising model. Some aspects of the theory of spin representations are developed as a matter of course, but for reasons of space we refer the reader to the literature for a more complete picture.

A.3 The Grassmann Calculus We begin by explaining the Grassmann calculus, which is at the heart of the Sato, Miwa, and Jimbo analysis. Suppose that Q is a polarization on W associated with the isotropic splitting W+ + W− and let Cliff(W± ) denote the subalgebra of Cliff(W ) generated by x ∈ W± . The multiplicative relations in Cliff(W± ) are precisely the same as in Alt(W ), so that there is an obvious algebraic isomorphism Alt(W± ) ≃ Cliff(W± ),

280


which allows us to identify the two spaces. For example, if wj ∈ W+ for j = 1, . . . , k, then w1 ∧ w2 ∧ · · · ∧ wk ≃ w1 w2 · · · wk . Associated with such an isotropic splitting of W there is a linear map (no longer an algebra homomorphism) opQ : Alt(W ) → Cliff(W ),

(A.10)

defined by linear extension of the map opQ (XY ) = XY

for X ∈ Alt(W+ ), Y ∈ Alt(W− ).

(A.11)

It is not hard to see that this map is well defined and gives a linear isomorphism between Alt(W ) and Cliff(W ), since the map is clearly surjective and the two spaces have the same dimension. We have labeled the map “op” to suggest an analogy between this situation and what is possibly a more familiar situation for the canonical commutation relations. There the map that takes symbols (for say, pseudodifferential operators) to operators has the same type ordering: the differential operator part acts before the multiplicative part. Of course, the symbols here are elements of the Grassmann algebra Alt(W ) rather than the symmetric tensor algebra, and the elements of Cliff(W ) are not really “operators” until we consider a representation of Cliff(W ). However, the Q Fock representation is natural in this regard. In the Fock representation the operator associated with a Grassmann symbol has the annihilation operators acting before the creation operators; this is, of course, how “normal ordered” products work. To further this analogy we denote the inverse of opQ by sbQ (sb is short for “symbol”); we also drop the Q subscripts when the polarization Q is understood. One useful property of the map opQ is FQ opQ (X) |0 = X

for X ∈ Alt(W+ ).

This is easy to check when X is a monomial,√and the reader can verify that it would not be true without additional factors of 2 if we worked with the Clifford relations that have the extra factor of 2 multiplying the bilinear form. Following Sato, Miwa, and Jimbo (though with somewhat different proofs) we determine what the symbols are for the elements g ∈ G in terms of their “induced rotations” T (g). Our first result is to show that if R is a suitable element of Alt2 (W ), then op(eR ) ∈ G and there is an explicit formula for the induced rotation. It is convenient to use the bilinear form on W to identify the space Alt2 (W ) with the space of skew-symmetric maps R : W → W . Suppose that {wk } is a basis for W and {wk∗ } is the dual basis with respect to the bilinear form (so (wj , wk∗ ) = δj k ). Then it is not hard to check that R := 21 k Rwk∗ ∧ wk ∈ Alt2 (W )


281

does not depend on the choice of the basis {wk }. We write R = Rˆ for this map. The factor of 21 is arbitrary, but it simplifies the relation between the induced rotation for op(eR ) and R. It isn’t necessary to choose R such that R = −R τ , but requiring this skew symmetry makes the representation of the element in Alt2 (W ) unique. Writing Rwk∗ = rj,k wj , where rj,k = (Rwk∗ , wj∗ ), we see that R=

1 2

j,k rj,k wj

∧ wk ,

and the matrix rj,k is skew-symmetric when R τ = −R. Observe, however, that the matrix with j, k entry rj,k is not the matrix of R in the basis {wj } unless the basis is self-dual (i.e., (wi , wj ) = δi,j ). The calculation of the induced rotation for op(eR ) (and later the expansion of op(eR ) itself) is much facilitated by choosing a basis for W that respects the isotropic splitting. Let {ek∗ } denote a basis for W+ and let {ek } denote the corresponding “dual” basis for W− (i.e., (ej∗ , ek ) = δj k ). Suppose that W has dimension 2n and define wk : = ek∗

wk+n : = ek

for k = 1, . . . , n,

for k = 1, . . . , n.

(A.12)

We say that a basis for W chosen in this fashion is compatible with the isotropic splitting W+ + W− . The reason for the funny labeling is that the elements of W+ are represented by creation operators and the elements of W− are represented by annihilation operators in the Fock representation. In the physics literature “creation operators” appear with a ∗ (or dagger †) superscript and annihilation operators appear without such superscripts. For example, with this notation the equality op(ej ∧ ek∗ ) = −ek∗ ej looks exactly like the “normal ordering” familar to physicists. The proof of our first result depends on a contraction defined on Alt(W ), which is suggestively written as a “derivative.” Suppose that R ∈ Alt(W ) and that a basis {wk } for W has been chosen. For each k from 1 to 2n we define a linear map δ : Alt(W ) → Alt(W ) δwk L as follows. Suppose that X is a monomial X = wk1 ∧ wk2 ∧ · · · ∧ wkm .

282


If X contains no factor wk , then

δ δwk

L

X = 0.

If X contains exactly one factor wk , then δ X = ±wk1 ∧ · · · ∧ wˆ k ∧ · · · ∧ wkm , δwk L where wˆ k means “omit” the factor wk and the plus or minus sign is determined by whether an even or an odd number of interchanges are needed to move wk from its original position to the leftmost position before dropping it (hence the subscript L for “left”). For example, δ w5 ∧ w3 ∧ w7 = −w5 ∧ w7 . δw3 L The operator δwδ k L is now defined by linear extension of these rules. If X contains more than one factor of wk , then of course, X = 0. However, we want to know that in this case it still makes sense to “calculate” δwδ k L X by the “signed Leibniz” rule. That is, the signed Leibniz rule should give 0 when applied to a monomial with more than one factor of wk . Thus, for example, δ w1 ∧ w3 ∧ w2 ∧ w3 = −w1 ∧ w2 ∧ w3 − w1 ∧ w3 ∧ w2 . (A.13) δw3 L In this example it is obvious that the right-hand side is 0, and in fact it is elementary to check that if X contains two factors wk , then the two terms that result from the application of the signed Leibniz rule precisely cancel out. If X has three or more factors, then each term that arises in the application of the signed Leibniz rule has at least two factors wk and hence vanishes in the Grassmann algebra. The following simple lemma is the key to our first result. Lemma A.3.1 Suppose that X is an even element in the Grassmann algebra Alt(W ) and that {wj } is a basis for W compatible with the isotropic splitting W+ + W− . Then δ [wk , op(X)] := wk op(X) − op(X)wk = op X . δwk∗ L Proof. It is easy to see that for m even, [wk , wk1 wk2 · · · wkm ] =

m j =1

(−1)j −1 {wk , wkj }wk1 · · · wˆ kj · · · wkm ,

(A.14)


283

where {wk , wj } := wk wj + wk wj = (wk , wj ) is the anticommutator. The anticommutator {wk , wkj } is 1 if wkj = wk∗ and 0 otherwise. If the indices are chosen in natural order, k1 < k2 < · · · < km , then op(wk1 ∧ wk2 ∧ · · · ∧ wkm ) = wk1 wk2 · · · wkm . Thus the result is true for monomials and by linear extension for all even X.

We obtain an interesting result by using this lemma to compute the commutator [wi , op(eR )], where R ∈ Alt2 (W ). As described above, we represent 1 2n ∗ R = 21 2n k=1 Rwk ∧ wk = 2 j,k=1 rj k wj ∧ wk ,

where rj k = (Rwk∗ , wj∗ ) is skew-symmetric. In the following sum the terms are nonvanishing only up to ℓ = n. Nonetheless, it is convenient to write the sum out to ℓ = n+1 for the purposes of the calculation that follows: n+1 Rℓ eR = . ℓ! ℓ=0

To find [wi , op(eR )] it suffices to calculate δ δ ℓ ℓ−1 [wi , op(Rℓ )] = op R = ℓop R R , δwi∗ L δwi∗ L

where the second equality follows from the signed Leibniz rule and the fact that R is even. Since rj,k = −rk,j , it is elementary that 2n δ 1 2n ∗ j,k=1 rj,k wj ∧ wk = k=1 ri ∗ ,k wk = −Rwi ∗ = −Rwi , ∗ 2 δwi L

where

i+n i∗ = i−n

for i = 1, . . . , n, for i = n + 1, . . . , 2n,

so that wi ∗ = wi∗ . Since Rwi = Q+ Rwi + Q− Rwi , the normal ordering prescription in op gives us n+1 op Rℓ−1 Rwi [wi , op(e )] = − = −(Q+ Rwi )op(eR )−op(eR )(Q− Rwi ), (ℓ − 1)! ℓ=1 R

or equivalently, (wi + Q+ Rwi )op(eR ) = op(eR )(wi − Q− Rwi ).

284


Multiplying this last equality by xi and summing, we obtain (writing x = (I + Q+ R)x op(eR ) = op(eR )(I − Q− R)x for x ∈ W.

i

xi wi )

(A.15)

We can use this to prove the following theorem. Theorem A.3.1 Suppose that R is skew-symmetric with respect to the bilinear form on W . If I − Q− R is invertible on W and ˆ R := R, then opQ (eR ) is an element of the Clifford group G with induced rotation T (opQ (eR )) = (I + Q+ R)(I − Q− R)−1 . Proof. The transpose of I − Q− R is I + RQ+ , since R τ = −R τ and Q±τ = Q∓ . Thus if I − Q− R is invertible, then so is I + RQ+ , and a little algebra shows that (I + RQ+ )(I − Q− R) = (I − RQ− )(I + Q+ R). Since both factors on the left are invertible, both factors on the right must be invertible as well, and we have (I + Q+ R)(I − Q− R)−1 = (I − RQ− )−1 (I + RQ+ ).

(A.16)

Define T = (I + Q+ R)(I − Q− R)−1 . Then T τ = (I + RQ+ )−1 (I − RQ− ), which is equal to T −1 by (A.16). Thus T ∈ O(W ) and by (A.15), (T x)op(eR ) = op(eR )x for all x ∈ W,

(A.17)

where the Q polarization is understood for op. Observe that in the Fock representation this last relation implies that the null space of F (op(eR )) is invariant under the action of F (x) for all x ∈ W . The Fock representation is irreducible [28], so the null space of F (op(eR )) must be either {0} or all of Alt(W+ ). The vacuum component of F (op(eR ))|0 is clearly nonzero, so the null space must be trivial. Thus F (op(eR )) is invertible, and this implies that op(eR ) is invertible, so that op(eR ) ∈ G with induced rotation given by T . Write |0 = 1 ⊕ 0 ⊕ · · · ⊕ 0 for the vacuum vector in Alt(W+ ) and define the dual vacuum 0| as the linear functional on Alt(W+ ) = C ⊕

n * k=1

Altk (W+ ),


285

C which is the identity on C and 0 on the complement, nk=1 Altk (W+ ). For X ∈ Cliff(W ) the vacuum expectation of X in the Q Fock representation is defined by XQ = 0|FQ (X)|0. The following result characterizes those elements of G that have nonvanishing vacuum expection in the Q Fock representation. Theorem A.3.2 Suppose that g ∈ G and write T := T (g). Then gQ = 0 if and only if Q− T + Q+ is invertible on W . If Q− T + Q+ is invertible, then R := (T − I )(Q− T + Q+ )−1

is skew-adjoint and where

g = gQ opQ (eR ),

(A.18)

ˆ R := R. Proof. Suppose to start that Q− T +Q+ is invertible. It is an elementary calculation using T τ = T −1 and Q±τ = Q∓ to show that R τ = (Q+ + T Q− )−1 (I − T ).

One can see that this is equal to (I −T )(Q− T +Q+ )−1 = −R by cross multiplying to eliminate the inverses. A direct calculation shows that I − Q− R = (Q− T + Q+ )−1 . Thus I − Q− R is invertible, and we can use the defining relation for R to solve for T : T = (I + Q+ R)(I − Q− R)−1 . The previous theorem now implies that there is a nonzero constant λ such that g = λ opQ (eR ). However, since opQ (eR )Q = 1, we find that λ = gQ . This finishes the proof in one direction. We defer the proof that Q− T + Q+ is invertible if gQ = 0 until later, when we have a formula that makes this obvious. Next we turn to a calculation of the symbol for an element g ∈ G when Q− T (g)+ Q+ is not invertible. It is useful to start with the simplest case. Suppose that w ∈ W and that w 2 = 12 (w, w) = 0. In this case w is an invertible element of Cliff(W ), and using the Clifford relations we obtain (x, w) wxw −1 = −xww −1 + (x, w)w −1 = − x − 2 w . (w, w) (x,w) The map x → rw x := x − 2 (w,w) w is orthogonal reflection in the hyperplane “perpendicular” to w (with respect to the bilinear form (·, ·), of course). Thus if (w, w) = 0 then w ∈ G with Tw = −rw .

286


Remark A.3.1. There is a theorem due to Cartan and Dieudonné that every orthogonal transformation is the product of reflections. This gives a slightly different perspective on the significance of Clifford algebras. In some (projective) representations of the orthogonal group (e.g., the Fock representations) the reflections are also the generators of an interesting algebra (the Clifford algebra) whose automorphisms determine the group representation. The strategy for dealing with orthogonals T for which Q− T + Q+ has a null space is simple. We apply reflections to T to kill off this null space. The reflections are covered by Clifford multiplication, and the calculation of the symbol is reduced to the case in which Q− T + Q+ is invertible by an inductive device. Before we introduce this device it is useful to make a few simple observations and to introduce some notation. The transpose of Q− T + Q+ is just T −1 Q+ + Q− = T −1 (T Q− + Q+ ). Since T is invertible, dim ker(Q− T + Q+ ) = dim ker(T Q− + Q+ ). If g ∈ G we write T (g) =

Ag Cg

Bg Dg

(A.19)

for the matrix of T (g) relative to a fixed polarization W+ + W− . Since τ Dg Bgτ τ T (g) = , Cgτ Aτg the relation T (g)τ T (g) = 1 translates into Dgτ Ag + Bgτ Cgτ = 1,

Cgτ Ag + Aτg Cg = 0,

Dgτ Bg + Bgτ Dg = 0, Cgτ Bg + Aτg Dg = 1.

(A.20)

It is also clear that ker(Q− T (g) + Q+ ) = ker Dg . The inductive device mentioned above is described in the following theorem. Theorem A.3.3 Suppose that g ∈ G and that Q is polarization on W . Suppose that dim ker Dg > 0. By (A.19) there exists a nonzero vector v ∈ ker(T Q− + Q+ ). Write v± = Q± v, and choose u− ∈ W− such that (u− , v+ ) = 1. Let u = v+ + u− and write h = ug ∈ Cliff(W ). Then ker Dh ⊂ ker Dg , dim ker Dh = dim ker Dg − 1, and g = v+ h − hv− .


287

Proof. Suppose to start that u ∈ W with (u, u) = 0 and write h = ug. It easy to see that w ∈ W− is in ker Dh if and only if Dg w − 2

(u, T (g)w) u− = 0. (u, u)

If u is chosen such that u− is not in the image im(Dg ), then it is evident that for w to be in ker Dh we must have w ∈ ker Dg and (u, T (g)w) = 0 for all w ∈ ker Dg . Thus ker Dh ⊂ ker Dg . Now choose a nonzero vector v such that (T (g)Q− + Q+ )v = 0. Then it is easy to see that Dg v− = 0 and v+ = −Bg v− . Thus v+ ∈ Bg ker(Dg ). However, the matrix form for T (g) and the fact that T (g) is invertible implies that ker Bg ∩ ker Dg = {0}. The relation Dgτ Bg + Bgτ Dg = 0 shows that Bg maps the kernel of Dg into the kernel of Dgτ . However, since Bg is injective on ker Dg and the two spaces ker Dg and ker Dgτ have the same dimension, it follows that Bg ker(Dg ) = ker Dgτ . Thus v+ ∈ ker Dgτ , and since im(Dg ) = (ker Dgτ )⊥ , it follows that if u− ∈ W− is chosen such that (u− , v+ ) = 1, then u− is not in the range of Dg . Thus for u := v+ + u− we have (u, u) = 0, and as noted above, ker Dh ⊂ ker Dg . However, recall that v− ∈ ker Dg and (u, T (g)v− ) = (u− , v+ ) = 1 = 0. Thus v− ∈ / ker(Dh ), and it follows that ker Dh is a proper subspace in ker Dg with codimension 1. Next we use the Clifford relations to do the calculation hv− = ugv− = uT (g)v− g = −uv+ g = v+ ug − (u, v+ )g = v+ h − g, from which it follows that g = v+ h − hv− .

