Renormalization: An Introduction (Theoretical and Mathematical Physics)

Manfred Salmhofer

Renormalization An Introduction

With 24 Figures

Springer

Prof. Dr. Manfred Salmhofer Mathematik ETH-Zentrum CH-8092 Zürich, Switzerland

Editors

Roger Balian

Nicolai Reshetikhin

CEA Service de Physique Théorique de Saclay F-91191 Gif-sur-Yvette, France

Department of Mathematics University of California Berkeley, CA 94720-3840, USA

Wolf Beiglbiick

Herbert Spohn

Institut ftir Angewandte Mathematik Universitüt Heidelberg Im Neuenheimer Feld 294 D-69120 Heidelberg, Germany

Theoretische Physik Ludwig-Maximilians- Universitüt München TheresienstraBe 37 D-80333 Mtinchen, Germany

Harald Grosse

Walter Thirring

Institut für Theoretische Physik Universitat Wien Boltzmanngasse 5 A-1090 Wien, Austria

Institut für Theoretische Physik Universitüt Wien Boltzmanngasse 5 A-1090 Wien, Austria

Elliott H. Lieb Jadwin Hall Princeton University, P. 0. Box 708 Princeton, NJ 08544-0708, USA

Library of Congress Cataloging-in-Publication Data Salmhofer, Manfred, 1964— Renormalization: an introduction / Manfred Salmhofer. p. cm . — (Texts and monographs in physics, ISSN 0172-5998) Includes bibliographical references and index. ISBN 3-540-64666-3 (alk. paper) I. Renormalization (Physics) I. Title. 11. Series. QC174.17.R46S35 1998 530.1'43—dc2 I 98-40042 CIP

ISSN 0172-5998 ISBN 3-540-64666-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Veriag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready copy from the author using a Springer TEX macro package Cover design: design & production GmbH, Heidelberg SPIN: 10652045 55/3144-5 4 3 2 10 - Printed on acid-free paper

Preface

Why another book on the renormalization of field theory? This book aims to contribute to the bridging of the gap between the treatments of renormalization in physics courses and the mathematically rigorous approach. It provides a simple but rigorous introduction to perturbative renormalization, and, in doing so, also equips the reader with some basic techniques which are a prerequisite for studying renormalization nonperturbatively. Beside these technical issues, it also contains a proof of renormalizability of 0 4 theory in d _ < 4 dimensions and a discussion of renormalization for systems with a Fermi surface, which are realistic models for electrons in metals. Like the two courses on which it is based, the book is intended to be easily accessible to mathematics and physics students from the third year on, and after going through it, one should be able to start reading the current literature on the subject, in particular on nonperturbative renormalization. Chapter 1 provides a brief motivation for studying quantum theory by functional integrals, as well as the setup. In Chap. 2, the techniques of Gaussian integration and Feynman graph expansions are introduced. I then give simple proofs of basic results, such as the theorem that the logarithm of the generating functional is a sum of values of connected Feynman graphs. In Chap. 3, the Wilson renormalization flow is defined, and perturbative renormalizability of 0 4 theory in d < 4 dimensions is proven using a renormalization group differential equation. The Feynman graph expansion of Chap. 2 is the explicit solution to this equation, but the analysis of the differential equation leads to a very simple renormalizability proof. It also brings out clearly that the renormalization subtractions really amount to a change of boundary conditions. In Chap. 4, a similar renormalization flow is applied to an infrared problem, that of many-fermion systems. Using the method of overlapping loops, I determine the leading contributions to the renormalization group flow to all orders of perturbation theory. This leading behaviour is then calculated in the simplest cases and its physical implications are discussed heuristically. Readers with a little experience in field theory can read Chap. 4 independently of Chaps. 1-3. I would like to thank Horst Kniirrer for the suggestion to write this text and for many valuable comments. I also thank Ldsz16 Erd6s and Pirmin Lemberger for reading carefully through Chaps. 1-3 and for suggesting var-

VI

Preface

ious improvements. Finally, I would like to thank Joel Feldman and Eugene Trubowitz for the collaboration which brought about many of the results about many-fermion systems described in Chap. 4 and for many discussions, Christian Lang and Erhard Seiler for teaching me a good part of what I know about quantum field theory, and Volker Bach, Georg Keller, Christoph Kopper, Walter Metzner, Edwin Langmann, and Christian Wieczerkowski for discussions. A saying attributed to Einstein goes 'Everything should be made as simple as possible, but not simpler.' I have made the proofs as clear and concise as I could without oversimplifying matters or discarding important details. Certainly, not everything here is as simple as it could possibly be, but I hope that it is at least readable.

Zurich, September 1998

Manfred Salmhofer

Table of Contents

1.

Field Theory 1.1 A Motivation for Path Integrals 1.2 Gaussian Integrals and Random Variables 1.2.1 Preliminaries 1.2.2 Gaussian Integrals in Finitely Many Variables 1.2.3 The Covariance Splitting Formula 1.3 Field Theory on a Lattice

1.4 Free Fields 1.5 Properties of the Free Covariance 1.6 Problems With and Without Cutoffs

1 6 6 7 10 11 12 15 17 19 21 23

Techniques 2.1 Integration by Parts 2.2 Wick Ordering 2.2.1 Definition and Main Properties 2.2.2 Further Results 2.3 Evaluation of Gaussian Integrals 2.3.1 Labelled Feynman Graphs 2.3.2 Symmetry Factors and Topological Feynman Graphs 2.3.3 Motivations for Taking the Logarithm 2.4 Polymer Systems 2.4.1 Preparation: Graphs and Partitions 2.4.2 The Logarithm of the Polymer Partition Function 2.5 The Effective Action and Connected Graphs 2.5.1 Definition and Semigroup Property 2.5.2 Derivation of the Graphical Representation 2.5.3 Result and Discussion 2.6 Graphical Representations: Conclusions

27 27 28 28 30 33 33 38 39 41 42 44 47 47 49 55 57

1.3.1 Discretization 1.3.2 The Partition Function and Correlations 1.3.3 The Ising Model

2.

1

VIII 3.

Table of Contents

The Renormalization Group

3.1 A Cutoff in Momentum Space 3.2 The Semigroup Structure of Renormalization 3.3 The Renormalization Group Equation 3.3.1 The Functional Form 3.3.2 The Component Form 3.4 The Structure of the RG equation 3.4.1 The Graphical Representation 3.4.2 The Relation to the Feynman Graph Expansion 3.4.3 The Continuum Limit at Fixed Ao 3.5 Differential Inequalities 3.6 Two Dimensions 3.6.1 Boundedness 3.6.2 01 3.6.3 Convergence 3.7 Three Dimensions 3.7.1 Power Counting for the Truncated Equation 3.7.2 Renormalization: A Change of Boundary Conditions 3.7.3 Renormalized 01 3.8 Four Dimensions 3.8.1 Counterterms in Second Order 3.8.2 Power Counting (Skeleton Flow) 3.8.3 The Boundary Conditions for Renormalization 3.8.4 Renormalized 0 4 Theory 3.9 The RG Flow in the Ladder Approximation 4.

The Fermi Surface Problem

4.1 Physical Motivation 4.2 Many-Fermion Systems on a Lattice 4.2.1 The Hamiltonian 4.2.2 The Grand Canonical Ensemble 4.2.3 The Fermi Gas 4.2.4 The Functional Integral Representation 4.2.5 RG Flow: Energy Scales 4.2.6 Model Assumptions 4.2.7 The Physical Significance of the Assumptions 4.2.8 The Role of the Initial Energy Scale 4.3 The Renormalization Group Differential Equation 4.3.1 The Effective Action 4.3.2 The RG Equation 4.3.3 The Component R.GE in Fourier Space 4.4 Power Counting for Skeletons 4.4.1 Bounds for the Infinite-Volume Propagator 4.4.2 Sup Norm Estimates 4.4.3 Estimates in L1 Norm

63 64 65 68 68 69 72 72 74 74 75 78 78 80 82 85 86 88 91 99 99 102 103 105 109 113 113 114 115 118 119 123 125 126 128 129 130 130 131 136 137 137 139 142

Table of Contents The Four-Point Function 4.5.1 Motivation 4.5.2 The Parquet Four-Point Function 4.5.3 The One-Loop Volume Bound 4.5.4 The Particle—Particle Flow 4.5.5 The Particle—Hole Flow 4.5.6 The Combined Flow 4.6 Improved Power Counting 4.6.1 Overlapping Loops 4.6.2 Volume Improvement from Overlapping Loops 4.6.3 Volume Improvement in the RGE 4.6.4 Bounds on the Non-Ladder Skeletons 4.6.5 The Derivatives of the Skeleton Selfenergy 4.7 Renormalization Subtractions 4.7.1 Motivation; the Counterterm 4.7.2 Full Amputation 4.7.3 Bounds for a Truncation 4.7.4 The Meaning of K 4.8 Conclusion 4.8.1 Summary 4.8.2 A Fermi Liquid Criterion 4.8.3 How the Curvature Sets a Scale 4.5

IX 145 146 147 149 151 155 157 157 157 160 161 162 165 167 167 168 170 173 175 175 176 178

A. Appendix to Chapters 1-3 A.1 A Topology on the Ring of Formal Power Series A.2 Fourier Transformation A.3 Properties of the Boson Propagator A.4 Wick Reordering for Bosons A.5 The Lower Bound for the Sunset Graph

181 181 181 184 185 189

B. Appendix to Chapter 4 B.1 Fermionic Fock Space B.2 Calculus on Grassmann Algebras B.3 Grassmann Gaussian Integrals B.4 Gram's Inequality; Bounds for Gaussian Integrals B.5 Grassmann Integrals for Fock Space Traces B.5.1 Delta Functions and Integral Kernels B.5.2 The Formula for the Trace B.5.3 The Time Continuum Limit B.5.4 Nambu Formalism B.5.5 Matsubara Frequencies B.6 Feynman Graph Expansions B.7 The Thermodynamic Limit in Perturbation Theory

191 191 192 196 198 201 202 204 205 212 213 214 217

X

Table of Contents

B.8 Volume Improvement Bounds B.8.1 The One-Loop Volume Bound B.8.2 The Two-Loop Volume Bound

220 220 222

References

227

Index

229

1. Field Theory

1.1 A Motivation for Path Integrals To motivate the study of field theory via the functional integral, we review the path integral of quantum mechanics, as discovered by Dirac [1] and Feynman [2]. We shall be brief and refer to the literature for many details because a number of excellent accounts (e.g. [3 ] ) deal with this topic in depth (and because these details will not be required for understanding the rest of the text). The presentation given here is taken from [3], it is included for the convenience of the reader and because, even though beautiful mathematical work has given a mathematically rigorous foundation for the functional integrals of quantum mechanics [3] and quantum field theory [6, 7, 5], many people still seem to believe that this is not the case. In fact, however, the Hamiltonian formalism of quantum field theory is so singular in more than two dimensions that the path integral seems a much better starting point. The classical Hamiltonian

Hc i =

p2

m

+ V(x)

becomes a quantum-mechanical Hamiltonian h2

H = — — 2i + V(x) 2m

(1.2)

under the replacement p —> hli 010x. States of spinless particles on Rd are described by vectors Iii E ii = L2 (Rd , c) , with the interpretation that for any measurable R c R d , fR itp(t, x) I 2 d d x i s the probability for finding the particle at time t in the region R. Given an initial state 00 in the domain D(H) of the Hamilitonian H at t = 0, the state at later times t > 0 is determined by the Schrödinger equation

alp at

ih— = HO

(1.3)

with initial condition 0(0, x) = 00 (x). If V is such that H is self-adjoint, then the time evolution t v.4 exp(—itH), t E R, is a strongly continuous semigroup, and for an 00 E 9-i, the state at time t> 0 is given by

2

1. Field Theory

= (e—itHlh,lpo)(x)

V(t, x)

= f dy K(t;x,Y) 00(Y)•

(1.4)

Rd

In Dirac's notation,

K (t; x, y) = (x IC iti I lh I y).

( 1 - 5)

Feynman's path integral formula is a rewriting of K in terms of an integral over paths, obtained from the Trotter product formula. Lemma 1.1. (Trotter product formula) If A and B are bounded opera-

tors, then e lt+B =

Lim

(eA/N eBIN )N

N—k>3

(1.6)

5

with the limit in the operator norm. This proof is taken from [3]. Let C = exp((A + B)/N) and D = eA lN eB IN . We have to show that IICN — DN II —> 0 as N > oc. By telescoping,

Proof.

—

N-1 cN _ DN =

Ls Ck(C D)DN -1— k .

(1.7)

k=0

Using the product inequality of the operator norm and

max{IIC11,11D11}
co of dig) exists as a measure on the set of all paths starting at xo . It is the conditional Wiener measure dpxo (X) describing Brownian motion, that is, the analogue of free quantum mechanical motion, but with the Schrbdinger equation replaced by the heat equation. For suitable [4], (1.27) converges as N —> co to the V, e.g. V E LP n.L', where p > (convergent) integral

g-

— 1 V(X(r))dr

(e —S11 0) (X0) =

f diixo (X)e 0

0 (X()).

(1.31)

17 00

Note that the improper integral f dx exp(ix 2 ) is convergent, but not absolutely convergent, which prevents a similar procedure in the real-time case. In fact, complex-valued Gaussian measures on infinite-dimensional spaces exist only under very restricted conditions, e.g. if the imaginary part of the covariance is relatively trace class to the real part. For determining spectral properties of H, e—SH is as good as eitil . Moreover, the path integral representation for the partition function for the canconical ensemble (in a finite volume) tre —'81-1 is obvious from (1.31). The Feynman - Kac formula (1.31) has many applications in quantum mechanics

[31In quantum field theory, one is not dealing with a single particle, but with infinitely many particles, because one has to account for the creation and annihilation of particles. One can formally write down a Hamiltonian, but it becomes very difficult to give a mathematical definition of it. We shall simply define the theory by the functional integral. The lesson from the quantum mechanical case is that free particles are described by a Gaussian measure. We shall therefore put the quadratic part of the action into the measure before taking the continuum limit that defines the field theory.

6

1. Field Theory

In the Minkowski spacetime, there are further problems: because of the indefinite metric, the kinetic term is not semibounded. For this reason, one studies the Euclidean version of the theory, i.e. the theory with imaginary time, and then tries to do the analytical continuation to real time. A historical introduction and physical motivations for doing quantum field theory, as well as a thorough development of the theory is, e.g., in the recent book by Weinberg [8].

1.2 Gaussian Integrals and Random Variables The fields in quantum field theory can be considered as random variables associated to a Gaussian measure that is perturbed by an interaction term. The Gaussian measure determines the properties of free particles. In this section, we give the elementary definitions and properties of Gaussian random variables indexed by a finite set. This is of relevance for quantum field theory because we shall define a regularized field theory on the lattice with finitely many points and then study the continuum limit. 1.2.1 Preliminaries All identities for Gaussian integration are based on the following two elementary formulas.

Remark 1.2. Let a > O. Then 00

L. e -2 " dx = f

(1.32)

and for all b E C

1(b) =

J.

e- i x2+bs dx = e

b2/2a

(1.33)

b2 1 I 2 + 2a Proof. Since --c2lx2 + bx = —1(x — i) a

1 (b) =--- e b2 /2 ' f dx C a ( x—b /a)2 /2 .

(1.34)

R

For b E JR, a shift in the integration variable proves (1.33). For ,t3 = 1m b 5L 0, the shift gives

1(b) =...,_ eb2 12a f dx e— a ( x— Wa)2 /2 . R

Since 1 e -a(x-ii3/a) 2 /21 = e -ax 2 /2 e 02 /2a ,

(1.35)

1.2 Gaussian Integrals and Random Variables

f Isi>/,

dx e-a(x -is/a) 2 / 2 < es2/2a

f

dx e—as2 / 2

---)

L-00

0,

7

(1.36)

lx1>i,

so it suffices to show that

L f dx e __

2 /2

---3 f dx e—as2 / 2 . L—oo

(1.37)

111

—L

Since the integrand is analytic in x, we may deform the contour to consist of a piece [—L, Li] + if3/a and two vertical pieces, going from ±L to ±L + i$/a. The contribution from [ — L, L ] + iiVa equals f LI, e —ax2 /2 dx, which converges to the desired result as L —> oo. The contribution from the two vertical pieces vanishes as L —> cx) because the integrand is exponentially small in L and the integration path is of fixed length 1/3/al. M

1.2.2 Gaussian Integrals in Finitely Many Variables

From now on, let N E N, A E MN(R), A = At > 0 (i.e., all eigenvalues of A are positive) and denote C = A -1 . Denote the scalar product on RN by

N

(0,0

=Eozoi

(1.38)

i=1

and

dNO

= 110i . • • dON -

Remark 1.3.

f e -1("4') dN O = (27) 1- det A-4 = det (270 i and for all

(1.39)

J E CN -105 ,A0)+G74)dNo = det (27C) 4 el (Jicv) . e f

Proof. Since A is symmetric, there is a V E

SO(N)

(1.40)

such that

V-1 AV = D = diagfai, • • • ,aN}

(1.41)

where ai are the eigenvalues of A. Since A is strictly positive, ai > 0 for all i E {1, ... , N } . Let V) = VO, then

(0 ) AO = (0 ) DO)

N = Eaioi2 i.i

(1.42)

8

1. Field Theory

and dNO = 1 detVidNO = eV). Thus by (1.32)

f

_1 E aop2

( 0 A ck ) d N ,

f

e ll) e i-i

N

f dIP i.i. 11

=

_ Similarly, (J, 0) = (J,V

-1

N

/2 7 I i V ai

la. 2

(1.43)

(27) 4 1 = det (27C)i. det A) i (

0) = (V J, 0), so, with b = V J E CN , and by

(1.33)

f dN o

f di/'e_

=

2 1P2 +bilP

i=1

= i=1

\i 22 eh i /2ai /

E

(det 27C) ei (J7cJ)

ai

(1.44)

because

z_s 2a. = i=i 2

— 2

(b,D -1 b) = — (J,17-1 D -1 VJ) = 2

(J,CJ).

(1.45)

•

Definition 1.4. Let C E GLN(R), C > O. The measure

dpc (0) = (det 27C)-i

(0,c -1 0) dN

(1.46)

is called the Gaussian measure on RN with mean zero and covariance matrix C. By definition, pc has total mass equal to one, i.e. f dpc (0) = 1. By Remark

1.3,

f dttc (0) ei( ") =

(1.47)

holds for all J E CN . This Fourier transform is often called the characteristic function of the measure pc. All moments of the measure pc can be calculated using (1.47) as polynomials in the covariance C, e.g.

f

cipcmo102 =

,2

, j2

I

(9 2

L aJiaJ2 =

f 0, (0)

e(J)(15) ] J=0

ei.(

_11=0

(1.48)

1.2 Gaussian Integrals and Random Variables

9

We shall see much more of that when calculating more complicated moments of pc as sums of values of Feynman graphs. Equation (1.47) already suggests that the limit N —> co may be taken under suitable conditions on C and J, e.g. that J is in the form domain of C. The existence of moments of pc in that limit then depends on further properties of C; this will be central in the discussion of renormalization. A natural interpretation from the probabilistic point of view (as well as that of field theory) is to regard not only the measure dpc as interesting, but also the variables 01, - - • , ON. The random variables 01, - - - , ON distributed according to (1.46) are called Gaussian random variables indexed by {1, ... , N}, that is, Gaussian fields. The right hand side of (1.47) is defined for C > 0 as well (in fact, for any C E MN(C)). As we shall show now, the f dpc on the left hand side of (1.47) is a measure for any C > 0 (i.e., where some of the eigenvalues vanish), but not for C G 0. To see this, let us go back to the one-dimensional case where the matrix C is a number e > 0, and where dpc (q5) =-- dOi gc (q51 ) with density

17.rc e—x2/2c. gc,(x) . \/..._

(1.49)

ji, gc(x)dx =

1 and gc (x) --- 0 for all x 0 0, gc is an approximate c—al identity, i.e. gc —> 8 in Si(R). Thus Since

dPc (0)

c, -->+0

00)4-

(1.50)

Similarly, the vanishing of an eigenvalue of C in the N-dimensional case simply implies that the measure contains a 8 function that restricts to the linear subspace orthogonal to the corresponding eigenvector. In this way, the measure dpc has a natural meaning also for C > 0, which we adopt from now on. It is natural because dpc for C > 0 can always be obtained by taking E -> 0+ of a measure dace with Ce > 0 for E > 0. With this definition, (1.47) holds for all C > 0, with a genuine measure pc on the left hand side. We shall need nonnegative covariances all the time in what follows because we shall impose strict cutoffs in momentum space. Finally, note that for C with some negative eigenvalues, the left hand side of (1.47) diverges: although for any C E GL N (JR) the linear functional

Lc(F) = (det 27- C ) f dN 0e 4( "15) F(0)

(1.51)

can be defined on a suitably restricted space of functions, such as CO° (RN), the inequality

Lc(F) < const Moo

(1.52)

holds only if C > 0. So there is no measure associated to the linear functional Lc if C has negative eigenvalues.

10

1. Field Theory

1.2.3 The Covariance Splitting Formula Theorem 1.5. Let Ci C2 E MN(R), G = Cif for i E {1,2 and C2 > O. Define C = C1+C2. Then for all F E L l ( PC)7 }

f dPc 6) f d 2 (02) F(01 + 02) f apc1 (01) f apc2(0 - 00F(o).

f diiC(0)F(0)

i(

>

a,

ttc

(1.53)

In other words, if C = C1+C2, the Gaussian random variable 0 is a sum of two independent Gaussian random variables, q5 = çSi + q52, and the Gaussian measure factors, dpcM = dpc i (Mdpc2 (02) •

Proof Let F(0) = ei (J,0 ). Then by (1.47)

f

dpc (0) e i J4) =

e —lp,c1,0-4(J,c2J)

(

= f dlic (01) ei(j 41) f =

f

dpc2 (02) e i(J'02) (1.54)

dPci ((bi) f dttc2 (02) ei(J,01+02)

so (1.53) holds for this case. Since a measure is uniquely defined by its Fourier transform, the measures dpc and dpc, * dpc2 are identical, therefore (1.53) holds.

Remark 1.6. One can derive (1.53) from (1.54) also more explicitly. By the explicit formula (1.54), both sides are analytic in J. Thus we may take derivatives with respect to J. If F is a polynomial, F(0) = P(0), with P (0)

=EE

(1.55)

where the inner sum runs over all sequences (ii, we have

f

dPc (0)

P(0) e i(J4)

, ji,) with jk E {1,..., N},

= f dpc P(.10

ei(J

= P(To ) f dtic ei(j'°) where the notation means that every 0i1 gets replaced by this becomes

(1.56)

k. Using (1.54),

1.3 Field Theory on a Lattice

a =

f

11

f apc,(01) f dlic2(o2)ei(J ,0 1+0 2)

2) . dlic1( 01 )f dpc2( 02) P ( 01 + çi,2 )ei ( -1,0 1 +0

(1.57)

The exchange of integration and differentiation with respect to J is justified because all integrals are still absolutely convergent. For the same reason, the limit J 0 exists, and it implies that (1.53) holds for all polynomials F. More general F can be approximated by polynomials. This technique of differentiating with respect to the source terms J will be a Leitmotiv in most of the technical discussion about Feynman graph representations.

Definition 1.7. Let F be a finite set, Irl = N, and let F = {0 : F —> 11} (then F RN). For 0,0 E F let (ç5, 1/)j-' = Es Er (15(x)0(x), let C be a linear map on F such that (0,C0) > 0 for all 0 0 0, and let A be its inverse. The Gaussian measure on F is defined by

d pc (0) -= (det 27C)

(10(x),

(4) 'A`b)

(1.58)

xEr Remark 1.8. If v : 11,

, n1 —> I' is bijective, d(ç5) can be written as (15 (v(0)Am(i)v(i)0(v(i)) N

dpc (0) = (det 27(C,( i

Hdo,(i) .

)) e ij1

(1.59)

i=1

dpc (0) is well-defined because all factors occuring in its definition are basis independent: the right hand side of (1.59) does not depend on v.

1.3 Field Theory on a Lattice We want to define a regularized field theory corresponding to the Euclidean action iii 2 =

fEdd d X

fEd

GVO(X)2

ddX [0(X)

0(X) 2 4 (X ) 4 )

+ 7710 2 )0X) + A O(X) 4 1

(1 . 60 )

where, in going from the first to the second line, we have assumed that 0 vanishes at infinity so fast that there are no boundary terms - this is justified because we have to assume enough decay of 0 and its derivatives to make the integral for S converge.