(A.21)

We can use this to prove the following result. Theorem A.3.4 Suppose that g ∈ G, Q is a polarization on W and dim ker Dg = k. Then there exist a constant c, vectors vj ∈ W , for j = 1, 2, . . . , k, and a ˆ skew-symmetric map R on W such that for R = R, sb(g) = cv1 ∧ · · · ∧ vk ∧ eR .

(A.22)

288


Proof. We use the preceding theorem to prove this by induction. If k = 0 then Theorem A.3.2 applies. If k = 1 then there exists a vector u such that for h = ug, Dh is invertible and so sb(h) = ceR . Recalling (A.21), g = v+ h − hv− , which implies sb(g) = v+ sb(h) − sb(h)v− = c(v+ − v− )eR , since eR is even. With v1 = v+ −v− this finishes the proof for k = 1. Now suppose that the theorem is known for some integer k and suppose that the dimension of ker Dg is k + 1. Then there exists a vector u such that for h = ug the dimension of ker Dh is k and g = v+ h − hv− .

The inductive hypothesis implies that

sb(h) = cv1 ∧ · · · ∧ vk eR , so that sb(g) = cv1 ∧ · · · ∧ vk ∧ vk+1 ∧ eR , where vk+1 = (−1)k v+ − v− . This finishes the proof.

It is useful for us to extend the Grassmann calculus to functionals on Cliff(W ) associated with skew maps Q on W that are not necessarily polarizations. These states are sometimes referred to as quasifree states because the 2k-point function for monomials in the Clifford algebra is combinatorially computable in terms of two-point functions in the same way that it is for Fock representations. The “free” in free states refers to the fact that in quantum field theory the free fields are constructed using Fock representations. Suppose that Q : W → W is a linear map with Qτ = −Q. We write Q± = 1 (1 ± Q) as before, although in this case the maps Q± are no longer projections. 2 Write W := W ⊕ W for the orthogonal space with the distinguished bilinear form

(x1 ⊕ y1 , x2 ⊕ y2 ) = (x1 , x2 ) − (y1 , y2 ). Note the minus sign! Then it is easy to check that Q 1+Q Q := 1 − Q −Q is a polarization on W. The map W ∋ x → x ⊕ 0 ∈ W


289

is obviously an orthogonal embedding and hence extends to a Clifford algebra embedding from Cliff(W ) to Cliff(W). For X ∈ Cliff(W ) we define the Q functional on X by XQ = XQ , where the X on the right is regarded as an element of Cliff(W) via the embedding described above. It is important for us that this consistently extends the Fock states. If Q is a polarization, then we want to check that w1 w2 · · · wk Q = (w1 ⊕ 0)(w2 ⊕ 0) · · · (wk ⊕ 0)Q .

(A.23)

Writing w1 = w1+ + w1− and using the Clifford relations yields w1 w2 · · · wk Q = w1− w2 · · · wk Q = =

k j =2

k j =2

(−1)j (w1− , wj )w2 · · · wˆ j · · · wk Q

(A.24)

(−1)j w1 wj Q w2 · · · wˆ j · · · wk Q .

This reduction formula shows that the Fock expectation of a monomial can be computed in terms of two-point functions. Since the functional on the right-hand side of (A.23) is also a Fock expectation, it has the same reduction formula, and all that needs to be checked to confirm the equality is the equality of the two-point functions. It is easy to see that Thus

Q− (w1 ⊕ 0) = w1− ⊕ (−w1− ).

(w1 ⊕ 0)(w2 ⊕ 0)Q = (w1− ⊕ (−w1− ), w2 ⊕ 0) = (w1− , w2 ) = w1 w2 Q .

Remark A.3.2. We are principally interested in the use of these generalized states to prove product formulas in ordinary (i.e., Fock) representations. However, it is worth noting that the case Q = 0 corresponds to what is called the trace functional on the Clifford algebra. For X ∈ Cliff(W ) we define Tr(X) = XQ=0 .

This functional plays an important role in the discussion of the Ising model with periodic boundary conditions. Suppose that {wj } is a self-dual basis of W (i.e., (wi , wj ) = δij ). Then for any product wα = wα1 wα2 · · · wαk with α1 < α2 < · · · < αk , Tr(wα ) := wα Q=0 = 0.

This is automatic when k is odd. When k is even all the pairings wαi wαj Q=0 = 1 (wαi , wαj ) are equal to 0. Since eQ=0 = 1, we find that the trace functional just 2 picks out X∅ , the coefficient of e in the representation X= X α wα . α

290


Next we introduce an extension of the Grassmann calculus for these states. Suppose that Q is skew-symmetric on W . It is a little easier to work with the Q state if we transform to a representation in which Q is diagonal. Define I I Q= , Q− −Q+ where we think of Q as a linear map Q : W → W ⊕ W. We have written W ⊕ W and not W since Q alters the bilinear form. In fact, Q is an orthogonal map from W to W ⊕ W with the bilinear form on W ⊕ W given by (x1 ⊕ y1 , x2 ⊕ y2 ) = (x1 , y2 ) + (x2 , y1 ). Since this bilinear form identifies the first component of W ⊕ W with the dual of the second component, it is natural to write W ∗ ⊕ W for the resulting orthogonal space. It is simple to verify that Q+ I −1 Q = , Q− −I and using this one easily checks that −1

QQQ

I = 0

0 := J. −I

Thus the orthogonal map Q : W → W∗ ⊕ W identifies Q with J on W ∗ ⊕ W with the space of “creation operators” Q+ W mapping into W ∗ ⊕ 0 and the space of annihilation operators Q− W mapping into 0 ⊕ W . The map Q extends to a Clifford algebra isomorphism from Cliff(W) to Cliff(W ∗ ⊕ W ), which we continue to denote by Q. It is clear that XQ = QXJ , so that we have an alternative representation for the Q state via this formula. We now define the symbol map associated with a skew-symmetric map Q on W as follows: Cliff(W ) ∋ X → sbQ (X) := FJ (QX)|0 ∈ Alt(W ∗ ⊕ 0) ≃ Alt(W ), where |0 is the vacuum vector in the J Fock representation of Cliff(W ∗ ⊕ W ). First we show that sbQ is a linear isomorphism and then we check that in case


291

Q is a polarization this definition reproduces the symbol map that was previously defined. Suppose that wj ∈ W for j = 1, . . . , k. Then sbQ (w1 w2 · · · wk ) = (c(w1 ) + a(w1− ))(c(w2 ) + a(w2− )) · · · (c(wk ) + a(wk− ))|0, since FJ (Qw) = c(w) + a(w− ) for w ∈ W. It is evident from this formula that the difference w1 ∧ w2 ∧ · · · ∧ wk −sbQ (w1 w2 · · · wk ) is a tensor of degree less than k. If we suppose for the sake of induction that all tensors of degree less than k are in the image of sbQ , then this fact shows that w1 ∧ w2 ∧ · · · ∧ wk is in the image of sbQ . Since sbQ e = 1 and sbQ (w) = w for w ∈ W , the induction is valid and sbQ is surjective. The dimensions of Cliff(W ) and Alt(W ) are the same, so sbQ is an isomorphism. We denote the inverse of sbQ by opQ (in Lemma A.4.1 below we give more-explicit formulas for these two maps). Now we check that this definition of sbQ agrees with the previous definition when Q2 = I . Let W+ + W− denote the isotropic splitting associated with such a Q. Let ej∗ ∈ W+ for j = 1, . . . , ℓ and ej ∈ W− for j = 1, . . . , k. Then the original definition of sbQ as the inverse of normal ordering gives sbQ (e1∗ e2∗ · · · eℓ∗ e1 e2 · · · ek ) = e1∗ ∧ · · · ∧ eℓ∗ ∧ e1 ∧ · · · ∧ ek .

(A.25)

On the other hand, FJ (Qej∗ ) = c(ej∗ ),

FJ (Qej ) = c(ej ) + a(ej ),

so that the new definition of sbQ gives sbQ (e1∗ e2∗ · · · eℓ∗ e1 e2 · · · ek ) = c(e1∗ ) · · · c(eℓ∗ )(c(e1 ) + a(e1 )) · · · (c(ek ) + a(ek ))|0. Since the annihilation operators a(ej ) don’t pair with any of creation operators c(ei ) (i.e., they have 0 anticommutators), the second definition of sbQ clearly reproduces (A.25). We are ready to prove a generalization of Theorem A.3.2. Theorem A.3.5 Suppose that g ∈ G and Q is skew-symmetric on W . Write T = T (g). Then the Q functional gQ is nonzero if and only if Q− T + Q+ is invertible. In case Q− T + Q+ is invertible, then R := (T − 1)(Q− T + Q+ )−1 is skew-symmetric and g = gQ opQ eR , where

ˆ R = R.

292


Proof. The proof given above that R is skew-symmetric depended only on the fact that Qτ± = Q∓ . Of course, this is still true if Q is merely skew-symmetric. Next we compute sbQ (g). We know from Theorem A.3.2 that g ∈ G is either even or odd. Suppose to begin that g is even. Then the embedding of g in Cliff(W) is an invertible element (which we continue to denote by g) and g(x⊕y)g −1 = T x⊕y since g is even. The orthogonal image g := Qg has induced rotation T Q+ + Q − T −I −1 , (A.26) T (g) = QT ⊕ I Q = Q− (T − I )Q+ Q− T + Q+ and consequently, R(g, J ) = (T (g) − I )(J− T (g) + J+ )−1 =

RQ+ Q− RQ+

R . Q− R

(A.27)

Then Theorem A.3.2 implies that g = gJ opJ eR(g,J ) ,

where R(g, J ) =

1 2

2n

j =1

R(g, J )wj∗ ∧ wj +

2n

j =1

R(g, J )wj ∧ wj∗ ,

(A.28)

since the dual of the basis {wj∗ , wj } for W ∗ ⊕ W is just {wj , wj∗ }. By definition, sbQ (g) = gJ FJ opJ eR(g,J ) |0,

where |0 is the J Fock vacuum. However, R(g, J ) = R + R′ , where R consists of those summands in R(g, J ) that are wedge products of two terms in W ∗ ⊕ 0 (the space of creation operators) and R′ consists of those terms in R(g, J ) that have at least one factor in 0 ⊕ W (the space of annihilation operators). Recalling the normal ordering prescription for opJ and using the fact that even elements in the Grassmann algebra commute with one another, we obtain # ′ $$ # |0 FJ opJ eR(g,J ) |0 = FJ opQ eR FJ opJ eR R = FJ opQ e |0 = eR . Consulting (A.27) and (A.28), we see that ∗ ˆ R = 21 2n j =1 Rwj ∧ wj = R.

Since gJ = gQ , we have finished the proof in one direction in case g is even. Now suppose that g is an odd element in G, with induced rotation T . Then g(x ⊕ y)g −1 = T x ⊕ (−y), where g in this formula is embedded in Cliff(W). Let {wj } denote an orthonormal basis for W (i.e., (wi , wj ) = δij ). Define h := (0 ⊕ w1 )(0 ⊕ w2 ) · · · (0 ⊕ w2n ).

A.4 Pfaffians

293

Since all but one of the factors in h anticommutes with (0 ⊕ wj ) and the remaining factor (0 ⊕ wj ) commutes with (0 ⊕ wj ), we obtain h(0 ⊕ wj )h−1 = 0 ⊕ (−wj ), so that T (h) = I ⊕ (−I ). Define g ′ = hg. Then T (g ′ ) = T ⊕ I, and if we write g′ = Qg ′ , then (A.26) implies that I 0 ′ . J− T (g ) + J+ = Q− (T − I )Q+ Q− T + Q+

(A.29)

Since Q− T + Q+ is invertible, by assumption it follows that J− T (g′ ) + J+ is also invertible. Theorem A.3.2 now implies that g′ is an even element of Cliff(W ∗ ⊕W ). This contradicts the assumption that g is odd and shows that Q− T + Q+ cannot be invertible when g is an odd element of G. As in Theorem A.3.2 we defer the proof that gQ = 0 implies that Q− T + Q+ is invertible until later.

A.4 Pfaffians In this appendix we develop formulas for vacuum expectations of products g1 g2 · · · gk , with gk ∈ G, the Clifford group. Suppose that gj Q := g j = 0 for j = 1, 2, . . . , k. Then TheoremA.3.2 implies that gj = gj op eRj and consequently, g1 g2 · · · gk =

k

j =1

, gj op eR1 · · · op eRk .

In this formula it is important for us that the one-point functions have been factored out. In the calculation of the scaling limits for the Ising model we have to divide the correlation functions by the product of the one-point functions to get a nonzero limit (this is the “wave function” renormalization of quantum field theory). To get an idea about what is involved in calculating the vacuum expectation of the normal ordered product on the right-hand side, suppose first that Xj ∈ Alt(W ) is a monomial for j = 1, . . . , k. The way in which one calculates op(X1 ) · · · op(Xk ) is to move the annihilation operators in each factor F (op(Xj )) to the right until they hit the vaccuum state and give 0. Each time an annihilation operator passes

294


a creation operator there is a “pairing”: a possibly nonzero result for the anticommutator. The vacuum expectation is then a sum of products of all possible pairings with appropriate signs. Notice that since each of the terms F (op(Xj )) is normally ordered, there are no “internal pairings” coming from the individual factors Xj . The precise combinatorial expression of this result is called Wick’s theorem and is one of the cornerstones of perturbation theory in quantum field theory. Wick’s theorem can be stated in terms of Pfaffians, and we do this. Our interest has to do with infinite-dimensional generalizations. Continuity properties of Pfaffians are analogous to the continuity properties of determinants, and these continuity results allow us to make a relatively painless transition to infinite dimensions. We begin then with an account of Pfaffians (following [126] quite closely). Suppose that r is a 2n × 2n skew-symmetric matrix with matrix elements rj,k . Let {ej } denote the standard basis of C2n . The Pfaffian, Pf(r) is defined by  n 2n 1  rj,k ej ∧ ek  = Pf(r)e1 ∧ · · · ∧ e2n , 2n n! j,k from which follows

Pf(r) =

1 sgn(σ )rσ (1)σ (2) rσ (3)σ (4) · · · rσ (2n−1)σ (2n) , 2n n! σ

where the sum is over permutations σ of the integers from 1 to 2n. Let r i,j denote the (2n − 2) × (2n − 2) skew-symmetric matrix obtained from r by removing the i and j columns and the i and j rows. We obtain a standard reduction formula for the Pfaffian by doing the following calculation:  n n  2n 2n 2n  r1,k e1 ∧ ek + rj,k ej ∧ ek  rj,k ej ∧ ek  = 2 j,k

j,k=1

k=2

= 2n = 2n = 2n

7 2n l=2

2n ℓ=2

8

r1,l e1 ∧ eℓ  

r1,ℓ e1 ∧ eℓ 

2n

j,k=1

2n

j,k=1,ℓ

n−1

rj,k ej ∧ ek 

n−1

rj,k ej ∧ ek 

2n (−1)ℓ r1,ℓ 2n−1 (n − 1)!Pf(r 1,ℓ ). ℓ=2

$n # = 0, only one = 0 for m > 1 and Since j,k =1 rj,k ej ∧ ek k r1,k e1 ∧ ek term survives in the binomial expansion of the right-hand side of the first equality. This gives the second equality. To obtain the last equality one need only observe that

m

e1 ∧ eℓ ∧ e2 ∧ · · · ∧ eˆℓ ∧ · · · ∧ e2n = (−1)ℓ e1 ∧ · · · ∧ e2n .

A.4 Pfaffians

295

We have proved the reduction formula Pf(r) =

2n (−1)ℓ r1ℓ Pf(r 1,ℓ ).

(A.30)

ℓ=2

Comparing this reduction formula with the reduction formula for the 2n-point function in a quasifree representation (A.24) and separately checking the result for the Pfaffian of a 2 × 2 matrix yields the following result.

Theorem A.4.1 Suppose vj ∈ W for j = 1, 2, . . . , 2k, and Q is skew-symmetric on W . Then v1 v2 · · · v2k Q = Pf(r),

(A.31)

where r is the skew-symmetric matrix with entries above the diagonal given by ri,j = (Q− vi , vj )

for i < j.

This is the simplest form of Wick’s theorem; all the normal ordered products are of length 1. We find the following generalization of this result useful. Theorem A.4.2 Suppose that Q is skew-symmetric on W and g ∈ G with gQ = 0. Let vj ∈ W for j = 1, 2, . . . , 2k and write R = (T − I )(Q− T + Q+ )−1 , where T := T (g). Then v1 v2 · · · v2k gQ = gQ Pf(r), where the skew-symmetric matrix r has i, j matrix element above the diagonal given by ri,j = (Q− vi , (I − RQ− )vj ) for i < j. Proof. An adaptation of the technique used to prove Theorem A.3.4 suffices to establish this result. Observe that v1 v2 · · · v2k gQ = Qv1 Qv2 · · · Qv2k QgJ .