12

1. Field Theory

1.3.1 Discretizat ion We use the lattice regularization only as a method to define the field theory. To analyze perturbation theory, we shall later take the continuum limit at a fixed ultraviolet cutoff in momentum space, to restore the Euclidean symmetries of the theory. The advantage of using the lattice is for us that we can use ordinary derivatives instead of functional derivatives (the latter can easily be defined in the continuum, but we shall not need that here and it loads the presentation with unnecessary discussions). However, the importance of the lattice theory goes well beyond the purpose for which it is used here. Up to now, the lattice regularization provides the only nonperturbative regularization of functional integrals that works independently of the coupling strength. Also, to this day it remains the only nonperturbative regularization that does not break gauge-invariance. We discretize space and time and introduce a finite volume. For convenience in using the Fourier transformation, we introduce the finite volume as a torus of sidelength L. We take a lattice of spacing E> 0, and define

F = re ,i, = Ezd 1 Lzd

(1.61)

where e > 0 is to be small, L» 1 and f', E N. When no confusion can arise, we simply denote FE ,L, by F. A particular choice of coordinates is to write F as

Ix

L

E EZ d : -2 < xk
oc, and the continuum limit E -) O. The theory where only E -> 0 is taken is a continuum field theory in a finite volume, the theory where only L —> oc is taken is a system of statistical mechanics. Since the number of lattice points goes to infinity in both of these limits by (1.63), both limits are systems with an infinite number of degrees of freedom, and hence interesting by themselves. The theory of critical phenomena, i.e. of the long-distance behaviour of the correlations of an infinite-volume system at fixed E is closely related to studying the continuum limit of field theory [14]. The continuum derivative of a field 0 : Rd --> R is discretized by

40(x) = ( 1 (0(x + eek) — 0(x))) E

(1.64) kE11,...41

where ek is the unit vector in k-direction, and the discretized action is

1.3 Field Theory on a Lattice 2 1 S r (0) = Ed (— (d e 0(1)) 2 ± -1( 27 4 . (5) 2 + A(x) 4 ) z, 2 2

13 (1.65)

where A> 0 and m> O. m is the mass and A the self-coupling. In all detail, SF

(0) = Ed

E ( .2x Er

If

'(0,(s

+ Ee k) — 0 (x)) 2 + *r P(s)2 + 4(44) .

k=1

0

0 r : F —3

F, OF

=.--

' is obtained by restricting (I) Ir, then

(1.66) E C ( /LZd ) to the lattice

--S (0) . ( 1.67) E–H) Many other choices for S have the same property, so the lattice action is not uniquely determined by the continuum action one wants to regularize. Whether different actions give the same continuum limit for the correlation functions of the statistical mechanical system given by S (defined in the next section) is a much more difficult question. We now streamline the notation to avoid too many E's and sums. We introduce the notation

151 FE ,z,

(4) Ts , L)

fF dx F(x) = E F(x), Ed

(1.68)

x Er

which is suggested by the convergence dx F(x) —+

E–>o

J

f

dx F(x)

(1.69)

RdiLzd

L

for any F E Cl(Rd/LZd, C), and define a scalar product on CF by

(q5, 7P) r = f dx 0(x)0(x).

(1.70)

F The backward derivative de*, defined by

(c/:(0) k (x) = –El (0(x –

Ee k

) — OW),

k E {1, ... ,c/}

(1.71)

is the adjoint of the forward derivative de :

(0, deOr = (c1:0,0)r -

(1.72)

Equation (1.72) is proven by summation by parts. There are no boundary terms because we chose the lattice as a torus. Introducing the lattice Laplacian A ,---- –4de (1.73) we can sum by parts in the sum for Sr, and get

14

1. Field Theory

Sr(40 ==

e d xer

2

dx =

0(x) (( —a + mo 2 )0)(x) + A0(x) 4 )

(1.74)

(0, ( — A ± rn0 2 )0)r + À f dx0(x) 4 .

The action of Laplacian is, more explicitly, d

1 20(X) — A0)(X) E =(

eek) 4 6(x

(1.75)

AO)r = (cle0, dE(15)r > 0,

(1.76)

k=1

Since for all

0, (0,

—

—d is a nonnegative operator, just like its continuum counterpart. Every linear operator on Cr is, when represented in a basis, simply a matrix in Mi r' (C). Choosing the orthonormal basis (Ex ) x er by Ex(Y) r (x , y), where (1.77) SF(X,Y) = E —d (5xy, we have 0(x) =

(Ex Or and

f

dy Ey (x)Ey (x`) = Sr(x, x 1 ),

SO

(A0)(x)

=

f

(1.78)

dy A(x , y)(P(x)

(1.79)

with the matrix representation (or "integral kernel")

A(x, y) = (Ex , AE y )

(1.80)

of A. In particular, the unit matrix has kernel 1(x, y) = (51 (x , y). The corresponding distributional kernel in the continuum is, of course, 5(x y). If A is invertible, the kernel of the inverse satisfies -,

—

dz A(x, z)(A -1 )(z ,y) =

Sr (x, y).

(1.81)

We also define a derivative

a

6 J(x)

-

a J(x)

(1.82)

so that J(x)

J(y) = 5(X, y)-

(1.83)

1.3 Field Theory on a Lattice

15

This also agrees with the formal continuum convention

6.1(x)

(1.84)

.1 (Y) = 5 (x — y).

With this, we have

6 e (J,(19». 6.1(x)

J(x) e(J'`15) r

(1.85)

-

The Laplacian has the kernel d

—

(x, y)

=

e —d-2 E fq kz, uzy

(5x+Eek,y

(1.86)

(5x Eek,Y) —

k=1 d

E -2

E

(26,(x,

y) — 6r (x + eek,y) — 6r(x — eek,Y))

k=1

(in the continuum, the distributional kernel would be (

-

1 6)(x

y)).

—

1.3.2 The Partition Function and Correlations With Dr0 = .., Ti xer d0(x), the functional integral with sources J : F is defined as (1.87) Zr,x(J) = f D rO e - sr(0)+GIMr. This integral converges if A> 0 and mo > 0, or if A> 0 and mo 2 = /./ E R is arbitrary. Expectation values of observables 0, that is, functions of the field 0, are defined by

( 0 ) = Zr,1),(0)

f Dr° e -sr(°) 0(0).

(1.88)

Only those 0 are admitted for which this integral converges; this includes, e.g., all polynomials in the fields, i.e., all moments of the measure with density Definition 1.9. The correlation functions, also called Green functions, are

(0( X 1) • • • (15 (X n)) =

1

Zrdk (0)

(1.89)

f DrO e—Sr(ç6) 110(X i) . k=1

These moments can be calculated as derivatives of Zr, x (J) with respect to J, via

(0(xi) • 0(xn))

[

6n 1 Zr x(J)] Zr,x(J) 6J(z i )...6.1(x n ) '

. J=0

(1.90)

1. Field Theory

16

Thus the partition function Zr, À (J) with source terms J contains all the information about the correlation functions. The derivatives 6n

[6.1(x 1 )...(5.1(x 7,)

log Zr x(J)1

'

(1.91) J=0

are the connected correlation functions. In Chap. 2, we shall prove that the connected correlation functions are sums of values of connected Feynman graphs. The significance of the Green functions is that the S-matrix can be constructed from them by the LSZ-formulas [9]. Note that in (1.88), we are integrating over all field configurations 0 : F R in the same way as we were integrating over all polygonal paths in the quantum-mechanical case. In the limit E --- 0, there is no reason to assume that these configurations arise from Cl fields 0 on i 'd/LZd. On the contrary, the typical field configurations are not Cl, just as the typical path contributing to the Wiener integral is not C'. A simple geometrical interpretation of the 0 field in d = 2 is that of the height of a surface, so Zr (J) describes a statistical mechanical ensemble of random surfaces. The probability density is given by the Boltzmann weight e - s/Z, so the action S effectively appears as an energy function. The form of S shows that large values of (40) 2 , of m0 2 02 , and of A04 are supressed. Note that, if e 0, the scale of smoothness, which is E, goes to zero, so there is no reason to expect the random surfaces to be Cl in the continuum limit. The particle physics interpretation of Z is that of a massive field theory with a repulsive self-interaction A > 0. For A = 0, the particles are free. Note that while the measure Dr0 has no limit as i F I oc, the Gaussian measure obtained by combining the quadratic part of the action with Dr0, and normalizing suitably, has a limit. This is a natural procedure from the physical point of view because the propagation of free particles is then given by the covariance of the Gaussian measure. The lattice Laplacian generates a random walk ensemble, which is yet another way of viewing the statistical mechanical system. Important nonperturbative results, like triviality of 04 in d> 5, are based on the random walk representation [10, 11]. In that language, m0 2 is the killing rate and A > 0 means that the random walks interact: they repel each other. For m0 2 > 0, —A + m0 2 > m0 2 > 0, so the Gaussian integral converges. Since A > 0, the 04 term only improves the convergence. The case m o = 0 (massless field) can also be defined (one has to take care of the zero modes of A, the "torons" (=constant configurations). This is no problem on but the limit L oc becomes more difficult in that case. The slow decay of (—A) -1 (x, y) in d = 2 causes the Mermin-Wagner theorem [15]. The limit E 0 and L oc does not exist for d = 2: there is no massless field in d = 2 [16]. We shall keep m o > 0 when analyzing perturbative field theory. --

1.3 Field Theory on a Lattice

17

1.3.3 The Ising Model

We show that the Ising model is a limit of the statistical mechanical system defined by our lattice field theory. The local term in the action is drawn in Fig. 1.1.

(a)

(b)

(c)

Fig. 1.1. The 04-potential for different values of mo 2 . (a) mo 2 > 0, (b) mo 2 =-- 0, (c) mo 2 < O.

Suppose that m0 2 = p < 0, A> 0, and let e = 1 for simplicity. Since 2

(1.92)

/±)2 _ 0 4A 7 16A

2

the local term in the action is of the form drawn in Fig. 1.1(c). We use

1 - - E(0(x + ek) - 0(x)) 2 2 x )k

. E 0 (x ) (t (x ± ek ) - d E 0(,)2 x,k

(1.93)

X

to rewrite the action in the exponent of the integrand for Zr (J) as

-Sr(4) =

E 0 (x) 0 (x + ek) -D( 122 + d)0(x) 2 - A0(x) 4 ) x

x,k

2 (kil+2d) 2 ( 4 (x) 2 ± ti + 2d) 16A 4A ±

I rl .

(1.94)

Zr(J) = Ki n f H ( c10(s ) e.1(x)0(x)--À(0(x) 2 -a2 ) 2 ) H eo(x)0(2,±ek)

(1.95)

(1)(x)0(s + ek) - A

= x,k

E x

If p + 2d < 0,

xEr

x,k

with

lp + 2di (1.96) K = exp( (11. + 2d)2), a = 16A 4A . The last factor in the integrand suppresses values of 14)(x)I far away from a. In the limit A oo, a kept fixed, the last factor becomes proportional to a sum of (5 functions because it is peaked at ±a, and because

18

1. Field Theory

f e _ot(02 _ a2,idO

f e-2a2

-ct)2

f e -2a2

2

(1.97)

2a2A. Therefore, the rescaled partition function

Zr (j) = Alf112 Z(J)

(1.98)

is a constant times

E

(J) =

eE

e

0(x)0(x+ek)

(1.99)

00E-q-ct,a}

Rescaling the fields, s(x) =

, and the sources, H(x) = as the partition function of the Ising model Zif,(H)

E

=

e--0E(8)

, we recognize (1.100)

sOE, { -1,+1 }

with the interaction

E(s) =

— E s(x)s(x

+ ek) —

H(X)S(X))

(1.101)

x ,k at inverse temperature

fl

IP(A) + 2d1 ' 4A )'-'°°

- a2- h im

(1.102)

The source term now appears as a spatially varying magnetic field H. The interaction is between nearest neighbours, and it is ferromagnetic, i.e. equal spins have bigger Boltzmann weight e0 s(x)s(x+ek) than opposite neighbouring spins. For this reason, the Laplacian is also called ferromagnetic. The quantities that can be put in relation to physical measurements are the correlation functions defined in Definition 1.9, i.e. the functions (F(s))

= 1

E

F(s)e- '3E(8) .

(1.103)

`j

For instance, (si) is the local magnetization and E y ((S x S y )— (S x )(S y )) is the magnetic susceptibility of the spin system. This is not the place to go into details about the Ising model and statistical mechanics, since there are many excellent references in the literature [17, 18, 19, 20, 21, 22]. We only mention very briefly some of the main results. The behaviour of the Ising model at H = 0 depends on 0: for small 3 (high temperature), the spins are disordered, and (s s) 0 exponentially fast as Ix-YI oo. For large 0,

1.4 Free Fields

(sz sy )

-Ix -y I -HD°

u(0) vanishes at a critical point

( (7 3 ) 2 0 0-

19

(1.104)

fie .

Note that for )3 > fie , the correlation functions depend on the boundary conditions. With our choice, we pick the symmetric phase where (si ) = 0. Had we chosen boundary conditions that favour spin alignment, such as the condition that all spins on the boundary are +1, we could have achieved (si ) = o- (0) so that

(s z sy ) - (s x )(sy )

--lx-Y1->00

0,

(1.105)

again, exponentially fast for 13 > fie . Only at )3 = 0,, one has a power law decay. It is at this point that one has to look for a continuum limit if one wants to construct a continuum theory without a momentum space cutoff directly 0 while keeping a finite from a lattice theory. As mentioned, we shall take E cutoff Ao in momentum space, and then study the limit Ao oc directly in the continuum (which is a different limit).

1.4 Free Fields For A = 0, the integral is Gaussian, and this corresponds to the case of free (noninteracting) particles. The free action is diagonal in Fourier space. We give only the main definitions of the Fourier transform here; details about the Fourier transform are in Appendix A.2. The dual lattice to F is F* = 2 /r zcii 2: zd . We define p - x = pi x i + .. . + ,.. 7 and 6(p,q) = Ld6pq . The Fourier Pdxd, fr. dPP(p) = IrdE pEr* P(P) transform of f : F

C is f : r*

C, given by

f(p) = f dx e -i" f (x). r The inverse relation is

f(x) = f - dp ei" r

i(P).

We denote the scalar product for functions i4, 7:b E Cr* by

(îko'k)r. = fdP (i5 (P)1 (P)

-

(1.108)

Fourier transformation is unitary: (0, Tp) r . = (0, T,b)r. The integral kernel of a linear operator transforms as

Â(p,q) = f dx f dy e -i"A(x, y)e i". r r

(1.109)

20

1. Field Theory

By(1.86) and (1.109), the Fourier transform of the kernel of the lattice Laplacian is

- .3(p, q) = 6r. (p, q) with

(1.110)

d

2

DE (p)

(p)

E(1

(1.111)

cos(epk )).

-

k=1

That is, the Laplacian is diagonal in the Fourier basis. Thus

(_ ± mo 2 ) 03, _ 6r. (A q) m In the limit E

(_ A rno 2)

0,

DE (p)

+1 DE (p) = 6r* (p, q) (p).

(1.112)

p2 , so

Lid(5_

1 (7 , q)

Pq

1 p2

rno 2

= r. (p, q) 0(p).

(1.113)

The quadratic form defined by the action is

=

dP (^b(P) (m0 2 + D (1)N(P)

fr *

=f

dp (7)(-p) (m 0 2 + De (pN(p)

(1.114)

Jr*

because q5(x) E

Vx implies

0(p) = ç( - p). Thus, the partition function is

Z r,À (J) = Z r » (0) f dpc r (0) e--À

0(x)4dx4-(J,(P)r

(1.115)

where the normalization constant of the Gaussian measure

Zr,0(0) = det(27rCr) 1/2

1

( (

k

-

Ed

L"

mo 2

± DE (p)

)1/2

(1.116)

pEr*

is unimportant because it drops out in the quotient defining the correlation functions. We can therefore redefine Zr,À(J) such that it does not contain that constant, and get

Zr,x(J) = f dpc, (0) e - w(0) + (J,O)r with V(0) = f 0(x) 4 dx The quadratic part of the action, which defines the covariance Cr of the Gaussian measure, describes free particles that propagate from x to y (graphically later denoted by a line connecting x and y). The interaction term describes the influence of particles on one another; for the 04 theory one may imagine it as an interaction between two particles, denoted by a vertex with four legs. The interaction is local. This is, in absence of other fields, the only way to maintain Lorentz invariance in the real-time theory.

1.5 Properties of the Free Covariance

21

1.5 Properties of the Free Covariance The properties of CF (X, y) in the limits L oo and E 0 are central to the question why renormalization is necessary. For e > 0 and L < oo,

fr. dp rno2eip(x-Y) ± imp) .

CF(X, y) =

(1.118)

For mo > 0, the denominator is always nonzero, so the summand is continuous in p on 1 1 (1 / E7rZd and one can take the limit L oo, in which the Riemann sum fr . dp becomes the integral T27 f8 dp where B = In other words, the integral is taken over [- 71e , 5-, ) d . Thus 1eiP(x - Y) Jim C(x,y) = (27)d Lddp , DE (p) rn o 2 -1-

The limit 6

( 1.119)

0, and is given by the integral

0 exists if x - y

1 C(x, y) = in_\d kh -11 1

ddp fRd

eiP( x - Y ) m 2 +p2 '

(1.120)

One can calculate C(x, y) in terms of a modified Bessel function d

C(X,y) = (27 ) —d / 2 (

774 ) 2 1 Kii _1(rnix — YI) -

IX — yl

(1.121)

In particular, for d = 3, 1

C(x, y) =

47rIx

e -inolx-Y1 .

(1.122)

- YI

Most properties of C can, however, be found without any explicit formulas. For instance, the exponential decay as Ix - yl oc follows directly from the analyticity of 1/(p2 -P 77102) for Ii < m. By translation and rotation invariance, C(x, y) is a function of ix - yl. Scaling out the factor mo gives ddp e ip(x- y) C(x, y) = f (2 70d rn0 2 ±p2 = rno d-20d(rno ix _ yo with

Cd(t) =

Lemma

1.10. (i) For (ii) For t > 1,

all t>

I

ddp eiPl t

(270d 1+p2

(1.123)

(1.124)

O:

Od(t) < const t- d;1 e-t

(1.125)

22

1. Field Theory

(iff) For t < 1,

> 33 if d> if d = 2

(d— 2)

Cd(t) = const { 'f log t The bounds become sharp as t

(1.126)

O. The constant depends only on d.

Proof. This is a standard proof [6]. For convenience of the reader, it is reproduced in Appendix A.3. •

Note that C(x, x) does not exist: there is a singularity in the propagator at short distances. This is the usual behaviour of a Green function (cf. the quantum mechanical propagator), but also the reason why renormalization is necessary. For large Ix — Y I, I C(x, 01 < const e - mo 1 x -Y 1 . If mo were zero, this would be replaced by a power law decay, and one would have an infrared problem as well. For E > 0, there is no singularity, but the propagator grows as 6 —. O. Remark 1.11. The lattice propagator obeys Cr e,L (X,

as e

X)

for d = 2 for d > 3

log E

r'd const

(1.127)

O. For d = 2, the constant depends on mo > O.

Proof Because

p2

2 Ez

(1 — cos(pe)) >

2 7r

we have

J

p2 _F ddm P 02

Hzi , if id

f m0

[__ t , lild

ddp

ddp 2 ± DE (p)

f [

m0 2 ±

p2 '

Z le , !ld 1E

Because [

:E 1 C

7 7r 6

1

d

{P : IPI < "\--Lr } E

6

(1.130)

we have

fP {7)415- Z76 }

dd p < 2 ± m0 2 —

Cr e,L (X,

X)


0, let us write this out as

Zr,),(J) --

1

:

Zr 0(0) --A fdx(4)(x) 4 -6Cr(x,x)0(x) 2 +3Cr(x,x) 2 )

By the lattice translation invariance, we have added the E-dependent term

6 ACF(0, 0)

f

Cr(x,x)

(1.145)

= Cr(0, 0) is a constant, so

dx 0(x) 2 — 3 A Cr(0, 0) 2

f

f

dx

(1.146)

r

to the action. The second of these terms is just a constant, which can be absorbed in a redefinition of Z. The first term is a correction of the mass term —6ACF (0, 0), 0 by (1.127). The interpretation of this term is the which diverges as E main point of field-theoretic renormalization: the extra term does not change the functional form of the action Sr (0), but only the value of the bare mass, that is, the mass parameter in the action, from m 1, 2 to mo 2 -6ACr (0,0). The bare mass is not observable because the particles are not free. The observed mass contains the interaction effects. There is, as we shall see, a one-to-one relationship between the bare and the observed mass, so one may fix the latter by experiment. This means that it is fixed as the value of a correlation function at a particular momentum. The simplest choice is

f dx (0(0)0(x)) = m102 .

(1.147)

Adjusting everything this way, one has removed the divergences in first order perturbation theory. The additional terms in Sr are local, so there is no problem with Lorentz invariance, once the correlation functions have been shown to converge. If one can absorb all divergent terms into a redefinition of finitely many parameters of the action (preferably, only few of them), such as mo2 and A, to any order in the expansion, then the theory is called perturbatively renormalizable. In such a theory, a finite number of parameters have to be fixed by experiment. Once this is done, one can predict the results of other experiments, by renorrnalized perturbation theory, in which no divergences appear, and compare them to measurements. Of course, the above discussion has not really proved anything of that kind. The divergences may reappear in higher orders, in particular because the terms added to make the first order finite may actually cause infinities in higher orders. It requires more insight into the structure of field theory to see that this is not so and to understand how renormalization really works (also, to say it again, Wick ordering is not sufficient for d> 2).

26

1. Field Theory

It will turn out that Wilson's form of the renormalization group, (see Chap. 3), provides a different point of view of this subtraction procedure, as a change of initial conditions for the sequence of effective actions. In particular, it can be formulated without any explicit reference to the bare parameters of the action, that is, to the very high energy behaviour, once the form of the action at that energy is determined. It relates only quantities at energy scales that are independent of, and much smaller than, the cutoff. In the next two chapters, we shall develop the formalism and prove some basic results about perturbative renormalization, that is, statements about the Green functions that hold in a formal power series expansion in A. The question about convergence or divergence of the expansion in A for the 0 4 theory is much harder and is discussed here only in a simplified case, namely the flow generated by "bubble" graphs, which leads to the so-called Landau pole in four dimensions (see the last section of Chap. 3).

2. Techniques

To get a deeper understanding, as well as the ability to do field theoretic calculations, we need to learn the standard techniques of Gaussian integration. Not all of this will be needed in Chap. 3, but these standard techniques are a prerequisite for reading most of the current literature in field theory. Most of this chapter is concerned with Feynman graph techniques. Emphasis is put on defining Feynman graphs in a simple but precise way and on giving an unambiguous meaning to the "sum over values of Feynman graphs". Although every statement about Feynman graphs can in principle be restated without a reference to a picture, their value for thinking about field-theoretic problems goes far beyond mnemotechnics. We shall see, for instance, that some combinatorial factors are very easily - and safely - calculated with the help of Feynman graphs. Needless to say, they are also the central tool of the detailed calculations that are the basis for the great success of perturbative quantum field theory in particle physics.

2.1 Integration by

Parts

Let A be a positive operator on RF and C its inverse. Recall that (50(x) 6 00(x) • Since fr dz C(x, z)A(z, y)=-- E -dSxy , and *OM = E -d 6xy , we have -

0(x) = f

dy Os, y)

6

1

(2.1)

SO

dy C(x, y) (5 ci("Or 5 0(Y) With this we get by integration by parts 0(x) e -

i (q5 'Aq5)r = -

Jrr

(2.2)

f Dr 0 F(0) 0(X) e - fr dy C(x, y) f Dr 0 F(0) 6 06

(Y)

1 6 dy C(x, y) f Dr q e- 1("49) F(0). 60(Y)

(2.3)

28

2. Techniques

Inserting the normalization, we get the integration by parts formula

f dpc, (0) 0(x)F (0) = f clY C(x, y) f dPc(0)

5F

(2.4)

r

2.2 Wick Ordering We have already seen that Wick ordering can cancel some divergences in the continuum limit. Since Wick ordering alone does not remove all divergences, this may not seem a big deal (and one can do renormalization without Wick ordering). However, Wick ordering is natural, because it simply corresponds to introducing orthogonal polynomials associated to the measure dpc given by the weight function e - i (0,A0 r , and it leads to substantial simplifications both in the Feynman graph representation and the renormalization group equation.

2.2.1 Definition and Main Properties Let

C = Cr be a nonnegative symmetric operator on Cr. For WF(- 1, A) (

J: F

= e i(JMr -#- 1(.1,C .1)r .