(A.32)

Let |0 denote the vacuum in the J Fock representation. Then in the notation of Theorem A.3.4 we calculated FJ (Qg)|0 = gQ FJ opJ eR(g,J ) |0 (A.33) = gQ FJ opJ eR |0, and if we let {wj } and {wj∗ } denote dual bases for W , then FJ opJ eR = eR ,

where R :=

1 2

2n

j =1

c(Rwj )c(wj∗ ).

296


With this substitution and FJ (Qvj ) = c(vj ) + a(vj− ) the representation (A.32) becomes − v1 v2 · · · v2k gQ = gQ 0|(c(v1 ) + a(v1− )) · · · ((c(v2k ) + a(v2k ))eR |0. (A.34)

Next we show that e−R ((c(vj ) + a(vj− ))eR = c(vj − Rvj− ) + a(vj− ).

(A.35)

Let adℓ (R)x = [R, [R, · · · [R, x] · · · ]] denote the ℓ-fold iterated commutator. Then the Taylor series expansion in λ gives e

−λR

xe

λR

=

∞ (−λ)m adℓ (R)x

ℓ!

ℓ=0

.

But it is a simple matter to use the anticommutation relations to prove that [R, c(vj ) + a(vj− )] = c(Rvj− ), which implies that all higher-order commutators are 0. One immediately deduces (A.35) (at λ = 1 of course). Since R τ is the sum of products of annihilation operators, it follows that 0|eR = 0|. Using these results to move eR to the left in (A.34) until it hits the dual vacuum, one obtains the representation , − − v1 v2 · · · v2k gQ = gQ (v1 − Rv1− ) ⊕ v1− · · · (v2k − Rv2k . ) ⊕ v2k J

But we can now use Theorem A.4.1 to evaluate the J Fock expectation in this last equation, , − − = Pf(r), ) ⊕ v2k (v1 − Rv1− ) ⊕ v1− · · · (v2k − Rv2k J where the i, j superdiagonal matrix element of r is given by (vi− , vj − Rvj− ) This finishes the proof.

for i < j.

We return to the consideration of Wick’s theorem once we have developed the SMJ. product deformation formalism and can present their generalization of this result. The remainder of this section is devoted to examining connections between Pfaffians and quasifree representations, culminating in a Pfaffian formula for relative Q functionals. It is useful to start by explaining the connection between Pfaffians and determinants. Suppose that A is a 2n × 2n nonsingular complex matrix and r is a 2n × 2n

A.4 Pfaffians

297

skew-symmetric matrix with complex entries. Write Aτ for the transpose of A. Then ArAτ is skew-symmetric, and writing = e1 ∧ e2 ∧ · · · ∧ e2n for the volume element, we have  n 1 Pf(ArAτ ) = n  ai,j rj,k aℓ,k ei ∧ eℓ  2 n! i,j,k,ℓ = Pf(r)Ae1 ∧ · · · ∧ Ae2n = det A Pf(r) .

Thus Pf(ArAτ ) = det A Pf(r).

Every skew-symmetric matrix on C2n is of the form ArAτ for special blockdiagonal skew-symmetric matrices r that have 0’s or 2 × 2 matrices of the form 2n 0 1 −1 0 on the diagonal (this is the standard representation of skew forms on C ). If any 0’s appear on the diagonal, then the reduction formula shows that Pf(r) = 0 and hence also Pf(ArAτ ) = 0, which clearly squares to the determinant det(ArAτ ) = (det A)2 det r = 0, 0 1 since det r = 0. If r consists of block entries −1 0 down the diagonal, then Pf(r) = 1 and we obtain Pf(ArAτ )2 = det(A)2 = det(ArAτ ). Thus for any skew-symmetric matrix r we have Pf(r)2 = det r. Next we present a formula for the Pfaffian of the sum of two matrices. We start with some notation. Let P denote the collection of subsets of {1, 2, . . . , 2n} (including, of course, the empty set ∅). For α ∈ P we write α = {α1 , α2 , . . . , αk }

with α1 < α2 < · · · < αk .

(A.36)

In this section we always suppose that subsets of the integers are indexed in natural order as in (A.36). For such an α we write #α = k for the number of elements in α and if r is a 2n × 2n matrix we write rα for the #α × #α submatrix of r made from the rows and columns of r indexed by α. That is, (rα )ij = rαi αj . If α is a subset of the integers we write α =β ⊔γ to signify a partition α = β ∪ γ with β ∩ γ = ∅. We write wα = wα1 wα2 · · · wαk

298


for a monomial in the Clifford algebra, and w∧α = wα1 ∧ wα2 ∧ · · · ∧ wαk for a monomial in the Grassmann algebra. Theorem A.4.3 Suppose that r and s are 2n × 2n skew-symmetric matrices with complex entries. Then Pf(r + s) = sgn(α, β)Pf(rα )Pf(sβ ), α⊔β=I2n

where IN = {1, 2, . . . , N} is the set of integers between 1 and N , and sgn(α, β) is the sign of the shuffle permutation (α, β) (i.e., α and β are already in natural order). By convention, Pf(r∅ ) = 1. Proof. Let {ej } denote the standard basis of C2n and write R = 12 j,k rj,k ej ∧ ek , S = 21 j,k sj,k ej ∧ ek .

Then

k n 1 Rn−k S n = + S) (R n n−k 2 n! 2 (n − k)! 2k k! k=0 = Pf(rα )e∧α ∧ Pf(sβ )e∧β , α⊔β=I2n

and since e∧α ∧ e∧β = sgn(α, β)e1 ∧ · · · ∧ e2n , we have proved the theorem.

If r and s are 2n×2n skew-symmetric matrices, then the determinant det(1+rs) has a canonical square root, which we denote by Pf(1 + rs). Since r 1 det = det(1 + rs), −1 s we can obtain such a square root as follows: r 1 Pf −1 s , Pf(I + rs) := 0 1 Pf −1 0

(A.37)

which is normalized to be equal to 1 when r = 0. The following expansion of Pf(1 − rs) is useful for us (note the minus sign in front of rs).

A.4 Pfaffians

299

Theorem A.4.4 Suppose that r and s are skew-symmetric 2n × 2n matrices. Then Pf(1 − rs) = Pf(rα )Pf(sα ), α∈P

where P is the collection of subsets of I2n . No odd subsets α contribute to this sum.

Proof. Let Ik denote the k × k identity matrix. Then it is a simple matter to use the reduction formula to show that 0 I2k Pf = (−1)k . −I2k 0 Thus using (A.37), −r I2n Pf(1 − rs) = (−1) Pf −I2n s 0 n = (−1) sgn(α, β)Pf −I2n

n

α⊔β=I4n

I2n 0

−r Pf 0 α

0 . (A.38) s β

In this last formula write β = γ ∪ (δ+2n) with γ , δ ⊂ I2n and δ + 2n defined as the subset of I4n obtained by adding 2n to each element of δ. Then −r 0 (−r)γ 0 Pf = Pf = Pf (−r)γ Pf (sδ ) , (A.39) 0 s β 0 sδ since

a Pf 0

0 = Pf(a)Pf(b) b

follows from an application of Theorem A.4.3 to a 0 a 0 0 0 = + . 0 b 0 0 0 b It is also easy to see that

0 I2n = 0, −I2n 0 α > ? > ? unless γ = δ. The matrix −I02n I2n is obtained from −I02n I2n by removing the 0 0 α rows and columns associated with the indices γ and δ + 2n. If the index γj is not matched by an index in δ, then the γj + 2n column that survives is a column with all zeros. A little thought shows that if α is the complement of γ ∪ (γ + 2n), then sgn(α, γ ∪ (γ + n)) = 1 and #γ 0 I2n Pf = (−1)n− 2 . −I2n 0 α Pf

#γ Substituting this and (A.39) in (A.38) and making use of (−1)− 2 Pf (−r)γ = Pf(rγ ), the theorem follows.

300


Now suppose that {wj } is a basis for W and ri,j is a skew-symmetric matrix. Define R = 21 i,j ri,j wi ∧ wj .

Recalling the definition of Pf(r) we find that for k = 1, . . . , n, Rk = Pf(rα )w∧α . k! α∈P,#α=2k It follows that

eR =

α∈P

Pf(rα )w∧α ,

(A.40)

where only α with #α even contribute to the sum. Theorem A.4.5 Suppose that R is skew-symmetric on W , Q is a polarization, and {wj } is a basis for W compatible with the polarization Q. Let ri,j = (Rwj∗ , wi∗ ), and as usual, R= Then

1 2

i,j ri,j wi

∧ wj .

opQ eR = Pf(rα,β )wα wβ ,

(A.41)

α,β

where α ⊂ {1, 2, . . . , n}, β ⊂ {n + 1, . . . , 2n},

wα = wα1 wα2 · · · wαk ,

wβ = wβ1 wβ2 · · · wβℓ ,

and the sum is over all such α and β with #α + #β even. Proof. This follows directly from the preceding expansion of eR and the fact that wedge products of elements from the basis {wj } taken in natural order are automatically normally ordered. We can now use this result to prove a Pfaffian formula for the ratio of Q functionals. Theorem A.4.6 Suppose that Q and Q′ are polarizations on W . Suppose that g ∈ G with gQ = 0, so that R = (T − 1)(Q+ T + Q− )−1 is well defined for T = T (g). Then gQ′ = Pf 1 − 21 (Q′ − Q)R . (A.42) gQ

A.4 Pfaffians

301

Proof. Let {wj } be a basis of W compatible with the polarization Q. Theorem A.3.4 and (A.40) imply that , , gQ′ = gQ opQ eR Q′ = gQ (A.43) Pf rα,β wα wβ Q′ . α,β

But Wick’s theorem implies , wα wβ Q′ = Pf(sα,β ),

(A.44)

where s is the skew-symmetric matrix with entries above the diagonal given by si,j = 12 (1 − Q′ )wi , wj for i < j.

Using the fact that Q and Q′ are skew-symmetric it is easy to check that si,j = 21 (Q − Q′ )wi , wj for all i, j. (A.45)

Combining (A.43) and (A.44) and using Theorem A.4.4 we see that gQ′ = gQ Pf(1 − sr).

This is essentially the formula we want, but at this point it is useful to explain the transition between the matrix form of the result we arrived at (involving the matrices s and r) and the operator form of the result stated in the theorem (involving Q, Q′ , and R). First observe that the i, k matrix element of the matrix product sr is just $ # ∗ 1 ∗ ′ = 21 (Q − Q′ )wi , Rwk∗ w , Rw (Q − Q )w , w i j j k j 2 = 12 (wi , (Q′ − Q)Rwk∗ ).

However, (wi , Awk∗ ) is the i, k matrix element of the operator A with respect to the basis {wj∗ }. Let A = 21 (Q′ − Q)R and introduce an arbitrary complex parameter λ; then we see that Pf(1 − λsr)2 = det(1 − λsr) = det(1 − λA). This shows that the entire function λ → det(1 − λA) has an analytic square root that is uniquely determined by fixing its value at λ = 0 equal to 1 (since any entire function that squares to a constant must itself be constant). Whenever λ → det(1 − λA) has an analytic square root we write Pf(1 − λA) for the analytic square root that is normalized to be 1 at λ = 0. With this new understanding of the Pfaffian we have shown that gQ′ = Pf(1 − 21 (Q′ − Q)R). gQ

302


However, note that both operators (Q′ − Q) and R have skew-symmetric matrices with respect to self-dual bases of W , so this formula also has a Pfaffian interpretation à la Theorem A.4.4 if such bases are used to represent (Q′ − Q) and R as matrices. We want to extend this theorem to general quasifree states. The extension is direct: Theorem A.4.7 Suppose that Q and Q′ are skew-symmetric on W . Suppose that g ∈ G with gQ = 0, so that R = (T − 1)(Q+ T + Q− )−1 is well defined for T = T (g). Then gQ′ = Pf 1 − 21 (Q′ − Q)R . (A.46) gQ

Proof. Suppose that {wj } is a self-dual (i.e., orthonormal) basis for W with respect to the distinguished bilinear form. Define R := Rˆ =

1 2

i,j ri,j wi

∧ wj ,

where in this case ri,j is the (skew-symmetric) matrix of R with respect to the basis {wj }. Theorem A.3.4 implies that g = gQ opQ eR . Using this formula to evaluate gQ′ , the proof proceeds exactly as in Theorem A.4.5 once we know the following fact: , opQ (wα ) Q′ = Pf 12 (Q′ − Q)α , (A.47) where 21 (Q′ − Q)α is the #α × #α skew-symmetric matrix with i, j entry 1 2

(Q′ − Q)wαi , wαj .

The following formula for opQ has (A.47) as a direct consequence (using the formula for the Pfaffian of a sum given above). Lemma A.4.1 Suppose that Q is a skew-symmetric operator on W . Suppose that {wj }kj =1 is a linearly independent subset of W . Then opQ (w1 ∧ · · · ∧ wk ) =

α⊔β=Ik

#α

(−1) 2 sgn(α, β)wα Q wβ .

(A.48)

We define w∅ = e. Proof. Suppose that α ⊂ Ik . Then by definition, sb(wα ) = FJ (wα1 ⊕ wα−1 ) · · · FJ (wαℓ ⊕ wα−ℓ )|0,

(A.49)

A.4 Pfaffians

303

where ℓ = #α and FJ (w ⊕ w− ) = c(w) + a(w− ). Suppose that α = β ⊔ γ is a partition of α. Then the coefficient of w∧γ in the expansion of sb(wα ) is given by sgn(β, γ )wβ . To see this, observe that the terms in (A.49) that contribute to w∧γ arise from choosing all the factors c(wγj ) in (A.49) without admitting any nonzero pairings with other elements in the product (recall that the set {wj }kj =1 is assumed to be independent). Shuffling these creation terms to the right to hit the vaccum makes a contribution sgn(α, β) to the sign of the result, and the constant term in the remaining multiplier is just : 9 , (wβ1 ⊕ wβ−1 ) · · · (wβm ⊕ wβ−m = wβ . J

Thus

sb(wα ) =

β⊔γ =α

sgn(β, γ )wβ w∧γ .

(A.50)

We now check (A.48) by applying sb to the right-hand side; we will see that the result is just w1 ∧w2 ∧· · ·∧wk , and since op is the inverse of sb, this verifies (A.48). Let α c denote the complement of α in Ik . Then we can rewrite the right-hand side of (A.48) as #α (−1) 2 sgn(α, α c ) wα wαc . α

Applying sb to this using (A.50) and writing α c = β ∪ γ for a partition of α c , we obtain #α sgn(β, γ )wβ w∧γ . (A.51) (−1) 2 sgn(α, α c )wα β⊔γ =α c

α

In this sum fix γ = Ik . The sum over α and β is then the sum over partitions α ∪ β = γ c of γ c (which is not empty). Since α c = β ⊔ γ , sgn(α, α c )sgn(β, γ ) = sgn(α, β), and we find that the coefficient of w∧γ is

α⊔β=γ c

#α

(−1) 2 sgn(α, β)wα wβ .

(A.52)

Let r denote the skew-symmetric matrix with entries above the diagonal given by ri,j = (Q− wi , wj ). Then Wick’s theorem implies wα = Pf(rα ).

304


Thus (A.51) becomes, #α (−1) 2 sgn(α, β)Pf(rα )Pf(rβ ) α⊔β=γ c

=

α⊔β=γ c

sgn(α, β)Pf((−r)α )Pf(rβ ) = Pf(rγ c − rγ c ) = 0.

Examining the special case γ = {1, 2, . . . , k}, we see that (A.51) becomes w1 ∧ w2 ∧ · · · ∧ wk . This finishes the proof of the lemma and hence also the proof of Theorem A.4.7.

A.5 Vacuum Expectations for Elements of G Our next goal is the determinant formula for g2Q in Theorem A.5.1 below. This formula, incidentally, supplies the missing details in the proofs of Theorems A.3.2 and A.3.4. It is known that T

C∗ → G → O(W ) is a topologically nontrivial extension; there is no continuous lifting of T (g) into G. For this reason there aren’t any nice normalizations for elements of G that would give formulas for the vacuum expectations gQ in terms of T (g). However, there is a (quadratic) homomorphism of G into C∗ , known as the spinor norm, that makes it possible to reduce the extension from an extension by C∗ to an extension by Z/2Z. This makes possible formulas for g2Q that depend only on the spinor norm of g and the induced rotation T (g). Our object in this section is to explain such a result. We start with a description of the spinor norm. Observe that there is a unique complex linear involution X → Xτ on Cliff(W ) that extends the identity on W . For example, τ wj1 wj2 · · · wjk = wjk · · · wj2 wj1 . Note that for X and Y in Cliff(W ) we have

(XY )τ = Y τ Xτ . If g ∈ G, then gxg −1 = T (g)x. Applying the involution to both sides yields g −τ xg τ = T (g)x,

A.5 Vacuum Expectations for Elements of G

305

from which it follows that g τ ∈ G and T (g τ ) = T (g)−1 . Thus T (g τ g) = I , which implies that g τ g commutes with the generators and hence all of Cliff(W ). Since Cliff(W ) is a simple algebra for even-dimensional W [28], we find that g τ g = nr(g)e for a scalar nr(g), called the spinor norm of g. Since g → g τ g is clearly a group homomorphism, it follows that g → nr(g) is also a group homomorphism (into the multiplicative group of nonzero complex numbers C∗ ). The kernel of this homorphism is a subgroup of G called Pin(W ) (the connected component of the identity in Pin(W ) is called Spin(W )). For g ∈ Pin(W ) we find formulas for g2Q that depend only on T (g). Before we show how to obtain these formulas it is useful to note a simple formula for nr(g) that also “explains” the name “spinor norm.” Suppose that {wj } is an orthogonal basis for W that is normalized so that wj2 = 1. Suppose g ∈ G has an expansion g= g α wα . α⊂I2n

Then nr(g) = To see this, note that by definition, nr(g)e =

gα2 .