C, let (2.5)

Definition 2.1. Let Pr be the algebra of polynomials in (0(x)),, E r. Wick ordering is the C-linear map flc :Pr Pr that takes the following values on the monomials: (2.6) r2c(1) = 1 ,

and for n > 1 and z 1 ,...,x,, E F (not necessarily distinct) n 1

Q c ((z) • - - 0(xn)) = [H

6

k=1 1

r (J, 0)]

Ik)

(2.7)

• J=0

Remark 2.2. flc is well-defined because for all permutations r E Sn

rIc (401) - - - 0(x,)) =--- f 2c (0(x n (i )) • • • 0(x r(n))) •

(2.8)

The term of highest degree in the polynomial f2c(0(x i )...0(x n )) is 0(x 1 )

• • • (15 (xn)

-

Proof. Let 7r E S. Since commute,

6 =E d a and since the partial derivatives

r2 c (0(x It (l )) .. - «x,.)) =

1-112-r

6

1

11=11 i 6 J(xir(k))

14)r(j'd

45x0 vvr ( j, 0) 1

.i.o

[pi li 6.

i J=0 QC ((XXI) • - . 0(Xn)) -

(2.9)

2.2 Wick Ordering

29

Thus f2c (0(xt) . . . (x)) is well-defined on the set of all monomials, a subset of which is a basis of the linear space Pr. f2c therefore defines a unique Clinear map from Pr to Pr. • Usually, the Wick ordered monomial flc(0(x 1 ) ... 0(x,)) is denoted by : (1)(s 1 ) ... q(s) :. In the following, it will be important to keep track of what covariance C is used for Wick ordering, therefore we use the notation Lk, • Of course, a Wick ordered monomial is a polynomial, and not a monomial, in the fields.

Theorem 2.3. Define, as a formal power series in J, c°

flc (e i(J' (P) r)

in

(2.10)

n=0

Then

(2.11)

Wr(J,0) = fk (e i(j)(15)r ) • Let a: F C be defined by n

a(x) =

E aor(x,sk) = { 60 - dak

k=1

if x = xk otherwise.

(2.12)

Then n

1

f/c (q5(xl) - - - 0(sn)) = [ (11 i k=1

a

ei(a4)r+i(a,Ca)r]

oak)

(2.13) cy=0

Proof. By Taylor expansion in J, and by definition of flc,

Wr (J, 0) — 1 n 1

7 = n>1 E7

II J(xlc) xi E

k=1 n

,...,x n Er

[ n

II aJPx,) VVF(J, 0)

k=1

= V 1 H Axic) fdx,...dxn n! n>1 k=1

[ft k=1

6 j (6x k

J=0

) VVF(J, 0) J=0

n

=

n E ni1 11 J (xk ) f dx 1 . . dxn in flc (11 4(xk)) n>1

= L_., 7.7.

k=1

k=1

fk ((J, (15)r')

n>1

= f2c (e i(jM r) — 1.

(2.14)

The last equality holds because flc is continuous in the formal power series topology (see Appendix A.1). (2.13) is obvious from the definition if one sets J(s) = e- da(x). •

30

2. Techniques

We shall use the suggestive notation flc (ei( ") r) for Wr (J, (to) since the power series is convergent for all 0 and J. The next theorem contains an alternative formula for the Wick ordered monomials that will be useful in Chap. 3. Theorem 2.4. Let

1 (5 1 ( 6 6 (5 ) C(x,y)6 o , C Tt) = - f dx f dy •ic. = -2 T 2 60(x) 0(Y) r r

(2.15)

Then Qc (46(xi)...0(x n )) = C A' 0(xi) ...0(xn).

(2.16)

In particular, if C depends differentiably on a parameter t, then

0 — Qc (0(xl) . • .0(sn)) = Ot

aAc

at

(-lc (0(xi) . . . cto(x n )) .

(2.17)

Proof. Consider C as a formal parameter. Then for any power series f (z) = Ek fk Zk

PAO ei(J'°' = f Gi (iJ ,C iJ)r) eiGIMP ,

(2.18)

e —Ac ei( " )r = el () r +1(JM r 7

(2.19)

SO

from which (2.16) follows by definition of flc, because derivatives with respect to J commute with Ac. If C depends differentiably on a parameter t, (2.17) follows by taking a derivative of (2.16) because Ac commutes with V-. •

2.2.2 Further Results

Section 2.2.1 contains all the statements about Wick ordering that we shall need for doing calculations. For completeness, and to verify that Wick ordering indeed gives the results stated in Sect. 1.6, we derive here the recursion relation and prove the orthogonality relations. Theorem 2.5. The Wick ordered monomials satisfy the recursion relation

(n+1 QC

(P(Sk))

=

n flc (11 0(xk)) Oxn+i) k=1

k=1

n

n

_ E c(sk ,xn+i )

f/c

k=1

(

11

(xi)

(2.20)

ig

Moreover,

f dttc(0) Qc (e i(jMr) = 1,

(2.21)

31

2.2 Wick Ordering

for

all n > 1,

f dpc(0) f 2c(0(xl) ...0(xn)) = 0,

(2.22)

and if n 1 n2 , n2

f (Lac (0) f2c ill (

flc

15 (S k))

(

Proof.

k=1

110(M))

=0.

(2.23)

(

Since

(5

'1

i 6J( sn 4-1 )

Wr (J, 0) = 0(xn-1-1)

_if

dx C(x n±i , x)J(x) Wr(J, 0),

r

(2.24)

(2.20) follows if we can show that n

. E 6 r (x , x k ) flc

1 6 11 i 6 j(xo J(x) Wr(J, 0)] n

[

k=1

J=0

( n

)

II q5(xi) z=i tore

k=1

(2.25) By the product rule, every derivative can act on J(/) or on Wr(J, 0). If ,y on J(/), we get 6(x,/ k ), and all other derivatives must act on (5. t5 acts Wr . This gives the sum on the right side of (2.25). If no derivative acts on J(x), evaluating at J =--- 0 gives zero, so (2.25) holds. (2.21) is obvious from doing the Gaussian integral: by (1.47),

f

cl A c ( 0 )

tk ( e v, o) r )

--72-

r f dc(q5) e i(J,49 )r e i(J,C J)

(2.26)

= 1.

Equation (2.22) follows from (2.21) by expansion in ni f CitiC(0)QC0) (H 40ric k=1

= li n

F(a, 13) = =

=

1=1

di i c f e

(n2

H 1=1

) O(Y1)

li k fi ai3i F(a, 0)]

[ k=1

where

J. By (2.13),

(2.27) a=0=0

( 0) ei(a4- 0,0P e i(a,Ca)r+i(0,CP)r

(a+13,C(a+0)) ,-.4 - i(a,Ca) r+ i(0 ,C 0) r

(2.28)

32

2. Techniques

Expanding the exponential, we see that every term has the same degree in a of as in O. Therefore taking derivatives and evaluating at a = fi = 0 gives zero if n i n 2 . • The recursion relation gives

0(40c(0(x)) - C(x,x)flc(1) 0(x) 2 - C (x , x)

=

fk(0(x) 2 )

(2.29)

and

A7(011 4)

ox) ,Qc(o(s) 3 )-3C(x,$)flc(0(x) 2 ) = q5(x) (0(x)flc (95(x) 2 ) - 2C(x, x) f c (0(x)))

=

- 3C(x, x) ,Qc (0(x) 2 ) = 0(x)4 - 6C (x , x)0(x) 2 + 3C (x, x) 2 ,

(2.30)

which are the identities used in the Sect. 1.6. The integration by parts formula gives

f

dPC(0)

n+1

Ec(xk,in+,) f d,c(0) 11 0(xi)

k=1

k=1

11 (too

(2.31)

z=1

zok

which is similar to the definition of the Wick ordered monomial, only that the terms that get added when one does integration by parts get subtracted in the Wick monomial. In one variable, the Wick ordered monomials are scaled Hermite polynomials: Calling (*I) = 0, C(xi, xi)_nflc c ,oneoigets Q(l) = 1, rk (0) = 0, and =C f-2c(on) (on+i) = j-) c for n > 1, n-1 ). Calling Pn ( 0) = flc (On), this is the recursion relation P

1 (5) = 0-Pn (0) - n C Pn-1(0) •

Thus

C n/2 \

Pn(çb) =

iin(f2-0)

(2.32)

(2.33)

where Ho (x) = 1, I-11 (x) = 2x, H1

(x)

= 2x H n (x)

-

2n Hn _ i (x)

(2.34)

are the usual Hermite polynomials. The generating function of the Hn is _ 82 == e 2xs

co

n=0

lin(X)sn.

n!

(2.35)

2.3 Evaluation of Gaussian Integrals

33

2.3 Evaluation of Gaussian Integrals 2.3.1 Labelled Feynman Graphs

In this section we evaluate Gaussian integrals of the form

( A,

, vp; 1p) = f dc(Ø)

(2.36)

Vk ± H k=1

where Vk (0) = f

.f

clxink vk(x l ,... ,smk )0(x i )... 4)(x mk )

(2.37)

and

11 q=1

I (V1 • • • ,Vpill)) = f dPC (0)

C (Vq(4 ) 0))*

(2.38)

I

We shall do the function in detail; the changes in the representation for Î will then be obvious. The result is going to be a sum of values of "Feynman graphs", which will be defined in detail below. However, we briefly indicate now how these graphs are going to arise, to make it easier to follow the derivation below. We shall visualize every vk as a vertex shown in Fig. 2.1, with mk external, numbered, legs. These legs will be joined to an external field IP or to another vertex by a line, which has an associated factor C(x i , x i ).

(k,m c l)

(k,2 (k,1)

(k,mk)

w

Fig. 2.1. A fully labelled vertex

A precise definition will follow below at the end of the derivation, but we shall discuss the graphical significance of various objects already when we introduce them. We introduce the notation Xk =

k ,1 • • •

Sk,mk ))

f dX k =

r

dzk1, • • •

r

dsk , mk

(2.39)

and k)

(2.40)

V,,(0) = f dX kV (X k)(1) Xk •

(2.41)

k

= (I)(X

AX k

C ,i) • • • (

so that

34

2. Techniques

The vk are assumed to be symmetric under permutations vk (xi, .. • , Sink) = Vk(Sii(1), • • • ,Sir(m),))

VT- E Smk ,

(2.42)

because any nonsymmetric part of vk would cancel in (2.41). Also, we define the set of legs of vertices determined by V1 , ... Vp as

Bo(p; mi • • Onp) = {(k,ik)

k E { 1, . • •

iP}iik E { 1, ...,rnk}}

(2.43)

on B0 by

and a partial ordering

(k , i)

(1, i) .#> k < 1.

(2.44)

Note that B0 specifies the set of vertices completely: every vertex has a number k E {1, ... ,p}, and every one of its legs is numbered as well. We now drop the arguments of B0, keeping in mind that it depends on p and m l , . . . , my . For 00Bc B0 let Lo (B) = {(b, b') =

b,

E B , b 11}

(2.45)

{((k,i), (i,j)) : (k , i) E B , (1, j) E B and k < l}.

We shall see that LO (B ') is the set of all internal lines that can be formed by pairing legs of vertices. Obviously, /(V1, , Vp; 0) is linear in every Vk• Since (Xk)k = (zb)bEB03

i(v1

, ..•

vp;

=

f II

dXb

ft

v(Xk)

Axi, • • • ,xp;//))

(2.46)

OC((4) 0) Xj0 •

(2.47)

k=1

bEB0

with J(Xl,

Xp;IP) = f dPC(0) k=1

Since the Xi will be fixed in the following, we write J(0) = J(Xl, • • • X,,; /'). We now use (2.13) in the form Mk

{ Qc ((o, + o xo = -.--....

i

n

1

U

(2.48)

ii •i aa k,j 11c (eir) j=1

) with

a (k) = 0 k,1 6F(x,xk,1) +

+ ak,mk 6 r(s,xk,mk)•

(2.49)

Note that even if the s k , r are not distinct, the a's are; this is a typical example of the combinatorial method of "giving identities" to identical objects that appear several times. Doing this for all k E { 1, ... ,p}, we arrive at

la

=

i act f

bEBo

b

flc(e,)]

dp,c(0) k=1

(2.50) aO

VbEBo

2.3 Evaluation of Gaussian Integrals

35

,0)) = ei(a (k) ,0)+I(0,(0 ,ca (k) ) , we get

Since Qc(e

P

1 a )0

J(0)=[(ll

P

(k) ) 1, \-L, (a (k),, a

7 Oat,

i Eca(k),o+o i

f Cl/IC(0)e

k=1

k=1

(2.51)

J a=0

b E Bo

By (1.40), the Gaussian integral is

f

i t(01(k/ i40-FIP) Cliic(0)e k= 1

i t(

p

P

) — 1( Ea (k), C E a (t) )

= e k=1

e

k=1

1=1

(2.52)

.

By the symmetry of C, (ci), Ca(') ) = (a(l) ,Ca(k) ), so

P ca") iDa(k),o), -E(a(k) , 1 a . e k=1 J(0) = [(H — —) e k 1 and f and g be polynomials in e = (el) ... ) en) Then

[f (;)e g( e)]

c.-0

=

[g ( 2ae) f

(el

c.0

.

(2.56)

36

2. Techniques

Proof. The identity is linear in f and g, so it suffices to verify it for monomials. For monomials, it is obvious. • By Lemma 2.6,

J(0) =

[exp (

E ,P(xb) aaab ± E *cfcsb,zo act

bEB0

=[

II

H )

(1 ± Oaba C (X1) ) S b i ) Oaa b i )

(ti,b 1 )E Lo(Bo)

bEB0

ab

I

(2.57) cc---,0

H ( 1+ 0(xo 0+) H ad cx=0 • bEB0

bEB0

We now apply the identity

H ar,

Il (1 + at.) = E

(2.58)

NM rEN

rEM

which holds for any finite family (a r ) rE m of commuting objects, to (2.57), take the derivatives and evaluate at zero. Since every ab appears only once in H bEBoab, every derivative can act at most once. If we apply

II

a

,

(

bE.130

i+cso — aab

)

=E

H

, IP(Xb)

ECB0 bEE

a

(2.59)

— 0ab

first, this leaves FL EA, \E ab for the other derivatives to act on. Because we evaluate at a = 0, all of these factors must be used up by derivatives (that is, all non-external legs of the vertices must be used to make internal lines of the graph), and we obtain

Alp) = E 11 Csb) J

(2.60)

ECB0 bEE

with

E

J =

{

H

aaab c(sb,z049c,,a b,

= E n Ip(xo EcB0

bEE

H

ada.0

bEB0\E

LCLo(Bo\E) (b,b')EL

E

fl

c(sb,sbi)

(2.61)

LEL(B o \E) (b,w)EL

where L E L(B) if L = {(b,b l ) E Lo(B) : IrilL and 7r2IL are bijective}.

(2.62)

Here 71-1 and 71-2 are the projections on the first and second components of a pair (b,b1 ): ri (b,b1 ) = b and T-2 (b, b') = b' . In other words, if L E L(B), then every e E B0 appears exactly once as a first entry b or a second entry h' in a pair (b, b') E L. The procedure of differentiating with respect to the ab and setting the a's equal to zero has the following natural graphical interpretation.

2.3 Evaluation of Gaussian Integrals

37

—The labelled vertex drawn in Fig. 2.1 represents the monomial vk(zk,i, • • sk,,,,) ak,1 • • • ak,mk , that is, leg number (k, 1) corresponds to the factor ak,i. —The action of1P(xk i )-d— is to make leg (k, i) an external leg of the graph: the factor aki is replaced by 1P(xki ). —The differential operator e;--0, C(s ki , removes the factors aki and . Accordingly, the legs (k, i) and (1, j) are joined to form a line of the graph. —The restriction that k < 1 following from the definition of L0 (B) means that only legs of distinct vertices are joined to form lines. —Evaluation at a = 0 implies that all terms that still contain a factor ab vanish. Thus all legs that are not assigned external fields must be paired to form internal lines of the graph. Examples are given in the next subsection. The precise definition of a Feynman graph is as follows. Definition 2.7. Let B o = Bo (p;m 1 ,... ,mx,), as defined in (2.43), and for O B C B o , let Lo (B) be defined as in (2.45) and L(B) be defined as in (2.62). Let E c Bo . A (labelled) Feynman graph for the vertex set determined by B o , and with external legs in E, is an element L E ,C(B 0 \ E).

Remark 2.8. By this definition, we have identifed a Feynman graph with its set L of internal lines. This is unambiguous since the vertices have already been fixed by specifying B o . The Feynman graphs are called labelled because every vertex and every leg of each vertex is numbered, and hence distinguishable from every other one. We specify "internal" lines because we sometimes also say external lines for the external legs (people not used to graphs having 'external legs' can think of the endpoint of the external leg as a vertex with incidence number one; this is unambiguous because such vertices do not occur as internal ones in our models). Note, however, that the external legs do not count as lines of the Feynman graph under Definition 2.7. This is a natural convention, as we shall see later on. Using the permutation symmetry (2.42) of the vk we obtain the following representation for AO). Theorem 2.9. Let I(0) = 1(171, ,V,0) be given by (2.38). Then

.40) =

E f del • • • dem ml) • • • 4)(m) Um (C; ci, . • • , em) m=orm

with rh =

>m

and

(2.63)

38

2. Techniques

v.

1

Um(C; 61, • • ,6m)

fr

111,•••,71m

E

111)k (Xk) k=1

dsb ll Bo\E c(x,,sb,)

(2.64)

LEL(Bo\E) (b,bs)EL

where the sum over iii,. , i E B0 is restricted such that the ri 's are pairwise distinct, E = { n' , . ohn } , and where x n, = for all 1 E {1, . . , TO. The Urn depend also on p and , VP.

Remark 2.10. If we had not used Wick ordering, we would have had an extra term f1

(2.65)

exP

k=1 aa which generates self-contractions of the vertices.

2.3.2 Symmetry Factors and Topological Feynman Graphs As an example, consider the integral =

f citt0(0) Qc(0(x) 4)Qc(0(0 4 ).

(2.66)

The graphical rules tell us that we have to join the legs of the two vertices shown in Fig. 2.2 in all possible ways to get the graphs. For instance, one can have the Feynman graphs in Fig. 2.3, with = {((1, 1), (2, 2)), ((1, 2), (2, 1)), ((1, 3), (2, 4)), ((1, 4), (2, 3))1 L2 = {((1,1), (2, 2)), ((1,2), (2,4)), ((1, 3), (2,1)), ((1, 4), (2, 3))}.(2.67)

(1,2)

(1,3)

(2,2)

(2,3)

(1,1)

(1,4)

(2,1)

(2,4)

Fig. 2.2. Two labelled vertices The value of each such graph is C(x, y) 4 , so to get the result we now have to count the number of Feynman graphs that one can make with these two vertices. There are four ways to join leg number (1,1) to any leg of vertex

2.3 Evaluation of Gaussian Integrals

Fig. 2.3. Two labelled Feynman graphs, corresponding to L1 and

39

L2

2. After that, there are three ways of joining (1,2) to any remaining leg of vertex 2, etc., so there are 4! different labelled graphs contributing to J. Thus J(, y) = 4! C(s, y) 4 .

(2.68)

Generalizing this, we can also write the Feynman graph representation in terms of Feynman graphs (see Fig. 2.4) where the legs of the vertices are no longer labelled. Then one has to include the correct symmetry factors, like the 4! in (2.68). Unlabelled Feynman graphs are also called topological Feynman graphs. If a labelled Feynman graph G' has been obtained from another labelled Feynman graph G by a reordering of the legs of a vertex, then G and G' give rise to the same topological Feynman graph. For instance, the two graphs drawn in Fig. 2.3 give rise to the same topological graph, drawn in Fig. 2.4. For small graphs, it is trivial to calculate the symmetry factors and thus to pass from one to the other representation. For large graphs, doing this can get rather involved. For the following graphical considerations, in particular the proof that taking the logarithm of the the partition function removes all disconnected Feynman graphs, working with labelled graphs is advantageous because every labelled Feynman graph appears exactly once in the sum and one does not have to think about any symmetry factors. 2.3.3 Motivations for Taking the Logarithm

The integral (2.66) occurs in the second order term of the expansion for

40

2. Techniques

Fig. 2.4. The topological Feynman graph corresponding to the graphs given by L1 and L2.

2r . fdp c (4)) e A2 = 1 + T

=1+

fr

dx f 2c (q5(x) 4 )

dx fr dy f dpc(0)Qc (0(x) 4 ) (20(4)(0 4 ) + O (A 4 )

12 2

4! f dx f dy CF(x, y) 4 + OW). F

(2.69)

F

Since CF depends only on z — y, and since fi-, dx = Ld, 2r =1+ A2 4! L d f dy C F (0, S) 4 ± OW). 2 r

(2.70)

The factor Ld was to be expected: in a statistical mechanical system, 2r . exp(L d fF) usually holds ("the free energy is a bulk quantity"), meaning that limL_÷ 00 fr,,,= f, : the free energy density stays finite in the thermodynamic limit. A proOf that this is indeed the case requires decay assumptions on the interaction and is not trivial. We shall do this proof in the next sections. One reason why this is nontrivial is that from fourth order on, disconnected graphs appear in the expansion for Z r . This gives rise to terms of order (Ld) 2 and even higher powers of Ld in higher orders, which is of course consistent with what one gets when expanding eL dfr , but it is clear that one has to prove finiteness of lim 0() fr . This proof is not by expansion in L d, of course, and one also knows from cluster expansions that the corrections to the bulk behaviour are not O(L) but more complicated functions of L [25]. By taking the logarithm, we shall be able to prove finiteness of the limit of the free energy density. The other problem is the limit e --4 0, in which the question of the convergence of fr dx CF (O , x) 4 for c --* 0 arises. If we take the continuum limit e -+ 0 naively, replacing the Riemann sum by the integral fR, dx C(0, x) 4 , and recall that

IC (0, x)i < const e —mixl

for II > 1,

(2.71)

we see that the integral over Rd \ Ix : II < 1} converges because m o > 0. But since

2.4 Polymer Systems

0 < C(0, x)

—

I ,I (d-2)

const { ' I log(x) I

if d> 3 if d = 2

41

(2.72)

as x -4 0, the integral

1 fix 1 3. Only for d = 2, 1

ddx C(0,x) 4 — f rdri log rI 4 3. To remove these divergences, we shall have to analyze the singularities in more detail. For this purpose, it is inconvenient to deal with connected and disconnected graphs at the same time. The most efficient way to proceed is to use Wilson's effective action, defined by

G erff(0) = log

f dp er,(0)e - w( 4)+0)

(2/5)

for V(0) bounded below. We shall prove in the next sections that Gerff (0) is the generating functional for the connected, amputated Green functions, and that Gerff (0) = Ldgerff (0) where gerff (0) has a finite limit as L -4 oc. The renormalization group is then an analysis where CF is split into many pieces, according to their distance from the singularity. The RG flow determines how singularities in Z and in the Green functions generated by geff build up in the limit E -4 O.

2.4 Polymer Systems In this section, we define the notion of an abstract polymer system and its partition function, and show that the logarithm of the partition function gets contributions from connected graphs only. This statement is of great use in statistical mechanics. It can be used to show that the free energy density indeed has a thermodynamic limit if the interaction decays sufficiently fast, and it is also useful to show exponential decay of correlations. The presentation in this section closely follows that in Erhard Seller's book [20].

42

2. Techniques

2.4.1 Preparation: Graphs and Partitions Let M c N be a finite set and

G(M) = P({(i,j) EMxM:i< j})

(2.76)

where P(A) denotes the set of all subsets of A. In other words, every G E Q(M) is a set of ordered pairs (i, j) EMXM with i < j. Every G E g(M) defines a graph with vertex set M by the definition that the line between k E M and 1 E M is a line of the graph if (k, 1) E G. We identify the graphical object with its set of lines G E G(M) by the following Definition 2.11. A graph G is an element G E

g(m). A subset H of G is

called a subgraph of G. For example, let M = {1,2, 4,5, 6, 7, 8} and G = 1(1,2), (1, 4), (2,5), (6, 8)}. The graph G is drawn in Fig. 2.5.

Fig. 2.5. An example of a graph In graph theory, a graph on M is usually defined as a set of unordered pairs {i, j} where i < j. This definition is obviously equivalent to the one given above. The notion of a graph is not the same as that of a Feynman graph because in a graph, two vertices can be joined by at most one line. The graph theoretical notation is, unfortunately, not completely standardized. In the nomenclature of [24], Feynman graphs would be called multigraphs and Feynman graphs without Wick ordering pseudo-multigraphs. In the graphical representation, the concept that a graph can be disconnected (as in the example drawn in Fig. 2.5) or connected is obvious. If G = 0, G is totally disconnected, i.e. it has I MI connected components, each consisting of one vertex i E M. Intuitively, a graph is connected if any two vertices i E M and j E M can be joined by a walk over lines of G. More formally, one defines for M = {m i , ... , mk} with m 1 < ... < mk that G E g(M) is connected if there is a permutation 7F E S(M) such that {(mi,m2), (m2, m3), • • • ) (mk-1, mk)} C {(r(i),r(j)) : (i,j) E G} .

(2.77)

An example for such a permutation is drawn in Fig. 2.6. We denote the set of connected graphs on M by G(M). If G E g(m) is disconnected, G induces a partition p(G) of M by grouping together the vertices in the connected components of G. More precisely:

2.4 Polymer Systems

(_. 1245

43

1111-...-111--411—AP

1425

4125

Fig. 2.6. Permuting a connected graph Definition 2.12. P is a partition of M 0 if P =

k E N and for all 1 E {1,...,k}:ONz C M, and Ni denote the set of partitions of M by P(M).