(A.53)

α

gα gβ wατ wβ .

α,β

One can isolate the coefficient of e by taking the trace. It is clear that Tr(wατ wβ ) = δα,β , and the result for the spinor norm follows. Next we describe a standard isomorphism between Cliff(W ⊕ W ) and the tensor product of two copies of Cliff(W ). Suppose as usual that W is a 2ndimensional space with the distinguished nondegenerate bilinear form (·, ·). Let {wj } be an orthogonal basis for W with (wj , wk ) = 2δj k . Observe that wj2 = 1 for j = 1, . . . , 2n. Let = (−1)n w1 · · · w2n be the “volume element” of W that is normalized so that 2 = 1. Then x ⊗ 1 + ⊗ y → x ⊕ y, for x, y ∈ W, extends to a Clifford algebra isomorphism Cliff(W ) ⊗ Cliff(W ) ≃ Cliff(W ⊕ W ), where W ⊕ W has the distinguished bilinear form (x1 ⊕ y1 , x2 ⊕ y2 ) = (x1 , x2 ) + (y1 , y2 ).

(A.54)

306


So W ⊕ W is just the direct sum of the orthogonal space W with itself. The factor

is easily seen to anticommute with each vector wk (since wk commutes with itself and anticommutes with the 2n − 1 remaining elements in the basis {wj }). Hence x = −x for all x ∈ W . This is all that is needed to see that the left-hand side of (A.54) satisfies the Clifford relations for W ⊕ W . Next we introduce a “square” homomorphism from the even subgroup of the Clifford group over W into the Clifford group over W ⊕ W by Geven (W ) ∋ g →sq(g) := g ⊗ g. The tensor products on the right are identified with elements in Cliff(W ⊕ W ) by the isomorphism (A.54). The reader can check that since g = g when g is even, the map sq is a homomorphism of groups with T (sq(g)) = T (g) ⊕ T (g). It is possible to extend this homomorphism to all of G in a way that makes Theorem A.4.7 work for odd g, but since gQ = 0 when g is odd and our principal result is Theorem A.5.1, we don’t bother. We are interested in sq(g) for two reasons. The first is that if Q is a polarization on W , that sq(g)Q⊕Q = g2Q . (A.55) To see this, observe that x ⊕ y → FQ (x) ⊗ 1 + FQ ( ) ⊗ FQ (y) gives a representation of Cliff(W ⊕ W ) on Alt(W+ ) ⊗ Alt(W+ ) that is equivalent to the Q ⊕ Q representation on Alt(W+ ⊕ W+ ) (the vacuum vector is |0 ⊗ |0). For g ∈ Geven , the vacuum expectation of FQ (g) ⊗ FQ (g) is clearly equal to g2Q . The second reason that we are interested in sq(g) is that the induced rotation T (sq(g)) = T (g) ⊕ T (g) commutes with the natural polarization 0 q= −i

i 0

on W ⊕ W. This makes possible an explicit calculation of sq(g)q . In fact, we find that when g is even, sq(g)q = nr(g). We can then use Theorem A.3.4 to write sq(g) = sq(g)q opq eRq = nr(g)opq eRq . This formula enables us to calculate g2Q = sq(g)Q⊕Q .


307

Theorem A.5.1 Suppose that g ∈ Geven . Then sq(g)q = nr(g). Proof. As in a previous calculation it is very helpful to diagonalize q. The reader can check that the map 1 −i 1 q := √ : W ⊕ W → W∗ ⊕ W 2 i 1 is an orthogonal transformation from W ⊕ W to W ∗ ⊕ W . Recall that W ∗ ⊕ W is the orthogonal space given by the direct sum W ⊕ W with the distinguished nondegenerate bilinear form (x1 ⊕ y1 , x2 ⊕ y2 ) = (x1 , y2 ) + (y1 , x2 ). The map q is chosen that ∗the polarization q on W ⊕ W is mapped into the 0 so on W ⊕W. Of course, q induces an algebra isomorphism polarization J = 10 −1 from Cliff(W ⊕W ) to Cliff(W ∗ ⊕W ), which we continue to denote by q. Evidently, sq(g)q = qsq(g)J , so that we can concentrate on calculating the right-hand side. Suppose that g α wα , (A.56) g= α

where {wj } is an orthonormal basis of W . Since g is even, we obtain qsq(g)J = gα gβ qwα ⊗ wβ J , α,β

where α and β are in natural order. Since the length ℓ of the multi-index β = (β1 , . . . , βℓ ) must be even, we have ℓ = 1 and hence wα ⊗ wβ = (wα1 ⊗ 1) · · · (wαk ⊗ 1)( ⊗ wβ1 ) · · · ( ⊗ wβℓ ) ≃ (wα1 ⊕ 0) · · · (wαk ⊕ 0)(0 ⊕ wβ1 ) · · · (0 ⊕ wβℓ ).

Apply q to this and take the vaccum expectation to obtain 2

k+ℓ 2

qwα ⊗ wβ J , = i k (−wα1 ⊕ wα1 ) · · · (−wαk ⊕ wαk )(wβ1 ⊕ wβ1 ) · · · (wβℓ ⊕ wβℓ ) J , = i k (0 ⊕ wα1 ) · · · (0 ⊕ wαk )(wβ1 ⊕ 0) · · · (wβℓ ⊕ 0) J = i k (−1)

ℓ(ℓ−1) 2 k,ℓ α,β

δ δ

k

(wαj , wβj ) = 2k δk,ℓ δα,β .

j =1

308


The second equality follows from the fact that the creation operators (−wαi ⊕ 0) can all be moved to the left past the annihilation operators (0 ⊕ wαj ) without picking up any nonzero anticomutators, and the annihilation operators (0 ⊕ wβi ) can all be moved to the right past the creation operators (wβj ⊕ 0) without picking up any nonzero anticommutators. The third equality follows from the fact that the vacuum expectation is zero unless the number of annihilation operators (i.e., k) matches the number of creation operators (i.e., ℓ). Since the basis {wj } is supposed to be orthogonal and both the multi-indices α and β are in natural order, the third ℓ(ℓ−1) ℓ vacuum expectation also vanishes unless α = β. The factor (−1) 2 = (−1) 2 (since ℓ is even) is just what is needed to reverse the order of the (wβj ⊕ 0) factors, and the last equality follows from our choice of normalization (wi , wj ) = 2δi,j . We have shown that q(wα ⊗ wβ )J = δα,β , and hence that qsq(g)J =

α

gα2 = nr(g).

We can now state the principal result of this section. Theorem A.5.2 Suppose that Q is skew-symmetric on W and g ∈ Geven with T = T (g). Then g2Q = nr(g) det (Q− T + Q+ ) . Proof. First suppose that Q is a polarization and let {wj } denote a basis for W with {wj∗ } the associated dual basis. For W ∗ ⊕ W we choose the basis {wj∗ , wk } so that the elements wj∗ ≃ wj∗ ⊕ 0 correspond to creation operators, and the elements wk ≃ 0 ⊕ wk correspond to annihilation operators in the J Fock representation of W ∗ ⊕ W . The induced rotation of qsq(g) is T ⊕ T , and one easily computes that RJ := RJ (qsq(g)) = (T − I ) ⊕ (I − T τ ). As usual we define RJ := Rˆ J =

1 2

4 2n 2n 3 ∗ ∗ ∗ k=1 RJ wk ∧ wk + RJ wk ∧ wk = k=1 (T − I )wk ∧ wk .

Then Theorem A.3.4 implies that

qsq(g) = qsq(g)J opJ eRJ = nr(g)opJ eRJ ,

since qsq(g)J = sq(g)q = nr(g). Thus

, g2Q = sq(g)Q⊕Q = qsq(g)Q⊕Q = nr(g) opJ eRJ Q⊕Q .


309

Writing S = T − I we obtain e RJ = 1 +

2n k(k−1) (−1) 2 Swα∗1 ∧ · · · ∧ Swα∗k ∧ wα1 ∧ · · · ∧ wαk . α, #α=k

k=1

Also, Wick’s theorem implies 0 sα = Pf , −sατ 0 Q⊕Q = Q− Swi∗ , wj by selecting the

9 # $: op Swα∗1 ∧ · · · ∧ Swα∗k ∧ wα1 ∧ · · · ∧ wαk

where sα is the k × k matrix obtained from si,j rows and columns with indices in α. Since k(k−1) 0 sα Pf = (−1) 2 det sα , −sατ 0 we find that

2n , opJ eRJ Q⊕Q = 1 + det sα . k=1 α, #α=k

Since s is the matrix of Q− S in the basis {wj∗ }, this last sum is just the Fredholm expansion [147], det(1 + A) = 1 +

2n

Tr(k A),

k=1

of det(1 + Q− S) = det(Q− T + Q+ ). Here k A is just the restriction of the k-fold tensor product A⊗k to Altk (W ). This finishes the proof when Q is a polarization. Now suppose that Q is merely skew-symmetric on W . Then using the result just proved for the polarization J , g2Q = Qg2J = nr(Qg) det(J− T (Qg) + J+ ).

(A.57)

However, if g=

g α wα ,

α

then Qg =

gα wα ,

α

where wj = Qwj . Since Q is orthogonal, wj extends to an orthogonal basis for W ∗ ⊕ W (with wj2 = 1), and it follows that nr(Qg) = gα2 = nr(g). α

310


As we calculated earlier,

I J− T (Qg) + J+ = Q− (T − I )Q+

0 , Q− T + Q +

so that det(J− T (Qg) + J+ ) = det(Q− T + Q+ ). These two observations and (A.57) finish the proof of the theorem.

Note that this result implies that if gQ = 0, it follows that det(Q− T +Q+ ) = 0 and hence that Q− T + Q+ is invertible. This finishes the proof of Theorems A.3.2 and A.3.4.

A.6 The SMJ Product Deformation Formalism In this section we introduce the product deformation formalism that is used to obtain formulas for vacuum expectations of products in spin representations in which the one-point functions are factored out. This isn’t, however, the only issue. There are other simpler formulas for the square of the vacuum expectation in which the one-point functions are also factored out that follow directly from Theorem A.5.1. However, for reasons that are explained later these formulas are not suitable for controlling the scaling limit of the Ising model (although they would be satisfactory for a treatment of the thermodynamic limit). Suppose that Xj ∈ Cliff(W ) for j = 1, 2, . . . , k. The vacuum expectation X1 X2 · · · Xk Q can be calculated by first expanding each Xj in monomials; Wick’s theorem applies to express each resulting term as sums of products of pairings with appropriate signs. Following SMJ we now introduce a formalism in which each of the factors Xj is associated with a separate copy of Cliff(W ) and the pairing between the i and j copies is given “weight” λij ∈ C (note that the “copies” won’t be independent unless λij = 0 for i = j ). We recover the original vacuum expectation in the limit λij → 1 for all i, j . We always suppose that the weight associated with internal pairings for each factor λjj is equal to 1 for j = 1, 2, . . . , k. Let denote a complex k × k symmetric matrix with matrix elements λij that has 1’s on the diagonal. Define an orthogonal space W () = W · · ⊕ W ⊕ · k summands

with symmetric bilinear form

(v1 ⊕ · · · ⊕ vk , w1 ⊕ · · · ⊕ wk ) =

i,j

λij (vi , wj ).

(A.58)

A.6 The SMJ Product Deformation Formalism

311

It is easy to see that if is nonsingular, then this bilinear form is nondegenerate. We always suppose that is chosen to be nonsingular (although we are interested in singular limits). The map from W to W () that injects W into the j th slot in W () (with 0’s in the other slots) is an orthogonal map (since λjj = 1) and consequently extends to an injection Ij of Cliff(W ) into Cliff(W ()). Suppose that Q is skew-symmetric on W ; we also write Q for the direct sum Q ⊕ Q ⊕ · · · ⊕ Q acting on W (). It is easy to see that this direct sum is also skew-symmetric on W (). Our interest in W () arises from the following result. Theorem A.6.1 Suppose that Xj ∈ Cliff(W ) for j = 1, 2, . . . , k. Suppose that Q is skew-symmetric on W . Then lim I1 (X1 )I2 (X2 ) · · · Ik (Xk )Q = X1 X2 · · · Xk Q ,

λij →1

(A.59)

where the limit refers to the limit in which λij → 1 for all pairs ij . Proof. It is understood in this result that for λij = 1 the matrix must be chosen nonsingular so that the bilinear form (A.58) is nondegenerate. By multilinearity it is enough to prove (A.59) when each Xj is a monomial. However, in this case both sides have the same Pfaffian expansions in terms of two-point functions. Thus it is enough to check the result for Xi , Xj ∈ W . But , , Ii (Xi )Ij (Xj ) Q = λij Q− Xi , Xj = λij Xi Xj Q , from which the result is obvious.

We want to compare X1 X2 · · · Xk Q with i = j , one obtains I1 (X1 )I2 (X2 ) · · · Ik (Xk )Q =

, j =1 Xj Q . Setting λij = 0 for

k

k , Xj Q

j =1

for λi,j = 0, i = j.

Unfortunately, it is not clear how to get formulas that relate vacuum expectations for Clifford algebras built on different orthogonal spaces. Instead, we follow SMJ [136] and introduce a “deformed” skew-symmetric Q() such that I1 (X1 )I2 (X2 ) · · · Ik (Xk )Q() =

k , Xj Q .

(A.60)

j =1

As before, the key to obtaining such a result is to ensure that it works for the two-point functions. Define   Q −λ12 ··· −λ1k   ..   Q . −1 λ21 . Q() =  .   .. Q −λk−1,k  λk1 · · · λk,k−1 Q

312


It is simple to check that Q() is skew-symmetric on W (), and since   Q− λ12 · · · λ1k   ..   Q− . −1  0 , Q()− =  .   .. Q− λk−1,k  0 ··· 0 Q− we obtain for Xi , Xj ∈ W , with i ≤ j , , , Ii (Xi )Ij (Xj ) Q() = δij Q− Xi , Xj = δij Xi Xj Q .

The desired result (A.60) is a straightforward consequence. Now we wish to use Theorem A.4.6 to compare the Q and Q() functionals on products I1 (g1 )I2 (g2 ) · · · Ik (gk ) for gj ∈ G. Our principal result is the following theorem (first proved by Sato, Miwa, and Jimbo [133]). Theorem A.6.2 , - Suppose that gj ∈ G for j = 1, 2, . . . , k, Q is skew-symmetric on W , and gj Q = 0 for j = 1, 2, . . . , k. Then for nonsingular, k , I1 (g1 )I2 (g2 ) · · · Ik (gk )Q = gj Q Pf (1 − A()R) , j =1

where



0

 −λ21 Q− A() =   ..  . −λk1 Q−

···

λ1k Q+ .. .

0 −λk,k−1 Q−

λk−1,k Q+ 0

λ12 Q+ 0 ···

and R is the diagonal matrix



R1 0  R= .  .. 0

0 R2 .. .

··· ···

0 0 .. .

0

···

Rk

     



  , 

with Rj = (Tj − I )(Q− Tj + Q+ )−1 and Tj := T (gj ). Proof. Theorem A.4.6 implies that I1 (g1 ) · · · Ik (gk )Q

= I1 (g1 ) · · · Ik (gk )Q() Pf 1 − 12 (Q − Q())R(g, ) k , = gj Q Pf 1 − 12 (Q − Q())R(g, ) , j =1

(A.61)


313

where R(g, ) = (T ()−I )(Q()− T ()+Q()+ )−1 , and T () is the induced rotation for I1 (g1 ) · · · Ik (gk ). It is an elementary calculation to see that 1 (Q 2

− Q()) = −1 A().