Nk}) with Nk = M. We

If M is finite, 'F'(M) is finite as well, because all N1 0. Lemma 2.13. p: g(m) -+P(M),G fore

E

p(G) is a surjective mapping, there-

E

F(G) =

E

F(G).

(2.78)

PEP(M) G:p(G)=P

GEg(m)

Proof. p is a well-defined map because the set IN1 , , NO, and not the sequence (N1 , , NO, appears in the definition of the partition (otherwise the correspondence wouldn't be unique because for example both {NI , N2 , . , Nk } and {N2 , Nk} are generated by the same G). Surjectivity is obvious. Remark 2.14. Let M be finite and G E g(M). If p(G) = On, 1N1, • • • then

G=

(2.79)

Gk k=1

with Gk E çc (Nk) for all k, and GIN, = Gk = {(i l i)EG:iENk and j E Nk}.

(2.80)

Moreover,

E PEP(m)

1M1

F(P) =

E m=1

J.

F(m,{N i ,.. ., Nm }).

771.

(2.81)

cm N 1 U...

6 N nt =.114

The is there because we now sum over all sequences (N1 , of all sets {N1 , , Nm }. Lemma 2.15. Let A c N be finite and nonempty. Then

,N) instead

44

2. Techniques

11 (1 + = E H ijE A

aii =

GEG(A) (i,j)EG

E 11 aii ) k=1 GEgc(NkHiMEG

Nk otvk

(2.82)

Ni U... U Pl tn =A

Proof. The

first equation is a consequence of the multinomial theorem 11 (1 + ar ) =

E 11 as

rER

SCR rES

with R = {(i, j) EAxA:i< j}. By Lemma 2.13 and Remark 2.14

EHaij = GEG(A) (i,j)EG iAi =

-11 i

E

E

Ni U... U N„,=A

G:p(G)= (m,{Ni,.••,Nm})

E rrt! IAI ,J.

E in m=1

PEP(A)

aii

(i,j)EG ni

m=1

=

H

E

>2

II

k=1 (iMEGINk

m

ri (

E

,

Ni U... U N„,=A

k=1

E GkEGc(Nk

11 )

(2.83)

ai i )

(i,j)EGh

• 2.4.2 The Logarithm of the Polymer Partition Function

ro

Definition 2.16. Let ro be a finite set, and be a symmetric relation on that is antireflesive, i.e. V-y E /70, -y Then we call the elements ey E ro . Let A be a commutative Cpolymers, and -y and 7' compatible if 7 algebra and a : a(7) a map. a(7) is called the activity of 7. * A , A set r c ro is called compatible if V-y, E r : -y The partition function of the polymer system is the finite sum

ro

—

Zro (a) =

E rcro

r compatible

a( -y) ^rEF

=l+ E rcro rot

H a(7).

(2.84)

-rEr

r compatible

The name originates from chemistry. Polymers are long chain molecules with a hard-core property: no two polymers can occupy the same region of space. This disjointness is an example for a compatibility relation, and it is obvious that any polymer must be incompatible to itself. Polymer systems arise in a natural way in expansions of the partition functions of statistical mechanics. For instance, the high-temperature expansion of

2.4 Polymer Systems

45

the Ising model can be brought in the form of a polymer expansion, with the polymers being connected clusters of bonds between nearest-neighbour sites on the lattice. Similarly, the low-temperature expansion of the Ising model can be rewritten that way, but with Bloch walls as the polymers [17, 18, 20]. One can make the compatibility constraint in (2.84) more explicit by introducing [20] geY , 7

')

ry 76 7'

01

={

(2.85)

otherwise

and u(71,...,ryn)= H (i±g(eYWYj)).

(2.86)

1. (k, 0 , and 6 i) = 0 otherwise. E I) is related to the incidence matrix of the graph. This is just a formal way of saying that eixk , sPi occurs if 1 joins leg number j of vertex k to a leg of vertex number k', with k' > k, etc. We can now do the x-integrals and get -

(Gih) )(q1 , • • • , qm) =

f

dPi

Oh(i)(PI)

tEL

with Pie = (A k),

y ( 1)

—

(2.171)

k=--1

as above,

the p

(k)

rif,k(pk)

,(b) + z_.,-(0) Pb

tEL

(2.172)

bE E

(note that by definition, only one term in this sum is nonzero). Less formally, but as precisely, one may say that the momentum pi that "flows" in line 1 = (b, b') is the momentum exiting from leg b (hence -FA k) if b = (k, 0) of vertex k, and it is the momentum entering leg b' of vertex lc' (hence —A k) exits from that leg of vertex k'). Thus the signs are due to the convention that every momentum is counted as exiting the vertex.

Theorem 2.26. Let Ch(x,y) be defined as in (2.169), with Fourier transform Ohm E C 8 (Rd I 2tr Zd ,C) and assume that for all vertices k E = Ld (

(k) 0 ic(A k) • • - , p$), ,

with fik E C8 ((Rd PilZ ( )mk -1 ,C). Then, for all E C Bo (with 1E1 = m) and all L E Gc(B0 E) : If m = 0 (E = 0), then

1-7(G) = Lde ) (L, C1, C2, V1, • - vi,) and limL,,, c, gih) g( h) exists. If m > 0

(2.174)

(E 0), then

fT(G.,h) )(qi, • • • , qm.) = L d8q1+...+q„,,0 -;'2: h) (q2, • • • qm)

(2.175)

with E C8 ((Rd /27rZ) m-1 ,C), and 7.4,:h) and its derivatives up to order s converge to functions 14,1,;?• E C 8 ((Rd /27rZ) m-1 ,C) as L oo. 1. If Oh(i) and the Ok are analytic in a strip around the real -(L,h) axis, rynir is also analytic in every D in the same strip.

Remark 2.27.

2.6 Graphical Representations: Conclusions

61

2. For the covariance given by 0(p) = (m0 2 +De (p)) -1 , all functions depend on E and they diverge as E O. They also depend on mo, and it is essential in the following proof that mo 2 > O. Proof. We do an induction in the number of vertices r of the graph. Recall that

1-7 (c4,h) )(qi, • • • qm)

f

HdpiOh (i) (pi)

11

t EL

k=1

70 ;i3k (Pk),

L d4529(1 k)

(2.176)

where, as before, Pk = (Ak) • • • Wdk) • If r = 1, all legs of the vertex vi are external legs of the graph, and f/'- (4) )(qi , . , q 1 ) = f)(qi , , qm ,), so the claim holds by the hypotheses of the theorem. Let r > 2, and assume the statement to be proven for all r' < r. There is a vertex k of GLh) such that the subgraph G', obtained by deleting the vertex k and all lines that end at k, is still connected. Without loss of generality, we may assume that k = 1. Let L1 be the set of lines 1 E L joining to this vertex. The graph is connected, so L 1 0 O. See also Fig. 2.7.

Fig. 2.7. The decomposition of G in terms of the vertex vi and the subgraph Then

17 ( 40)(qi, • . •

,qm) = f II dp,oh(,),b(pof (Gq.,\ LELi

1 )(-P1 )

(2.177)

where Pi, = — (Pi)a if (a, b) E L1. Denote 13' = ' ,). The number of , pm external legs m' of G' must of course be at least one since otherwise G' could not be joined to vertex 1. The inductive hypothesis applies to G', so

=

6p/I +...+0 ".Ytn i1

(A)

6p ( 1 ) +...„(1 )

(P21) • • •

(2.178)

• • •

By our hypotheses,

ii(p(1 1) ,

•,

)=

where i3 E C 8 , and 5,1 converges to a

C8 function

)

as L oo.

(2.179)

62

2. Techniques

Suppose first, that (as drawn Fig. 2.7) some legs of vi and some external legs of G' are external legs of G. Then (p i , . . ,pm ' ,) contains those qm . that enter G', and ) contains those qm that enter vertex 1, and therefore the product of the two 6's is L'1.5

P1 1

(1)

r dx

L..

Pm'

0=

rdx

rdx P1

(1) n •

(2.180)

I P7711 1`;

If no legs of vertex 1 are external legs of the graph G, then (pti , . . , pm ` f) = (1) so (2.180) still holds. Since pi(1) m mi +p = IELi E pj + Q where Q = Em qm the sum running only over those external moments for which qm is associated to an external leg of vertex 1 (Q = 0 if vertex 1 joins only to internal lines). ,

1-7* (Gi(Lh) )(D,

(2.181)

qm )

= fiELi

H dpi

Oh) (pi) L d 6

E

,r-1( 131 ).

IELi

If m = 0, (Sql+ ...+0 is replaced by 1 and Q by 0 in this formula. Remove the summation fr. = Ird E pi I Er. of one line /1 E L1 by the delta function, then (2.182) (G )(qi , , qm ) = L d (Sql +.„-EOW(q2) ...34m) 'i,h) where

W(q2,

.,q) =

r !EL I

dp,-61(,)(pi))0h(,)(—

Q) '51Tre,r—i (P i )

1 0ii

(2.183) with qi = qk. This is the approximation, by a Riemann sum, of an integral over the bounded region [-71- /E, r/E]d(i Lli -1 ) . The integrand is e Zd bounded because Ohm E Cs (Rd/ zd , c\) for all t and because Rd P2-r is compact, and the same holds for ;y" by the inductive hypothesis. By the dominated convergence theorem, the limit L oo is finite and (by repeating this argument after taking up to s derivatives) C8 in qi, qm- • At this point, we have reached the stage that we know the preliminaries for reading at least part of the current literature on field theory without too much trouble, for instance [26, 27, 28, 44, 45, 46, 47, 58, 59]. In the next chapter, we introduce a renormalization group differential equation that allows us to analyze the renormalization problem in a very efficient way.

3. The Renormalization Group

We now turn to the problem of renormalizing our field theory to all orders in perturbation theory. To this end, we shall define a flow of effective actions, by defining an effective action that depends on a scale parameter A. The differential equation obtained by applying 8/8A to the functional integral for the effective action is the renormalization group (RG) equation. The graphical expansion derived in Chap. 2 provides an explicit solution of the RG equation. However, the differential equation itself is very useful for deriving bounds for the solutions without using the details of the graphical representation. We shall learn more about the connection between the two approaches as we develop the theory. The main ideas for the renormalizability proofs using the RG differential equation are due to Polchinski [29]. His proof, which uses a non-Wick—ordered form of the equation, was further simplified in [31] and then extended to gauge theories by Keller and Kopper [32]; a brief but very readable review of that work appeared in [34]. Here, we shall use the Wick—ordered form of the differential equation which was introduced soon after Polchinski's work by Wieczerkowski [30]. Using this form of the equation makes the inductive proofs still simpler. The renormalization group is in truth not a group, but a semigroup, namely the semigroup of effective actions introduced in Sect. 2.5. It is only a semigroup because Gaussian integrals are defined only for nonnegative covariances. Thus integrating over fields with covariance C > 0, which is a convolution with the Gaussian measure dpc, cannot be undone by another convolution with the measure associated to —C because the latter does not exist. If a 04 term is present, one may argue that the integral may still be convergent for C with negative eigenvalues, but then one has to prove that the sign of the 04 coupling does not change when integration is done, which is not obvious. When considering formal power series expansions of the effective action, the argument that the 04 -term controls the growth of the quadratic part does not apply anyway).

64

3. The Renormalization Group

3.1 A Cutoff in Momentum Space To motivate the choice of flow parameter, recall that the cause of the divergences was the singularity of C(x, y) = lirnE ,0 Cre. ,, (x, y) at x = y. This singularity is due to the slow decay of the propagator in momentum space: Oe (P)

=

1

—4' E --+o

mo 2 + D 6 (A

1 p2 ± mo 2

= 6 l (19)'

(3.1)

In particular, 0 V Ll (Rd) and thus C(x, x) is divergent. In principle, we could now study how the singularity builds up as E O. This is, however, messy, because the theory on a lattice has no rotation symmetry. Instead, we shall take a continuum limit at a fixed momentum space cutoff Ao > 1, which is possible on the level of perturbation theory as a formal power series, and then renormalize directly in the continuum. As mentioned, the main motivation for introducing a lattice in the present setting was to give clean definitions of Gaussian integrals and objects like without going through easy but lengthy (and in the end, irrelevant) functional analytical details of defining functional derivatives in the continuum. Let K E C"(RZIF , [0,11) obey

4

(3.2)

K (x) = {9

and K ' (x) < O for x E (1, 4). For convenience, we may choose K such that

ii/Cii00

5_ 1.