(A.62)

To finish the proof we need to calculate R(g, ), and we start by determining the induced rotation of T (Ij (gj )). Since gj is an even element in G, Ij (gj ) commutes with the vectors in W () that are orthogonal to the vectors Ij (w) for w ∈ W . Thus T (Ij (gj )) is the identity on such vectors. It is also clear that T (Ij (gj )) = T (gj ) := Tj on Ij (W ). Observe that Ii (v) − Ij (λij v), Ij (w) = 0 for v, w ∈ W. Since every vector w = w1 ⊕ · · · ⊕ wk can be written w= Ii (wi ) − Ij (λij wi ) + Ij (wj ) + Ij (λij wi ), i =j

i =j

we find that   T (Ij (gj ))w = Ii (wi ) − Ij (λij wi ) + Ij Tj wj + Tj λij wi  . i =j

i =j

Thus



0  ..  .  Tj () := T (Ij (gj )) = I +  Sj λj 1  ..  . 0

···

···

 0 ..  .   Sj λj k  , ..  .  0

(A.63)

where Sj := Tj − I and the nonzero entries in the matrix occur in the j th row. It is convenient to introduce the matrices     0 ··· 0 S1 0 · · · 0  .. ..    . ..  .   0 S2    .    λj k  S :=  . and j = λj 1  ,  .  .. ..  0  .  . .  0 ··· 0 Sk 0 ··· 0 so that Tj () = I + Sj . Next we calculate R(g, ). Since T () = T1 () · · · Tk (), we see that

T () − I = (T1 () − I )T2 () · · · Tk () + (T2 () − I )T3 () · · · Tk () + · · · + Tk () − I = SG(),

(A.64)

314


where G() := 1 T2 () · · · Tk () + 2 T3 () · · · Tk () + · · · + k . Next we factor Q()− T () + Q()+ . Define 

  λ21 Lj :=   .  .. λj 1  Q−   0 Mk−j :=   .  .. 0

and 

  D=  

Then

0

Q+

···

Q+ ···

Q+ λj,j −1

λj +1,2

···

Q− ···

Q− T1 + Q+

0

0 .. .

Q− T 2 + Q +

0

···

Q− 0

 0 ..  .  ,  0  Q+  λj +1,k ..  .  ,  λk−1,k  Q−

0



0 .. .

···

0 Q− Tk + Q+

  .  

Q()− T () + Q()+ = −1 (Mk T () + Lk ), and it is not difficult to check that −Lj +1 −Lj 0 Tj +1 () = Dj +1 + 0 0 Mk−j

0 Mk−j −1

for j = 0, . . . , k. Inserting these identities successively (j = 0, . . . , k) into Mk T () + Lk yields Mn T1 () · · · Tk () + Ln = D1 T2 () · · · Tk () −L1 0 + T () · · · Tk () 0 Mk−1 2

= D (1 T2 () · · · Tk () + 2 T3 () · · · Tk () + · · · + k ) = DG().

Thus Q()− T () + Q()+ = −1 DG().

(A.65)


315

The left-hand side is invertible, since g1 g2 · · · gk Q()

k , gj Q = 0, = j =1

and because D is also invertible it follows that G() is invertible (this can also be independently verified by solving (A.64) for G()). Combining (A.64) and (A.65), one obtains R(g, ) = (T () − I )(Q()− T () + Q()+ )−1 = SD −1 = R. Substituting this and (A.62) into (A.61), we obtain I1 (g1 )I2 (g2 ) · · · Ik (gk )Q =

k , gj Q Pf 1 − −1 A()R .

j =1

Recalling the characterization of this Pfaffian as a canonical square root of the corresponding determinant, it is clear that it is a similarity invariant and the theorem follows. Passing to the limit λi,j → 1 (through invertible ) one obtains [133]

Theorem , - A.6.3 Suppose gj ∈ G for j = 1, 2, . . . , k, Q is skew-symmetric on W , and gj Q = 0 for j = 1, 2, . . . , k. Then g1 g2 · · · gk Q =

k , gj Q Pf (1 − AR) ,

(A.66)

j =1

where 

0

 −Q− A :=   ..  . −Q−

Q+

···

0 ···

0 −Q−

 Q+ ..  .    Q+  0



R1 0  and R =  .  .. 0

with Rj = (Tj − I )(Q− Tj + Q+ )−1 and Tj = T (gj ).

0 R2 .. .

··· ···

0 0 .. .

0

···

Rk



  , 

(A.67)

In order to deal with the Ising correlations above Tc we need a generalization of Theorems A.4.1 and A.4.2. For simplicity we do not give the most general result but only the version that is required for the application to the Ising model (see [133], and [110] for the general case). However, before we prove such a result it is convenient to do a “perturbation” calculation for R matrices. Suppose that Qj for j = 1, 2 are skew-symmetric maps on W . Suppose that T is an orthogonal + map on W and that (Q− j T + Qj ) is invertible on W . Define + −1 Rj = (T − I )(Q− j T + Qj ) .

316


Then

R2 = R1 (1 − 12 (Q2 − Q1 )R1 )−1 .

(A.68)

To see this, observe that

+ − + −1 R2 = R1 (Q− 1 T + Q1 )(Q2 T + Q2 ) .

But the inverse of the final two factors is − + −1 + − + −1 + 1 (Q− = (Q− 1 T +Q1 − 2 (Q2 −Q1 )(T −I ))(Q1 T +Q1 ) . 2 T +Q2 )(Q1 T +Q1 )

Equation (A.68) follows at once. Theorem A.6.4 Suppose Q is skew-symmetric and Rj is skew-symmetric for j = 1, 2, . . . , 2k. Write gj = op eRj , where Rj = Rˆ j , and suppose that uj ∈ W for j = 1, 2, . . . , 2k. Write hj = op uj eRj . Then if g1 g2 · · · g2k Q = 0,

h1 h2 · · · h2k Q = g1 g2 · · · g2k Q Pf(u),

(A.69)

where u is the 2k × 2k skew-symmetric matrix with i, j entry above the diagonal given by ui,j = ((1 − AR)−1 AIi ui , Ij uj ), where A and R are defined in Theorem A.6.2, Ii is the injection of W into the ith slot in the direct sum W (2k) = W · · ⊕ W, ⊕ · 2k

and the symmetric bilinear form on W distinguished form on W .

(2k)

is just the 2k-fold direct sum of the

We first establish a representation of h1 h2 · · · h2k Q to which we can apply Theorem A.4.2. Observe that both sides of (A.69) are linear in each of the vectors uj . Thus we can prove (A.69) by selecting the uj from the elements of a basis for W . Let {vj } be an orthogonal basis for W normalized so that vj2 = 1. Suppose uj = vaj

for j = 1, 2, . . . , 2k.

The aj need not be distinct or occur in natural order. Let W ⊕ W denote the orthogonal space with distinguished bilinear form (x1 ⊕ y1 , x2 ⊕ y2 ) = (x1 , x2 ) + (y1 , y2 ). Write

Q Q = 0 ′

0 0

and

Rj′

Rj = −I

I . 0


317

Then both Q′ and Rj′ are skew-symmetric on W ⊕ W. Let R′j = Rˆ j′ and define # ′$ R gj′ = opQ′ e j .

Now write vj = vj⊕ 0 and wj = 0 ⊕ vj so that {vi , wj } is a basis for W ⊕ W. w R′ Using the fact that √vi2 , √j2 is an orthonormal basis to calculate e j , one obtains e

R′j

=

2n ℓ(ℓ−1) 1 (−1) 2 w∧α ∧ v∧α eRj , ℓ 2 #α=ℓ ℓ=0

where the factor (ℓ!)−1 that would otherwise appear is eliminated by our convention that the multi-index α is summed only over occurrences in natural order α1 < α2 < · · · < αℓ . Since the Q′ pairings between wi and wj (for i = j ) and between wi and vj are 0, it follows that gj′ =

2n ℓ(ℓ−1) 1 (−1) 2 wα opQ v∧α eRj . ℓ 2 #α=ℓ ℓ=0

(A.70)

Next we introduce the deformation space W ′ (). It is the direct sum of 2k copies of W ⊕ W with the distinguished bilinear form ′ ′ x x1 x2k x1 , ′ ⊕ · · · ⊕ 2k ⊕ ··· ⊕ ′ y2k y1 y2k y1 =

2k

i,j =1

λij (xi , xj′ ) +

2k

(yi , yi′ ),

i=1

where we make the usual assumptions that λii = 1 and that the 2k × 2k matrix with entries λij is symmetric and invertible. Let Ij′ denote the orthogonal injection of W ⊕ W into the j th slot in W ′ (). As we did above, let Q′ denote the 2k-fold direct sum Q′ ⊕ · · · ⊕ Q′ on W ′ (). The representation for h1 h2 · · · h2k Q of interest to us is , ′ ′ ′ h1 h2 · · · h2k Q = 22k lim I1′ (wa1 ) · · · I2k (wa2k )I1′ (g1′ ) · · · I2k (g2k ) Q′ . (A.71) λij →1

To obtain this result first substitute (A.70) into the right-hand side and then do the Q′ contractions of the wj factors (it is just the trace functional for products of the wj ). The right-hand side becomes lim I1 (h1 ) · · · I2k (h2k )Q ,

λij →1

which is just h1 · · · h2k Q , as noted in Theorem A.6.1. In order to use Theorem A.4.2 to calculate the right-hand side of (A.71) we need to know the R matrix for T (g ′ ), where ′ ). g ′ := I1 (g1′ ) · · · I2k (g2k

318


For this purpose we use the perturbation formula (A.68). It is also convenient at this point to change coordinates on W ′ () and regard it as the space W () ⊕ W (2k) , where W () is the deformation space introduced in (A.61) and W (2k) is the 2k-fold direct sum of W with itself. In this representation, Q 0 ′ Q = , 0 0 where each of the entries is a 2k × 2k matrix. Now introduce Q() 0 ′ Q () = , 0 0 in the notation of Theorem A.6.2. The calculation of the R matrix R1 for T (g ′ ) relative to Q′ () follows the calculation in Theorem A.6.3 and one obtains R I 0 R1 = , (A.72) −I 0 0 I where R is the diagonal matrix with iith entry Ri = (Ti −I )(Q− Ti +Q+ )−1 . The R matrix R2 of T (g ′ ) relative to Q′ is then given by the perturbation formula (A.68), R2 = R1 (1 − 12 (Q′ − Q′ ())R1 )−1 . However, 1−

1 (Q′ 2

0 − Q ())R1 = 0 I ′

−1

I − A()R 0

−A() I

0

0 . I

Thus

R(I − A()R)−1 R2 = −(I − A()R)−1

(I − RA())−1 (1 − A()R)−1 A() 0

0 . I

(A.73)

To do this calculation we need to know that 1−A()R is invertible. Theorem A.6.3 implies that 1 − AR is invertible since g1 g2 · · · g2k Q = 0. Thus 1 − A()R is invertible as long as all the λij are sufficiently close to 1. Since we need these formulas only in the limit λij → 1 this suffices. Applying Theorem A.4.2 to the right-hand side of (A.71) yields , , ′ ) Q′ = g ′ Q′ Pf(w()), (A.74) I1 (wa1 ) · · · I2k (wa2k )I1 (g1′ ) · · · I2k (g2k where the i, j matrix element of w() is given by D E 0 Q− 0 Q− 0 0 0 − R , . 2 1 1 Ij waj Ii wai Ij wj 0 0 2 2 W ′ ()

A.7 Infinite Dimensions

319

Note that in this formula Ii wai is just the injection of wai in the ith slot in W (2k) . Combining this with (A.73), we see that w()i,j = 14 Ii wai , (I − A()R)−1 A()Ij waj .

Multiply (A.74) by 22k and use the evident scaling properties of Pfaffians and (A.71) to get , h1 h2 · · · h2k Q = lim g ′ Q′ Pf(u()), λij →1

where

u()i,j = Ii wai , (I − A()R)−1 A()Ij waj .

However, it is easy to see that , ′g Q′ = I1 (g1 ) · · · I2k (g2k )Q , and the theorem now follows from Theorem A.6.1.

Note that the same proof adapts to provide Pfaffian formulas when the vectors uj are replaced by monomials in Alt(W ), but the notation is necessarily more complicated. The result is a generalization of Wick’s theorem that specializes for gj = 1 to the usual form of Wick’s theorem. Now we turn to infinite-dimensional generalizations of the results we need for the Ising model calculations.

A.7 Infinite Dimensions In this section we suppose that W is a complex Hilbert space with an inner product x, y → x, y that is conjugate linear in the first slot. We suppose there is a distinguished conjugation P on W . A conjugation is a conjugate linear map on W such that P 2 = I . Often we write x¯ := P x. The real subspace WR for P is the set of vectors x ∈ W with x¯ = x. The reader can check that the existence of a conjugation on W is equivalent to the existence of a real Hilbert space H , so that W is naturally the complexification of H . In some ways it is illuminating to rework the constructions that follow in terms of H , but to avoid confusion we won’t do more than mention this here in passing. The bilinear form defined by (x, y) = x, ¯ y

for x, y ∈ W

is easily seen to be nondegenerate on W . An isotropic splitting of W is a continuous splitting W = W+ + W− into (closed) isotropic subspaces W± . We could develop the theory for such splittings, but it does simplify matters to further restrict our attention to Hermitian polarizations W+ ⊕ W− in which the isotropic subspaces W± are perpendicular to one another with respect to the Hermitian inner product

320


·, ·. If A is a bounded linear map on W we write A∗ for the adjoint of A with respect to the Hermitian inner product on W . It is easy to check that the ±1 eigenspaces of an operator Q with Q2 = I give a Hermitian polarization of W if and only if Q∗ = Q and QP + P Q = 0. We also call this sort of operator a Hermitian polarization. Because W now has two natural forms, (·, ·) and ·, ·, there is possible confusion regarding notions of orthogonality etc. We use the adjective “complex” to refer to the complex bilinear form (·, ·) and “Hermitian” to refer to the Hermitian symmetric form ·, · (so, for example, v is complex orthogonal to w means (v, w) = 0). The algebraic Clifford algebra Cliff0 (W ) consists of finite sums of finite products of elements from W subject to the Clifford relations xy + yx = (x, y)e, where e is the identity in the algebra. The algebraic Clifford group G0 is the subgroup of elements in g ∈ Cliff0 (W ) such that gxg −1 = T (g)x

for x ∈ W.

Our strategy for obtaining results in infinite dimensions is straightforward. We first generalize the finite-dimensional Fock representations to representations associated with Hermitian polarizations of the infinite-dimensional Hilbert spaces W . We then translate the results of interest for the Ising model analysis to these Fock representations of G0 . Finally, we introduce a closure of the action of G0 in Fock representations, and by suitably modifying the results we obtain extensions that are valid for elements of the closure. The material on Fock representations in infinite dimensions is very well known, so we will be brief. Suppose that Q is a Hermitian polarization and W+ ⊕ W− is the associated splitting. The space Altk (W+ ) is defined as the completion of the set of algebraic tensors Altk0 (W+ ) with respect to a suitable inner product. Algebraic tensors are finite sums of finite products w 1 ∧ w 2 · · · ∧ wk

with wj ∈ W.

The inner product with respect to which the completion is defined is the linear extension of u1 ∧ u2 ∧ · · · ∧ uk , v1 ∧ v2 ∧ · · · ∧ vk = det (u, v) , , where u, v is the k ×k matrix with i, j matrix element ui , vj . Wedge products of orthonormal bases are again orthonormal. The algebraic tensor algebra Alt0 (W+ ) consists of finite sums of elements taken from the spaces Altk0 (W+ ) for a finite selection of values for k. The Fock space Alt(W+ ) is now the infinite direct sum of Hilbert spaces Alt(W+ ) := C ⊕

∞ k=1

⊕Altk (W+ ).

A process of completion is required to define this infinite sum as a Hilbert space.


321

We now introduce more-conventional notation for creation and annihilation operators (notation that does not make much sense for a space Alt(W+ ) without a Hermitian inner product and is strange in any case given that it is the annihilation operators that are naturally defined as Hermitian adjoints). For x ∈ W+ and w ∈ Alt0 (W+ ), define the creation operator a ∗ (x)w = x ∧ w. It is not hard to show that the operator norm of a ∗ (x) is bounded by x, and so it has a unique continuous extension to all of Alt(W+ ). We define an annihilation operator a(x) = a ∗ (x)∗ for x ∈ W+ as the adjoint of a ∗ (x) acting on the Hilbert space Alt(W+ ). Notice that the argument for this version of an annihilation operator is in W+ and not W− . The relation between the two different notions of annihilation operators is a(x) ¯ = a(x)

for x ∈ W− .

Since P anticommutes with Q it maps W− into W+ . It is up to the reader to make distinctions based on the location of the argument for a. However, since in this last section we use only the version with argument in W+ , this should not be a source of confusion. The Fock representation of the Clifford relations for the Hermitian polarization W+ ⊕ W− is F (x) = a ∗ (x+ ) + a (x¯− )

for x = x+ + x− ∈ W+ ⊕ W− .

One can confirm that F extends to a representation of Cliff0 (W ) using the canonical anticommutation relations a ∗ (x)a ∗ (y) + a ∗ (y)a ∗ (x) = 0, a(x)a(y) + a(y)a(x) = 0, a(x)a ∗ (y) + a ∗ (y)a(x) = x, y I,

(A.75)

for x, y ∈ W+ . We leave the confirmation of these relations to the reader. The map P extends from W to a conjugate linear algebra automorphism of ¯ Combining this with the transpose, we Cliff0 (W ), which we write as P X = X. define the conjugate linear involution X → X ∗ by X∗ := X¯ τ . It is clear from the definitions that the Fock representation associated with a Hermitian polarization is a ∗ representation in the sense that F (X ∗ ) = F (X)∗

for X ∈ Cliff0 (W ).

322


It is useful at this point to describe an extension of the action of G0 to a larger group. We start with a normalization of the action of G0 that was already described. Let Pin0 (W ) denote the kernel of the homomorphism nr, that is, the set of g ∈ G0 with nr(g) = 1. Let Spin0 (W ) denote the subgroup of elements g ∈ Pin0 (W ) such that g is an even element in Cliff0 (W ). There is an exact sequence T

0 → Z/2Z → Spin0 (W ) → SO0 (W ) → 0,

(A.76)

where SO0 (W ) consists of the orthogonal transformations on W that are finite-rank perturbations of the identity I + F with det(I + F ) = 1. We extend the action of Spin0 (W ) to a bigger group SpinQ (W ), so that a version of the exact sequence (A.76) survives. We then introduce a “diagonal” action by the general linear group GL(W+ ), which acts by automorphisms on SpinQ (W ). The semidirect product of these two actions is big enough to contain both the action of the spin operator below Tc and the transfer matrix for the thermodynamic limit of the Ising model. Extending the action by the Clifford generators F (w), one obtains operators that include the disorder operators for the Ising model used to represent correlations above Tc in the thermodynamic limit. To get an idea of what sort of groups are involved in this scheme we first calculate the Hermitian norm of the vector F (g)|0 for g ∈ Spin0 (W ) with gQ = 0. The linear functional 0| is clearly represented by an inner product with the vector |0 in Alt(W+ ), and as a consequence the square of the norm of F (g)|0 is given by , 0|F (g)∗ F (g)|0 = 0|F (g ∗ )F (g)|0 = g ∗ g Q .

To evaluate this we want to apply Theorem A.6.3. Suppose that g is in the Clifford algebra generated by the finite-dimensional subspace W0 . Since we can always add the finite-dimensional subspaces QW0 and P W0 to W0 , it is possible to choose W0 so that it is invariant under both Q and P (recall that Q and P anticommute). On such a subspace the bilinear form (x, y) equals x, ¯ y and so is clearly nondegenerate. Also, since the subspace W0 is invariant under both Q and P , we have Alt(W+ ) ≃ Alt W0+ ⊗ Alt W⊥+ , where W⊥ is the Hermitian orthogonal complement of W0 . The Fock representation on Alt(W+ ) is equivalent to W0 ⊕ W⊥ ∋ x ⊕ y → F (x) ⊗ 1 + (−1)N ⊗ F (y) acting on Alt(W0+ ) ⊗ Alt(W⊥+ ). The operator N acts on Alt(W0 ) as the number operator. Hence g ∗ gQ can be evaluated in the Q Fock representation of Cliff(W0 ) on Alt(W0+ ) (since g is even it acts as the identity on Alt(W⊥+ )). We obtain , ∗ , g g Q = g ∗ Q gQ Pf(1 − AR) = | gQ |2 Pf(1 − AR),

where

0 A= −Q−

Q+ 0

and

R(g ∗ , Q) 0 R= . 0 R(g, Q)


323

To simplify the Pfaffian term further, suppose that the matrix of T (g) relative to the W0+ ⊕ W0− splitting is A B T (g) = . C D Since Q+ = 10 00 and Q− = 00 01 , one finds that A − 1 − BD −1 C BD −1 −1 R(g, Q) = (T − I )(Q− T + Q+ ) = . D −1 C 1 − D −1 However, (A.20) implies that A − BD −1 C = D −τ .

(A.77)

To see this, multiply both sides of (A.77) by D τ and evaluate D τ A − D τ BD −1 C, using the relations D τ A = 1 − B τ C and D τ B = −B τ D, to get 1. Thus −τ D − 1 BD −1 R(g, Q) = . D −1 C 1 − D −1

(A.78)

Also, since

A∗ T (g ) = T (g) = B∗ ∗

∗

C∗ , D∗

we obtain

D¯ −1 − 1 R(g , Q) = D ∗−1 B ∗ ∗

Thus



0 0 0

 AR =   −D ∗−1 B ∗

C ∗ D ∗−1 . 1 − D ∗−1

0 0 0 1 − D ∗−1

 D −τ − 1 BD −1 0 0  . 0 0  0 0

(A.79)

Introducing the complex parameter λ, we find after making a similarity transform in (A.79) that interchanges the second and fourth rows and columns,   1 λBD −1 λ(D −τ − 1) 0 −λD ∗−1 B ∗ 1 0 λ(1 − D ∗−1 )  det(1 − λAR) = det    0 0 1 0 0 0 0 1 = det(1 + λ2 D ∗−1 B ∗ BD −1 ).

324


The Pfaffian Pf(1 − λAR) is the holomorphic square root of λ → det(1 + λ2 |BD −1 |2 ), normalized to be equal to 1 at λ = 0. Evidently, 1

F (g)|02 = | g |2 det(1 + |BD −1 |2 ) 2 ,

(A.80)

where the square root is the positive square root of the determinant. Now suppose that gn ∈ Spin0 (W ) is a sequence such that F (gn )|0 converges in the Hilbert space Alt(W+ ). Then gn also converges, and if it converges to a nonzero value, (A.80) implies that the limiting value of the determinant det(1 + |BD −1 |2 ) is finite as well. The determinant det(1 + |BD −1 |2 ) is absolutely convergent if |BD −1 |2 is in the trace class [147]. The operator |BD −1 |2 is in the trace class provided BD −1 , and hence B is in the Schmidt class. Since nr(gn ) = 1, Theorem A.5.2 implies that gn 2 = det Dn , where Dn is the D matrix element of gn . To ensure the convergence of gn , this last equation suggests that Dn ought to converge in trace norm to an operator of the form I + trace class. Since g → g τ exchanges the roles of the A and D matrix elements and g → g ∗ exchanges the roles of the B and C matrix elements, the following definition is natural. Let SOQ (W ) denote the group B of complex orthogonal operators T on W whose matrix representation T = CA D in the W+ ⊕ W− splitting of W has the property that A − I and D − I are trace class operators and B and C are Schmidt class operators. To see that this is a group, one needs to know that the trace class and the Schmidt class are ideals in the ring of bounded operators and that the product of two Schmidt class operators is a trace class operator (see [147]). The following theorem is proved in [36]. Theorem A.7.1 There exist a topological group SpinQ (W ) and a continuous homomorphism T onto SOQ (W ) such that T

0 → Z/2Z → SpinQ (W ) → SOQ (W ) → 0 is an exact sequence of groups. There are a dense linear domain D ⊂ Alt(W+ ) that is invariant under the action of Cliff0 (W ) and a linear action of SpinQ (W ) on D such that gF (w)u = F (T (g)w)gu for w ∈ W, u ∈ D. Furthermore, if g ∈ SpinQ (W ), there exists a sequence of elements gn ∈ Spin0 (W ) such that An →A, and Dn → D in trace norm, Bn →B and Cn → C in Schmidt norm, (A.81) gn u→gu in Hermitian norm for u ∈ D,


where T (gn ) =

An Cn

Bn Dn

and T (g) =

A C

B D

325

are the matrices of T (gn ) and T (g) relative to the Q polarization of W . We won’t give the proof of this theorem here but just sketch some of the ideas. The group SpinQ (W ) is obtained by completion from the two-sheeted covering Spin0 (W ) of SO0 (W ). It is still useful to think of Spin0 (W ) as an “abstract” subgroup of Cliff0 (W ), but we regard SpinQ (W ) as a group of linear transformations acting on D. So for example, we write gu for the action of g ∈ SpinQ (W ) acting on u ∈ D. However, it is still convenient to write gQ = 0|g|0 when g ∈ SpinQ (W ), even though in this usage ·Q does not take an element of the abstract algebra Cliff0 (W ) as an argument. As explained above, trace class convergence for the diagonal elements A and D and Schmidt class convergence for the off-diagonal elements B and C provides a suitable topology for the base SO0 (W ) with respect to the closure of the action of Spin0 (W ). We need only provide a way to identify the “sheet” in which convergence over the base takes place. The vacuum expectation gQ gives such control when it is nonzero. However, even when gQ = 0, it is possible to perturb Q by a finite-rank operator to give a new Hermitian polarization Q′ for which gQ′ = 0. The Pfaffian formula for the ratio of Q functionals shows that the extended notion of convergence that includes convergence of some Q′ functional does not really depend on which such Q′ functional is used to fix the lift from SO0 (W ). Another way to say the same thing is that every finite-rank perturbation Q′ of Q determines a trivialization of the bundle Spin0 (W ) → SO0 (W ). For the set of T ∈ SO0 (W ) such that g2Q′ = det D ′ = 0 there are just two elements in Spin0 (W ) covering T , corresponding to the two square roots of det D ′ . The Pfaffian formula for the ratio of Q functionals shows how these square roots are related to one another in different trivializations. There is yet another way to look at this, which is simpler for some purposes. Suppose gn is a sequence of elements in Spin0 (W ) such that T (gn ) converges in SOQ (W ). Then gn Q′ converges to a nonzero value for some Q′ that is a finiterank perturbation of Q if and only if the sequence of vectors gn |0 converges in Alt(W+ ) (see [36]). In general, the elements of SpinQ (W ) are represented by unbounded operators on Alt(W+ ). The domain D is obtained by letting Cliff0 (W ) act on a suitable closure of the orbit of Spin0 (W ) acting on the vacuum |0 (see [36]). The existence of a sequence gn ∈ Spin0 (W ) converging to g ∈ SpinQ (W ) strongly on D is a by-product of the construction of SpinQ (W ). Observe that (A.78) implies that the R matrix of an element of SOQ (W ) is trace class on the diagonal and Schmidt class on the off-diagonal. The trace class is the closure of the finite-rank operators in the trace norm, as is the Schmidt class. This makes it

326


trivial to obtain finite-rank approximations of such R matrices; furthermore, since the complex skew-symmetry condition is a linear condition, it is easy to maintain the connection with the group Spin0 (W ). Some complication arises from the need to choose a Hermitian polarization Q′ such that the matrix element D ′ is invertible, but this is not serious. Now we introduce the diagonal action associated with the general linear group GL(W+ ), the group of bounded invertible linear maps on W+ . Suppose that G ∈ GL(W+ ) and define Ŵ(G) = I ⊕ G ⊕ G⊗2 ⊕ · · · ⊕ G⊗n ⊕ · · · .

(A.82)

Then Ŵ(G) is a well-defined linear map on the algebraic tensors Alt0 (W+ ). In general, it does not extend continuously to all of Alt(W+ ) unless G is a contraction. However, one may check that on the algebraic tensors, Ŵ(G)F (x) = F (G ⊕ G−τ x)Ŵ(G),

(A.83)

¯ where G = G is the transpose (since P anticommutes with Q it follows that G and also Gτ act on W− ). In fact, it is not difficult to show that Ŵ(G) leaves the subspace D invariant and that τ

¯∗

SpinQ (W ) ∋ g → Ŵ(G)gŴ(G−1 ) ∈ SpinQ (W ) is an automorphism of groups, which we denote by α(G). Thus the semidirect product SpinQ (W ) ×α GL(W+ ) acts on D. Let ker denote the kernel of this action on D and define Spin(W, Q) = SpinQ (W ) ×α GL(W+ )/ker. From (A.83) one sees that for (g, G) ∈ Spin(W, Q) and w ∈ W, (g, G)F (w) = gŴ(G)F (w) = F (T (g)G ⊕ G−τ w)(g, G) as an equality of maps acting on D. It is shown in [36] that if we define T (g, G) := T (g)G ⊕ G−τ , then one has the exact sequence of groups T

0 → C∗ → Spin(W, Q) → SOres (W ) → 0.

(A.84)

The group SOres (W ) is the connected component of the identity in the group Ores (W ), which in turn is defined as the group of invertible complex orthogonals on W that have off -diagonal matrix elements B and C in the Schmidt class (the diagonal elements are automatically Fredholm but not necessarily compact perturbations of the identity). Combining the action of Spin(W, Q) with the action of the Clifford generators F (w), one can realize automorphisms F (x) → ±F (T x) for T ∈ Ores (W ) by a similarity transform with an invertible linear map on D.


327

The following result is used to evaluate the spontaneous magnetization for the Ising model. Theorem A.7.2 Suppose that Q is a Hermitian polarization on W and g ∈ Spin(W, Q) extends to a unitary map on Alt(W+ ). Then if D(g) is invertible, | gQ |2 = det |D(g)|. Proof. Each g ∈ Spin(W, Q) has a factorization g = hŴ(G) with h ∈ SpinQ (W ) and G ∈ GL(W+ ). There exists a sequence hn ∈ Spin0 (W ) that converges strongly to h on D and such that T (hn ) converges to T (h) in trace norm on the diagonal and Schmidt norm off the diagonal. Note that since D(g) is supposed to be invertible and D(g) = D(h)G−τ , the operator D(h) must also be invertible. Since D(hn ) converges to D(h) in operator norm, it follows that for sufficiently large n, D(hn ) is invertible. Recalling (A.80), we see that for n large enough, 1 hn |02 = | hn |2 det 1 + |Bn Dn−1 |2 2 ,

where Bn = B(hn ) and Dn = D(hn ). Passing to the limit n → ∞ using (A.81) and the continuity of the determinant in the trace norm yields 1 h|02 = | h |2 det 1 + |BD −1 |2 2 ,

(A.85)

where B = B(h) and D = D(h). However, since Ŵ(G)|0 = |0, we find that g|0 = h|0, and since B(g) = B(h)G−τ and D(g) = D(h)G−τ , it follows that B(g)D(g)−1 = B(h)D(h)−1 . Thus since we suppose that g is unitary, 1 g|02 = 1 = | g |2 det 1 + |B(g)D(g)|2 2 .

(A.86)

Since g is unitary, the induced rotation T (g) is unitary as well. The matrix form of T (g)∗ T (g) = 1 implies in particular B(g)∗ B(g) + D(g)∗ D(g) = 1, so that 1 + D(g)∗−1 B(g)∗ B(g)D(g)−1 = D(g)∗−1 D(g)−1 .

Thus (A.86) becomes

1

| gQ |2 = det(D(g)D(g)∗ ) 2 = det |D(g)|. Note that it was part of the proof that |D(g)| has a determinant. The operator D(g) won’t have a determinant unless g ∈ SpinQ (W ). Next we want to generalize Theorem A.6.3. Suppose that g ∈ Spin(W, Q) and write A B T (g) = C D

328


for the matrix of T (g) relative to the Q polarization. Suppose that D is invertible and factor T (g) recalling (A.20) as A B AD τ BD −1 D −τ 0 = C D CD τ 1 0 D −τ τ −1 1 − BC BD D 0 = . CD τ 1 0 D However, since B and C are Schmidt class, the operator BC τ is in the trace class and BD −1 and CD τ are in the Schmidt class. Thus 1 − BC τ BD −1 ∈ SOQ (W ). CD τ 1 We know that there is an h ∈ SpinQ (W ) with

1 − BC τ T (h) = CD τ

BD −1 . 1

There are actually two such elements in SpinQ (W ), but it is immaterial which we choose. The exact sequence (A.83) implies that there exists a nonzero complex constant c such that (A.87) g = chŴ(D −τ ). This is the representation we use to prove generalized product formulas. Before we turn to this application we calculate the action of g on the vacuum vector. It is useful to start by figuring the R matrix for h, AD τ − 1 − BD −1 CD τ BD −1 R(h) = CD τ 0 (A.88) 0 BD −1 = , CD τ 0 since AD τ − 1 = −BC τ and CD τ = −DC τ . We now use Theorem A.4.5 to derive a formula for g|0 that is used in Chapter 1 to compute the two-point spin correlations of the Ising model. Theorem A.7.3 Suppose that g ∈ Spin(W, Q) and that gQ = 0. As above, write T (g) =

A C

B . D

Suppose that ej ,j = 1, 2, . . ., is an orthonormal basis for W+ . Then g|0 = g

α

Pf(rα )e∧α ,


329

where the sum is over all multi-indices α1 < α2 < · · · < αℓ and (rα )i,j = eαi , BD −1 e¯αj . Proof. We can write g = chŴ(D −τ ), where c ∈ C, h ∈ SpinQ (W ). Since gQ = chQ = 0, we can normalize h so that hQ = 1. Note that for this choice of h we have c = gQ . Let PN denote the Hermitian orthogonal projection on the span of {e1 , . . . , eN } ∪ {e¯1 , . . . , e¯N }, and write

RN = PN R(h)PN .

Let RN denote the associated element of the Grassmann algebra (see Theorem (A.4.5) and define hN = opQ eRN ∈ Spin0 (W ).

Then hN converges strongly to h on the dense domain D. In particular, we can use Theorem A.4.5 to compute hN |0 = Pf rαN e∧α , |α|≤N

where N ri,j = (RN ej∗ , ei∗ ) = ei , R e¯j for i, j ≤ N.

Then since we know that hN |0 converges to h|0 as N → ∞, the theorem now follows by passing to the limit N → ∞. Now we turn to the generalization of the product formula in theorem A.6.3. Suppose that gj ∈ Spin(W, Q) and Dj := D(gj ) is invertible for j = 1, 2, . . . , n. The vacuum expectation we wish to consider is gn gn−1 · · · g1 Q .

(A.89)

The reversal of the ordering for the product is deliberate. It is done for reasons that are explained more fully in remarks concerning the transfer formalism and time ordering that can be found above in Chapter 4. This switch causes us a small amount of grief at this stage, but it makes it somewhat easier to relate the formulas of this chapter with the time-ordered products of Chapter 1. Our first step in dealing with (A.89) is to substitute the representation (A.87), gj = cj hj Ŵ(Dj−τ ), in (A.89), where

Aj T (gj ) = Cj

Bj Dj

330


and R(hj ) = Introduce the notation

0 Cj Djτ

Di,j = Di Di−1 · · · Dj

Bj Dj−1 . 0 for i > j,

with Dii = Di in particular. Then since Ŵ(G)∗ leaves the vacuum invariant, we see that , gn gn−1 · · · g1 Q = cj h′n h′n−1 · · · h′1 Q , (A.90) j

where

h′j = Ŵ(Djτ 1 )hj Ŵ(Dj−τ 1 )

(A.91)

and τ Di,j = Djτ · · · Diτ

−τ τ −1 with Di,j = (Di,j ) .

The elements h′j are all in SpinQ (W ), so we can use a limiting procedure to evaluate (A.90) (although it is a little tricky to do so right away since we can’t vary hj independently from Dj in this result). Suppose then to start with that all the elements gj are in Spin0 (W ). Then all the elements h′j are also in Spin0 (W ), and Theorem A.6.3 implies that , ′ ′ - , ′ hn hn−1 · · · h′1 = (A.92) hj Pf(1 − AR ′ ), j

where A is given by (A.67) and R ′ is the diagonal matrix with j th entry R(h′n−j +1 ). Using (A.88) and (A.91) we obtain (for j = 1, . . . , n) 0 βn−j +1 ′ Rj = , (A.93) γn−j +1 0 where τ βj = Dj,1 Bj Dj−1 Dj,1 , −1 −τ γj = Dj,1 Cj Djτ Dj,1 .

(A.94)

To take advantage of all the zeros in the A and R ′ matrices it is convenient to reorganize the deformation space W ⊕n as W ⊕n = W+⊕n ⊕ W−⊕n . The matrix of A relative to this decomposition of W ⊕n is U 0 A= , 0 −L

(A.95)


331

where  0  0 U = .  ..

0

1 0

 1 ..  .   1 0

··· .. . 0 0

···

and



0

0

···

 1 0 L= . . ..  .. 1 ···

0 1

The matrix of R ′ relative to the decomposition W+⊕n ⊕ W−⊕n is

β , 0

0 R = γ ′

where β and γ are the n × n diagonal matrices    γn βn    .. and γ := β :=    .

..



.

β1

Now the Pfaffian in (A.92) becomes 1 Pf Lγ

 0 ..  . .  0 0

γ1

 .

−Uβ . 1

This Pfaffian is the analytic square root of 1 λ → det λLγ

−λUβ 1

(A.96)

,

evaluated at λ = 1. Looking ahead to the scaling limit calculation for the Ising correlations, we wish to transform this last determinant so that only factors of Dj−1 and Dj−τ and no factors Dj or Djτ appear in it. In the Ising calculation Dj−1 and Dj−τ both scale to bounded operators, but Dj and Djτ scale to unbounded operators. In the calculations that follow it is useful to introduce a number of diagonal matrices. Define n × n diagonal matrices  Bn Dn−1   B :=   

  C :=  

..



. B2 D2−1

B1 D1−1



Dn−1 Cn ..

. D2−1 C2

  , 

D1−1 C1

  , 

(A.97)

332


and 

  D0 :=  



Dn,1 ..

. D2,1 D1





Dn−1,1 ..

  , 

  D1 :=  

and

−τ γ = D−1 1 CD1 .

. D1

With these definitions it straightforward that β = Dτ0 BD0 Thus

1 Lγ

1 −U Dτ0 BD0 −Uβ = −τ 1 LD−1 1 1 CD1 τ 1 −D−τ D1 0 U Dτ0 B D−τ 1 1 = 0 0 D−1 D0 LD−1 1 0 1 C

  .  1

0 . D0

Since the Pfaffian is a similarity invariant, we see that 1 −UB gn gn−1 · · · g1 Q = gj Q Pf , LC 1 j

where 

and

−τ 0 1 Dn−1 0 0 1    τ  U := D−τ 1 U D0 =  .  ..   0 ···



L := D0 LD−1 1

0 1

  −1  Dn−1  =    ..  . −1 Dn−1,2

0 0 1 −1 Dn−2

−τ Dn−2 .. .

··· ··· .. ..

···

−τ Dn−1,2

..

.. .

.

1 0 0

D2−τ 1 0

···

0

.

. ···

1 D2−1

         



    ..  , .   0 0 1 0

and we also used the fact that 9 : , , gj Q = cj h′j Ŵ(Dj−τ ) = cj h′j Q .

(A.98)

(A.99)

Q

We should clarify the structure of the matrices U and L. The non zero entries in −r a row of U above the superdiagonal all have the form Dn−j,k where j fixes the


333

row and k runs from n − j to 2. The non zero entries in a column of L below the −1 subdiagonal all have the form Dn−j,k where j fixes the column and k runs from n − j to 2. Before we turn to the infinite-dimensional extension of this result it is useful to observe that 1 −λUB det = det 1 + λ2 UBLC . λLC 1

The Pfaffian of interest to us is the holomorphic square root of λ → det 1 + λ2 UBLC .

An argument based on this observation shows that the Pfaffian 1 −UB Pf LC 1

is continuous in the Schmidt norm in UB and LC basically because the determinant A → det(1 + A) is continuous in the trace norm for A. We have proved the following theorem in the special case in which gj ∈ Spin0 (W ). This infinite-dimensional result is used to control the thermodynamic and scaling limits for the Ising correlations below Tc .

Theorem A.7.4 Suppose that gj ∈ Spin(W, Q) for j = 1, . . . , n and the matrix of T (gj ) with respect to the Q polarization of W is given by Aj Bj . T (gj ) = Cj D j If Dj is invertible for j = 1, . . . , n, then gn gn−1 · · · g1 Q =

n , 1 gj Q Pf LC

j =1

−UB , 1

(A.100)

where B, C, U, and L are given by (A.97), (A.98), and (A.99). Proof. Suppose that gj = cj hj Ŵ(Ej−τ ) for cj ∈ C, hj ∈ Spin0 (W ), and Ej ∈ GL(W+ ). No relation is assumed between Ej and the matrix elements of Aj Bj . T (hj ) = Cj Dj

Write Ei,j = Ei Ei+1 · · · Ej for i > j with Eii = Ei as above. Define Then

−τ τ h′j := Ŵ(Ej,1 )hj Ŵ(Ej,1 ) ∈ Spin0 (W ).

gn gn−1 · · · g1 Q =

n

j =1

, cj h′n h′n−1 · · · h′1 Q

n , 1 gj Q Pf ′ ′ = LC j =1

−U ′ B ′ , 1

(A.101)

334


where B ′ , C ′ , U ′ , and L′ are calculated as B, C, U, and L are in (A.97), (A.98), and (A.99) but with the replacements τ Bj ←⊣ Bj′ := Ej,1 Bj Ej,1 ,

−1 −τ Cj ←⊣ Cj′ := Ej,1 Cj Ej,1 ,

Dj ←⊣ Dj′

:=

(A.102)

−1 Ej,1 Dj Ej,1 .

) Now fix the choice of Ej for j = 1, . . . , n and replace h1 by a sequence h(N that 1 converges as N → ∞ to h1 ∈ SpinQ (W ) (with Dk invertible, of course) and write 9 : ) (N ) −τ converges to g1 Q as N → ∞ since g1(N ) |0 g1(N ) = c1 h(N 1 Ŵ(E1 ). Then g1 Q

converges to a vector in D. This also implies 9 : N →∞ gn gn−1 · · · g1(N ) −→ gn gn−1 · · · g1 Q ,

(A.103)

Q

although one must be a little careful since gj for j = 2, . . . , n need not be bounded because of the factors Ŵ(Ej−τ ). However, h′j ∈ Spin0 (W ) is bounded for j = 2, . . . , n and (A.103) can be confirmed by passing to the h′j representation in (A.101). On the other hand, B1(N ) and C1(N ) converge in Schmidt norm and D1(N ) converges in trace norm. Thus D1(N ) is invertible for all N large enough, and also D1(N )−1 converges uniformly to D1−1 . Since the Pfaffian on the right-hand side of (A.100) is continuous for Schmidt norm convergence in UB and LC, it converges to the appropriate limit as N → ∞. Thus (A.101) extends to h1 ∈ SpinQ (W ). The same argument allows us to extend (A.101) to h2 ∈ SpinQ (W ), and continuing in sequence, the result is true if hj ∈ SpinQ (W ) for j = 1, 2, . . . , n. Now return to the original situation in which gj = cj hj gŴ(Dj−τ ), Aj Bj ∈ SOres (W ), T (gj ) = Cj Dj 1 − Bj Cjτ Bj Dj−1 T (hj ) = , 1 Cj Djτ and Ej = Dj . Then one may verify that τ Bj′ = Dj,1 Bj Dj −1,1 ,

−1 Cj′ = Dj,1 Cj Dj−τ −1,1 ,

Dj′

(A.104)

= 1.

It follows that for the standard factorization of gj we have    0 0 0 1 ··· 1   ..  . .  0 0 . .  and L′ = 1 0 U′ =    .. . .  .. . . . 0 1 1 ··· 0 ··· 0 0

··· 0 1

 0 ..  . ,  0 0


and a little calculation shows that τ D1 1 −U ′ B ′ = L′ C ′ 1 0

0 D−1 0

1 LC

−UB 1

−τ D1 0

335

0 . D0

The Pfaffian is invariant under similarity by bounded invertible transformations even in infinite dimensions, and so n , 1 gn gn−1 · · · g1 Q = gj Q Pf LC j =1

−UB . 1

This finishes the proof.

The generalization of Theorem A.6.4 that we require to deal with the Ising correlations above Tc is accomplished in much the same manner. First suppose that g ∈ Spin(W, Q) and w ∈ W. We write op(ug) = u+ g + gu− ,

(A.105)

and we think of u+ and u− as acting by F (u+ ) and F (u− ) in the Fock representation (it is natural to regard g ∈ Spin(W, Q) as “even” elements). This is not exactly consistent with our previous usage, where we would have written F (op(ug)) = F (u+ )F (g) + F (g)F (u− ) for g ∈ Spin0 (W ), but it is convenient and it does help lighten the notational burden. Theorem A.7.5 Suppose that for j = 1, . . . , n, gj ∈ Spin(W, Q) with Dj = D(gj ) invertible, and uj ∈ W . Then if gn · · · g1 Q = 0, op(un gn ) · · · op(u1 g1 )Q = gn · · · g1 Q Pf(u),

(A.106)

where u is the n × n skew-symmetric matrix with i, j entry above the diagonal given by 7 + 8 −1 u 1 −UB Uu+ i ui,j = , j− , (A.107) LC 1 −Lu− u i j n

with the understanding that rows and columns of u are labeled in reverse order. For n = 4, for example,   0 u4,3 u4,2 u4,1 −u4,3 0 u3,2 u3,1  . u= −u4,2 −u3,2 0 u2,1  −u4,1 −u3,1 −u2,1 0

The matrices B, C, U, and L are given by, (A.97), (A.98), and (A.99), the bilinear form (·, ·)n on W ⊕n is the direct sum of n copies of the distinguished bilinear form ⊕n on W , and finally, the vectors u± are the injections of u± i ∈ W i into the ith ± ± ⊕n ± summand of W± = Wn ⊕ · · · ⊕ W2 ⊕ W1 (note the reverse order).

336


Proof. The strategy of the proof is the same as for the preceeding theorem. First we consider the case gj ∈ Spin0 (W ) for j = 1, . . . , n. Since we suppose that Dj is invertible, we can factor gj = cj hj Ŵ(Dj−τ ). Substituting (A.105) into the left-hand side of (A.106) and moving the factors Ŵ(Dj−τ ) to the left until they hit the vacuum, we obtain , op(un gn ) · · · op(u1 g1 )Q = (A.108) cj op(u′n h′n ) · · · op(u′1 h′1 ) Q , j

where h′j is given by (A.91) and − D1τ u+ ′ 1 + u1 uj = + − τ Dj,1 uj + Dj−1 −1,1 uj

for j = 1, for j > 1.

(A.109)

We can apply Theorem A.6.4 to the right-hand side of (A.108). Splitting the space W ⊕n = W+⊕n ⊕ W−⊕n as before and relabeling yields , , op(u′n h′n ) · · · op(u′1 h′1 ) Q = h′n · · · h′1 Q Pf(u),

where the i, j matrix element of u above the diagonal is (see (A.104) and the subsequent display) 7 −1 ′ τ + τ + 8 D0 uj D0 ui 0 1 −U ′ B ′ U ui,j = . (A.110) − , −1 − L′ C ′ 1 0 −L′ D−1 u D i 1 uj 1 n

In this formula u± i

is injected into the ith summand of W±⊕n

Wn± ⊕· · ·⊕W1± , and

= the matrix of u has row labels starting with n on the upper left and column labels starting with n on the upper left. Thus the first row of u is [0, un,n−1 , . . . , un,1 ], etc. This, of course, adjusts for the right-to-left ordering in the summands of W±⊕n . Recalling (A.98) and (A.99) and making use of τ −τ τ D0 0 D1 0 , = 0 D0 0 D−1 1 we see that (A.110) can be rewritten 7 −1 1 −UB U ui,j = LC 1 0

+ + 8 u 0 ui . , j− u −L u− j i n

Since it is clear that gn · · · g1 Q =

j

, cj h′n · · · h′1 Q ,

this finishes the proof for gj ∈ Spin0 (W ). The rest of the proof follows the proof of the preceding theorem so closely that there is no need to repeat it here.

Appendix B Relations Between Transforms

In this appendix we collect the transformations that are useful in understanding different aspects of the Ising model.

B.1 Boltzmann Weights Let Kj =

Jj kT

for j = 1, 2,

where J1 is the horizontal and J2 the vertical interaction strength, k is the Boltzmann constant, and T is the temperature. The hyperbolic parametrization of the Boltzmann weights is cj = ch(2Kj ), sj = sh(2Kj ), where ch = cosh and sh = sinh and j = 1, 2. The dual interaction constants Kj∗ are defined by sh(2Kj ) sh(2Kj∗ ) = 1. We write cj∗ := ch(2Kj∗ ) = sj∗ := sh(2Kj∗ ) =

cj sj 1 sj

, ,

338

Appendix B. Relations Between Transforms

for j = 1, 2. The spectral curve is a branched cover of P1 with branch points αj± for j = 1, 2. The Boltzmann parametrization of the branch cuts is ∗

α1 = (c1∗ − s1∗ )(c2 + s2 ) = e2(K2 −K1 ) , ∗ α2 = (c1∗ + s1∗ )(c2 + s2 ) = e2(K2 +K1 ) . An elliptic parametrization that meshes well with the uniformization of the spectral curve for T < Tc is 1 k = s1∗ s2∗ = s1 s2 for the elliptic modulus and (writing sn(u, k) = sn(u) and etc.) c1 = cn(2ia), s1 = −isn(2ia), c2 = ik −1 ds(2ia), s2 = ik −1 ns(2ia), and c1∗ s1∗ c2∗ s2∗

= ics(2ia), = ins(2ia), = dn(2ia), = −iksn(2ia).

For the Boltzmann weights to be real and positive we choose 0 < 2a < K ′ . The branch point α2 is α2 = −k −1 ns2 (ia).

B.2 Representations of the Complex Orthogonal Space W 1. W ≃ ℓ2 (Z + 12 ). Write v=

k∈Z+

pk qk xk (v) √ + yk (v) √ . 2 2 1 2

Then the Hermitian inner product is u, v =

1 k∈Z+ 2

x k (u)xk (v) + y k (u)yk (v),

B.2 Representations of the Complex Orthogonal Space W

339

and the distinguished bilinear form is (u, v) = xk (u)xk (v) + yk (u)yk (v). k∈Z+

1 2

The conjugation that relates the two is and yk → y k .

xk → x k

2. W ≃ L2 (S 1 , C2 ). Write 1 x(z) k− 12 xk , z =√ yk y(z) 2π 1 k∈Z+

2

where z = eiθ . Then the Hermitian inner product k x k xk′ + y k yk′ becomes ) dz (x(z)x ′ (z) + y(z)y ′ (z)) , 1 iz S and the complex bilinear form k xk xk′ + yk yk′ becomes ) dz (x(z)x ′ (z−1 ) + y(z)y ′ (z−1 )) . 1 iz S The distinguished conjugation that relates the two is −1 ) x(z) x(z ¯ → . y(z ¯ −1 ) y(z) 3. W ≃ L2 (S 1 , C2 ), the “Q representation” for T < Tc . This is obtained from the preceding representation by the action of the unitary multiplication operator 1 1 −w U=√ , 1 2 w so f (z) = U

x(z) , y(z)

where w(z) = −iw+ (z)w− (z), 1 (z) , w+ (z) = A A (z) 2

w− (z) =

A2 (z−1 ) , A1 (z−1 )

1

Aj (z) := (αj − z) 2 .

340


For T < Tc we have α2 > α1 > 1 and Aj is continuous on the unit circle and normalized so that Aj (1) > 0. The map U diagonalizes Q, so that 1 0 . U QU ∗ = 0 −1 The map U is unitary, so the Hermitian inner product remains as above. However, 0 −w(z) τ −1 −1 , = U (z )U (z) w(z) 0 so the distinguished bilinear form is ) dz 0 −w(z) τ . f (z) g z−1 (f, g) = w(z) 0 1 iz S

This representation is useful for doing Wiener–Hopf-type calculations. 4. W ≃ L2 (S 1 ) ⊕ L2 (S 1 ), the “hyperbolic spectral transform.” This is obtained from item 2 above by the matrix-valued multiplication operator S(z) =

1 2 A ¯ 1 A2 (z) iA1 A¯ 2 (z) , A¯ 1 A2 (z) −iA1 A¯ 2 (z) α1 α2 − 1 z

where the square root is normalized to be negative at z = 1 and is branched along the negative axis. The hyperbolic spectral transform f (z) is x(z) f+ (z) = S(z) . f (z) := f− (z) y(z) A calculation shows that −1 = S(z)S(z)∗

s1 1 2 sh γ (z) 0

0 , 1

so that the Hermitian inner product is ) f¯+ (z)g+ (z) + f¯− (z)g− (z) f, g = S1

s1 dz . 2iz sh γ (z)

It diagonalizes Q, so

SQS ∗ =

1 0

0 . −1

One finds that −1 = S(z−1 )S(z)τ

s1 0 2 sh γ (z) 1

1 , 0


so the distinguished bilinear pairing is ) (f, g) = f+ (z)g− (z−1 ) + f− (z)g+ (z−1 ) S1

341


5. The “elliptic spectral representation” W ≃ L2 (0, 2K) ⊕ L2 (0, 2K) is obtained from the hyperbolic spectral representation by an elliptic substitution. Define z(x) :=

1 + α2 x . x + α2

Then the elliptic substitution x = ksn2 (u, k) becomes for the z variable z = ksn(u + ia)sn(u − ia). The elliptic spectral representation is then F+ (u) f (ksn(u+ + ia)sn(u+ − ia)) := + , f− (ksn(u− + ia)sn(u− − ia)) F− (u) where iK ′ 2

u± := u ±

with u ∈ [0, 2K],

and K ′ is half the imaginary period of sn(u, k). The Hermitian inner product is F, G =

2K

)

0

(F¯+ (u)G+ (u) + F¯− (u)G− (u)) du,

where K is half the real period of sn(u, k). The complex bilinear form is given by (F, G) =

)

0

2K

(F− (u)G+ (u) + F+ (u)G− (u))du.

This last result is worth commenting on. The bilinear form in the hyperbolic spectral transform is the sum of two terms, ) s1 dz f+ (z)g− (z−1 ) 1 2iz sh γ (z) S and )

S1

f− (z)g+ (z−1 )


342


In the first of these make the substitution z = ksn(u+ + ia)sn(u+ − ia) and use the identity ksn(u + iK ′ )sn(u) = 1 to see that the integral becomes ) 2K 0

F+ (u)G− (u)du,

since du =

s1 dz . 2iz sh(γ (z))

In the second integral we want to make the substitution z = ksn(u− + ia)sn(u− − ia) with u ∈ [0, K]. However, this parametrization is a clockwise parametrization of the circle, so we need to adjust the integral by a sign when we make this substitution. Since s1 dz , du = − 2iz sh(γ (z)) for this parametrization we find that the second integral becomes ) 2K F− (u)G+ (u)du. 0

6. In order to make the connection between the scaling limit calculations in Chapter 2 and the tau function calculations in Chapter 3 it is useful to compare the scaling limit of the hyperbolic spectral transform with the spectral transform for the Dirac operator of Chapter 3. We first recall the formal scaling of Fourier series to the Fourier transform. Suppose that 1 1 ei(k− 2 )θ fk . f (θ) = √ 2π 1 k∈Z+ 2

Suppose that m1 > 0 (the notation for the horizontal scaling mass of Chapter 2). 1 Introduce p = m−1 1 θ and xj = (j − 2 )m1 and define 1

F(p) := m12 f (m1 p), −1

F (xk ) := m1 2 fk . Then it is easy to check that ) π/m1

−π/m1

2

|F(p)| dp =

)

π

−π

|f (θ)|2 dθ

and k

|F (xk |2 xk =

k

|fk |2 ,


343

where xk = m1 . Also, 1 ixk p F(p) = √ e F (xk )xk . 2π k Thus one sees that the Fourier series scales as m1 → 0 to ) 1 eixp F (x) dx. F(p) = √ 2π R In terms of the standard Fourier transform Fˆ , we see that F(p) = Fˆ (−p). We know from (4.11) that the change of basis in C2 , 1 f1 = √ (e1 + e2 ), 2 1 f2 = √ (−e1 + e2 ), 2 puts the formal scaling limit for the transfer matrix into the standard form for the Dirac equation that is used in Chapter 3. The new coordinates ψj (x) defined by ψ1 (x)f1 + ψ2 (x)f2 = F1 (x)e1 + F2 (x)e2 are related to F (x) by 1 1 1 F (x) = √ ψ(x). 2 −1 1 As noted in Chapter 3, √ S(m1 p) √1 − ip = √ 1 − ip m1 →0 m1 lim

√ i √1 + ip . −i 1 + ip

Thus the scaling limit of the of the hyperbolic spectral transform in the ψ-coordinates is √ √ 1 1 1 ˆ 1 − ip i √1 + ip ψ(−p). √ √ 1 − ip −i 1 + ip −1 1 2 To compare this with the spectral transform for the Dirac operator (4.74) we introduce the change of variables r=

ω(p) + p m

344


and use $ 1 − i # −1 1 r 2 + ir 2 , 2 $ ( i − 1 # −1 1 r 2 − ir 2 . i 1 + ip = 2 (

1 − ip =

The scaled hyperbolic spectral transform followed by the change of variables r = m−1 (ω(p) + p) becomes 1 1 1 − i r − 2 ir 2 ˆ m(r −1 − r) . ψ √ 1 1 2 ir 2 r − 2 √ times the spectral transform of ψ Recalling (4.74), we see that this is just 1−i 2 −1 evaluated at r rather than r. This inversion just straightens out the mismatch in the scaling of Fourier series compared to the standard Fourier integral transform (p → −p).

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List of Notation

M,N , , 1 E (σ ), 1 Z , 2 σA , 2 σA , 2 + Z ,4 σA + , 4 V1 (σ ), V2 (ρ, σ ), 4 σj , 6 V1 , V2 , 6 VM , 8 pk , qk , 9 xk , yk , 10 V1 , V2 , 12 TM , WM , 15 ± WM , 15 |0M , 17 sj , 17 A(z), B(z), C(z), 24 α1 , α2 , 25 γ (z), v(z), 25 Tc , 26 T (V ), Tz (V ), 31 w(z), 32 Aj (z), 32

w± (z), 33 W± , 33 Q, Q± , 33 A(si ), B(si ), C(si ), D(si ), 34 A • B, 45 Z(, K), 46 rk , 53 µk , 54, 55 M± , 64 z(x), 67 K, K ′ , 71 z(u, a), λ(u, a), 71 α(θ), 73 q(x), 73 F (x), 75 F± (x), 76 G± (u), 77 Sf (u), 78 S(z), 83 m1 , m2 , 107 τ (a; T ), 108 τ (a), 108 ¯ j , y¯j , 109 T (σj ), Aj , Bj , Cj , Dj , 109 XT± (θ, θ ′ ), 113

358

List of Notation

oT (p), oT∗ (p), 117 X± (p, p ′ ), 119 B, C, L, U, 124 µj , 127 Ti,j , 128 ei (k, m), fi (k, m), 151 ψ(x), 154 D, 154 J , 157 G(x, y), 158 R, infinitesimal rotation, 160 u, u−1 , 162 e(x.u), 163 Cj (b, ǫ), 164 wℓ (x), 165 wℓ (x), Bessel function representation, 166 ∗ (x), 167 w−ℓ wℓ∗ (x), Bessel function representation, 167 wˆ ℓ (x), 167 cℓ (x), cℓ∗ (x), 168 cˆℓ , 171 cℓ (w), cℓ∗ (w), contour integral representation, 175 p(r), ω(r), 185 Q± (r), 185 e± (r), 185 f± , 185 α, β, γ , δ, 194 Sj , Uj , Lj , 199 Uj , Lj , 199 Pr(aj ), 200 [−] (aj ), 200 Wint [−] (a), 200 Wint Wext , 201 Gr(W0 ), 201 P0 , 201 P[−] int (a), 202 Pr(a), 202 [−] (a)), 202 δ(Wint

[−] (a)), 202 τ (Wint P(a), 206 ˆ − a), 211 w(x w(x, a), 212 F ± (a), 212 [+] Wint (a), 212 [+] Pr (a), 213 [+] δ(Wint (a)), 213 [+] τ (Wint (a)), 213 Wj , Tij , 213 F(a), 214 O0 , Oi , O∞ , 224 wj , wj∗ , 240 a, cℓ , cℓ∗ , 253 F, G, 255 a∗ , A∗ , 256 , 256 A, 257 H , 263 ψ, H , 267 Altk (V ), 273 v1 ∧ v2 ∧ · · · ∧ vk , 275 c(v), c(v)τ , 276 |0, 276 a(ϕ), 277 Cliff(W ), 277 Q, isotropic splitting, 278 FQ (x), Fock representation, 278 G, T (g), 278 O(W ), 279 opQ , 280 sbQ , 280 ˆ 281 R, XQ , 285 W, 288 Q, 288 XQ , 289 Q, 290 J , 290 Pf(r), 294 β ⊔ γ , 297 wα , w∧α , 297 Pin(W ), Spin(W ), nr(g), 305

List of Notation

sq(g), 306 Q(), 311 A(), 312 a(x), a ∗ (x), 321 SpinQ (W ), SOQ (W ), 324

Spin0 (W ), SO0 (W ), 325 Ŵ(G), 326 Spin(W, Q), SOres (W ), 326 B, C, 331 U, L, 332

359

Index

Abraham, xii, 83 Abraham and Martin-Löf, xi, 4, 5, 9, 15, 19, 21 addition theorem, 77 admissible frame, 201 annihilation operators, 276 anticommutation relations, 278 Avogadro’s number, 2 Basor and Tracy, 144 Bessel function, 100, 163, 166, 175, 177, 229 Boltzmann weight, vi, 2, 69 branched, 159 Brauer–Weyl, 9, 15 broken symmetry, vii canonical, 227 canonical basis, 232, 253 canonical section, 213 Cartan and Dieudonné, 286 Cheng and Wu, 96 Clifford algebra, 277 Clifford group, 278 Clifford relations, 9–11

cluster decomposition, 57 complex orthogonal, 11, 279 configuration energy, 1 conformal field theory, 140 conformal regime, 106 conjugation, 167, 178 contour integral representation, 164, 180 convolution, 79 corner transfer matrix, xv correlation functions, vi correlation length, 96, 105, 106 creation operators, 276 critical correlations, 58 critical exponent, 106 critical temperature, vii, 26 critical two-point function, 140 Curie, 3 cylinder measures, 135 cylinder sets, 132 deformation equations, 223, 255, 260 determinant bundle, 198 Dirac equation, 232 Dirac operator, 161 discriminant, 65

362

Index

disorder operator, 89 disorder variables, 51 DLR equations, 131 Dobrushin, 131 duality, 45 elliptic integral, 69 expansion at infinity, 172 ferromagnetic, v, 3 Flaschka and Newell, 270 Fock representation, 14, 278 Fourier transform, 161 free Fermion, 147 Fuchs, 270 Gambier, 270 Garnier, 270 Gaussian domination, 136 Gibbs random field, 131 Gibbs state, vi, 4, 132 GKS inequalities, 145 graph coordinates, 205 Grassmann calculus, 279 Grassmannian, 201 Green function, 151, 175, 178, 179, 236 Griffith’s inequalities, 55, 126, 135 Haag, 2 Hardy space, 34 Helmholtz equation, 227 Hermitian polarization, 319 high-temperature expansion, 47 holomorphic differential, 65 holonomic quantum field, 147 holonomy, 148 induced rotations, 11 infinite-volume limit, 8 Ising, ix

isotropic, 277 isotropic splitting, 13 Jacobian elliptic function, 68 Kadanoff and Ceva, xi, 49 Kadanoff and Kamoto, ix Kato, 37 Kaufmann, x, 9 KdV, ix Kovalevskaya, 270 Kramers and Wannier, ix Kramers–Wannier duality, 21, 29 Lanford and Ruelle, 131 Laplace method, 99 Lenz, ix line bundle, 169 local eigenfunctions, 232 local expansion, 170 local operator product, 160 localization, 181, 188, 234 long-range order, 105 low-temperature expansion, 47 Luther and Peschel, xv, 271 Malgrange, 270 McCoy and Wu, vii McCoy, Wu, and Perk, ix Minkowski, 155 Minlos, xii, 132 Miwa, Jimbo, and Ueno, 270 modern analysis, xi monodromy-preserving deformation, 148 Montroll, Potts, and Ward, xi, 59, 63, 101 nearest neighbors, 1 number operator, 19 one point function, 42 Onsager, ix, xi, 3 Onsager and Kaufmann, ix Onsager–Yang, 80

Index

open boundary conditions, 45, 51 orthogonal reflection, 285 Painlevé, viii, 223 Painlevé equation, 270 Painlevé property, 270 Palmer and Tracy, xi partition function, 2 Peierls, ix Pfaffian, 87, 294, 301, 324, 333 Pfaffian bundle, 199 plus boundary conditions, 3, 9 Poincaré, xv polarization, 13, 278 Potts and Ward, xi, 59, 60, 101 product deformation, 310 projection, 189 pure states, 57 quantum inverse scattering, xv quasifree state, 288 ramification, 65 rapidity, 161 rational parametrization, 162 renormalization, 293 rotational invariance, 265 Ruelle, 4 Sato, Miwa, and Jimbo, viii, xv, 270 scaling limit, 108, 124, 128 Schlesinger, 270 Schmidt class, 36 Segal and Pressley, x Segal–Wilson, ix, xiv Sobolev space, 182 spectral curve, 64, 72, 76, 129, 161 spectral representation, 185, 189 spectral transform, 78, 186 spin configuration, 1 spin correlation, 2 spin operator, 6, 11, 33 spin representation, 279

363

spontaneous magnetization, 63, 81 Stokes’s theorem, 158 supersymmetric, xv susceptibility, xv, 140, 143 symbol map, 290 symbols, 280 symmetries, 261 Szego˝ , xii, 59 Szego˝ theorem, 102 tau functions, 147, 198 two-dimensional Ising model, vii thermodynamic limit, vi time-ordered, 156, 159 time ordering, 155 Toeplitz, xii, 43, 59, 60 Tracy, 140, 144, 271 Tracy and Widom, 140, 271 transfer matrix, 32 transverse, 235 trivialization, 213 two-point function, 88, 95, 142 two-point scaling, 143 uniformization, 68, 69, 72, 76, 161, 162 vacuum vector, 14, 276 wave functions, 160 wedge product, 274 Whittaker and Watson, xi, 163 Wick’s theorem, 294 Widom and Tracy, 144 Wiener–Hopf, 33–35 Wu, 96 Wu, McCoy, Tracy, and Barouch, viii, xv, 223, 270 Yamada, xii, 89, 96 Yang, xi, 43, 63 Z invariant, xv

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