Let

ei1L(x-11) K( = f dp 0 m 2 + D e (p) r.

(3.3) p2 — -

).

(3.4)

The function K (p 2 I Ag) cuts off large values of p because, by definition of K, it vanishes whenever II > 2A0 . In the thermodynamic limit we get

Co 1; 0' ') (x , y) =

2 ddp eiP(x–Y) P (27r)d mo 2 + D,(p) K ( A8 )

(3.5)

which converges for all z, y, in particular for x = y by the dominated convergence theorem because the integrand converges pointwise and is bounded by the function --LANA, which is integrable. Thus ino -

II)

e rL Co,A 0 = Ern lim Co ,Ào

6-+0 L--+oo

exists and is a bounded function. For x ,---- y,

(3.6)

3.2 The Semigroup Structure of Renormalization

=

Co,no (x, x)

f

1 (27r )d 00

=

65

K -2 -) ddp p 2 ± ° m 0 2 ( A

d-1

r

2

Ad f r2 ±rno2 K — A2 )dr

(3.7)

o with Ad =

. Thus co

(1

Co,A 0 (x,x)

=

r r2 ±

Ad

, dr +

r d-1

72-)dr) f r 2 ± mo 2 K( All 1

0 2Ao

< Ad(v(rn 0) + f rd-3 dr) 1 log 2A0 < Adv(m q5) + Ad { 1 ( 7 &Id-2 d-2

\''''11)

d=2 d > 3.

(3.8)

This bound gives the correct behaviour for large Ao . The function v(m) is uniformly bounded in m 1 for all d > 3, but not for d = 2 (however, mo > is fixed). The continuum propagator CO 3A0 (x, y) is rotation invariant because a rotation R(x — y) can be undone by a change in the integration variable from p to R -1 p since p2 and the measure ddp are rotation invariant. Before taking the continuum limit E 0 of the effective action at fixed Ao, we derive the RG equation.

3.2 The Semigroup Structure of Renormalization Let 0 < A1 oo, and as e —> O. The coefficients GA!A,..° have two complementary interpretations:

(i) GAm'A,..° is the order r (in A) connected, and amputated m-point function with infrared cutoff A and ultraviolett cutoff 2A0. This follows from the graphical representation derived in Chap. 2. The ultraviolet cutoff is 2A0 because values 1/31 > 2A0 are forbidden in all loop integrals. Similarly, the propagators vanish when II 0, CAI , A0 > 0 because A < A'. So, by the covariance splitting formula,

g(4), C

— AV (A°) ) =

Ç(,

C A,A 1

C Al ,A 0 , — AV (A°) ))

(3.15)

3.2 The Semigroup Structure of Renormalization

67

holds (for details, see (2.105)). This identity is the semigroup property. Together with the graphical representation, it implies that the coefficient functions G° also appear as effective interaction vertices for a field theory with Ultraviolet cutoff A' (and an infrared cutoff A < A'). Note that the semigroup property also shows that it is natural that amputated Green functions appear as effective vertices: in the graphical representation of the G; in terms of the G0 the propagator CAA , pairs of legs of the vertices defined by the G 1° are joined to lines associated to propagators CA,Ai Thus it is natural that no propagators CAI , A 0 are associated to these legs of vertices. Note also that the Wick ordering with respect to CA A ', as implied by the representation given in Theorem 2.23, is also natural in this context: Wick ordering is always done with respect to the covariance of the fields that have not yet been integrated over. For A = Ao, CA O ,A 0 = 0, and the effective action indeed consists only of the vertices originally present in V. For A = 0, all fields have been integrated over, thus one has a theory without infrared cutoff, only with the ultraviolet cutoff Ao. ,

.

This has the following significance for the problem of perturbative renormalization, which we may now formulate. We say that the theory is perturbatively renormalizable if we can adjust a finite number of parameters in V(A0) such that for all m and r, the continuum Green functions GM° have finite limits as Ao —> oo. This is physically sensible because the can be used to construct the S-matrix and hence to predict the outcome of experiments. If only a finite number of parameters has to be fixed, the theory remains predictive. Moreover, the details of the interaction at very high energies A o —> oo cannot be observed, so changing V( A0) is allowed. Remark 3.1. The importance of the semigroup property is that it allows us to separate low and high energy scales in a clear way: Suppose we know that "° has a finite limit as Ao oo. Then the same holds for G°5 GA 171 5 T m,Ar° because g(0)CO,A01 - AV (11°) ) = g(01CO,A'

AV (A°) )).

(3.16)

In other words, once we know that the effective vertices G0 for the fields with energy scales below A 1 are finite, the same holds for GM.° simply because they are given by finite-dimensional Feynman integrals over IpI < 2A1 and with bounded integrands. In fact, once the limit Ao —> 00 has been taken in G."°, any reference to Ao is removed, so the limiting effective action G m il l,.A° = GAml-;:l° contains all the information about the theory at energies below A 1 . In more physical jargon, the effective action represents all the data that are accessible to us at low energies. We show finiteness of GAm1 ;(1 0 for fixed A 1 > 0 by use of the RG differential equation.

3. The Renormalization Group

68

3.3 The Renormalization Group Equation 3.3.1 The Functional Form The RG equation is obtained by taking a derivative of g(Ø, CAA) , — ) V (A° )) with respect to A. To do this efficiently, we introduce sources J, (J(x))xEr and write eg(0,cAA0,-w) =

f dpcAA0(x) e

-XV (x+95) (3.17)

= [e —AV (4'67) f d liCAA 0 (X)

J=0

{e -Wqr) e G1,0) e (.1,cAitc, .7)

This identity holds in a formal power series expansion in A because V is a polynomial, so in every order in A, the differential operator is just a polynomial in k 6 J Differentiating with respect to A, we get

a eg (

= e _ w ( h ) e (J,0) e i(J,CA A 0 .0 _1 (j acAA 0 J)]

OA -

2

an

. J=0 (3.18)

Rewriting

1

°CAA

-2-(j'

an

1. 6 acAA 0 6 ) e (.1,0) , ° J) e(j4) = 2( 50' an 60

(3.19)

and taking out the 0-derivatives, we get

a

1 = o y, e g(0,c„A

an -

6 acAA 0 (5 aA 60)eg(0,cAA0,v).

2 (4 '

(3.20)

This is a heat equation for eg:

a gA (o) _ a2LA,A0 gA (o) — an e an e with

=

1, 6

6

(3.21)

(3.22)

the Laplacian associated to CA,A0 in field space. In the formal continuum limit, 2iA,A 0 becomes a functional Laplacian. For E > 0 and L < oo, the number of field variables is finite, and therefore 2iA,A0 is an ordinary Laplacian in finitely many (but very many) variables (0(x)) sEre ,,, . Doing the derivatives, we get 1 6g acAA0 6 g .92iA (3.23) an = an' A ° g 2 ( JO' an so ) •

a0

3.3 The Renormalization Group Equation

69

Remark 3.2. It is obvious that the form of this equation does not depend on the nature of the parameter with respect to which we differentiate. That is, as long as only the covariance, but not the the initial interaction, depends on a parameter t, application of -g-i will lead to an equation of the same form as (3.23). Remark

3.3.

e g(o,cA,A o ,— A V)

= e ,AaA,A0

(3.24)

Proof. The right hand side of (3.24) also satisfies the heat equation (3.21) because 2icA , 40 and its derivative with respect to A commute. Since 2ic40 ,40 = 0, both functions coincide at A = Ao. By the uniqueness of the solution to the heat equation, they must be equal for all A. • We know already that the solutions to this differential equation are given by (3.13), but the differential equation itself will be very useful for proving bounds for the GA' A0. MO"

3.3.2 The Component Form We now derive the component form of the RG equation, that is, the corresponding equation for the Gf,?1,.,A °, simply by inserting the expansion (3.13) into (3.23). We begin by motivating how the flow equation will look like in terms of the functions af,,°. If we denote the associated polynomial graphically by the vertex shown in Fig. 3.1, where every leg carries a field variable 0(x), then the two terms on the right hand side of (3.23) can be visualized as in Fig. 3.2, where the slash on the internal line indicates that it carries propagator an CA,Ao•

Fig. 3.1. The vertex associated to Gnm'rn ° At this point, there are two different ways to proceed: one can expand in the ordinary monomial basis (RA) ... Ckpm ), as done in Polchinski's original work [29] and in [31], or one can expand in Wick ordered monomials, as done in [30]. We follow the second route and use (3.13). This removes the linear

70

3. The Renormalization Group

Fig. 3.2. The right hand side of the non-Wick-ordered RG equation

containing the self-contraction, and thereby simplifies the inductive proofs (as we shall see). The details of the calculation are similar to those of the derivation of the Feynman graph expansion, only a little simpler. We show why the term in (3.23) that is linear in g is removed, and defer the details of rearrangement of flc îk(p i ))(2c (n •- (qi )) occuring in the quadratic term to the appendix, where we also give the formulas in position space. will produce two terms, one It is evident from (3.13) that a derivative appears, and one where the Wick ordered monomial gets differwhere entiated. This second term cancels the first term in (3.23): by Theorem 2.4, A and because aAco,A = —aAcA,A0 implies A -A—o,A = term, i.e. the one

w

((pi) ...(pm))

— aAA0,A 9CO 3 A(s(PI) • • • j) (Pm))

(3.25)

aAAA,A0 0co,A(41) • •

When multiplied with (Sr. (p i + .. . + pm, 0)GAm' ,Ar° (P2 , . • , pm) and integrated over Pi,.. ,pm , this gives exactly the same as the first term on the right hand side of (3.23). Thus, the flow equation (3.23) now reads rit(r)

00

m

E E f H dki (Sr. (ki + .. . + km, 0) nco,A (;( 1e1), • • • , Ar

(k m ))

r=1 m=0 j=i

• aA GL'i^.A0(k 2 ,..., km)

g ( (56(/) ,

,( r) ög, ,A0

If we rewrite the right hand side in the same form as the left hand side, , «k m )) to get the component we can compare coefficients of fleo A ( form of the RG equation. When comparing coefficients, we have to remember that (lc° A (j)(k 1 ), . , «k m )) is invariant under permutations of {1, , m}, so only the symmetric part of the coefficients is nonzero. This means that we have to symmetrize the coefficients. For a function F of (k2, , km ), let

3.3 The Renormalization Group Equation 1

(SmF)(k2

,•••,

km)

=

71 (3.27)

E Fuc,r(2),•••,kr(m))

In ' rEsm m

E ki wherever it appears

where k1 = —

as kir ( i) for some 1 E {2, ... ,m}.

j=2

In Appendix A.4, it is shown that, as a formal power series in A, (.., (5g ,

aAc (r,A) (5g. , )r. =

00

- A

50

° 60

(3.28) 7.1=12

where Q(rr,A,A0)

f

f 11 -rr

La

kw)

dki 5 r. (E ki, 0) Qri'llriA° (IS) f2 Co ,A i=1

i=1

m•=0

m

m

in

fn(r) ..i.‘ _ V-•

(11 (Ici))

i=1

(3.29)

with Is = (k2, . . . , km ) E (Rd) m 1

(3.30)

and mt, (p, le) G,°7.,, (—p, Qm rArA ° (ls) = f dK nir f dwA(p) aACAA0 (po) GA

ID

.

(3.31)

In this expression, the integral f dkrar is written for a weighted sum

f

dkmr (m i , m", r', r", 1) F(m l , m", r', r", 1)

E

,

T., +T."-, r i > l,r fi >1

(3.32)

km , ,,,,, i F(m.' ,7fl",r',r",1)

E (m',m",1)EMrirdini

where Ni 7, r i , In

=

{(ne, m", /) : 1> 0, m` E {1, . . . ,

m" E {1, . . . , fil (r" )} , m` + m" = m + 2 1 + 2}

(3.33)

and where the weight kmi mui is the combinatorial factor kmi mili = m' m" 1!

— 1) (m' — 1) (m" ) . / ) /

(3.34)

Moreover, we use the notations p .

(pi, • • • ,P1)

E (R1 1

Ill

k

(—Ek 3 ,k2,... ,km , --1-1) 8=2 k" = (kne _i,... , km ) .

(3.35)

72

3. The Renormalizat ion Group

and po is given by m'-- 1--1 PO

—

pi —

I

E

=-Epi +

j=1

j=1

m

m'---1-1

j=2

E

kg,.

(3.36)

j=1

Finally, ddpi „ I ) dwA (p = .1* y (27)d U0A

(3.37)

p, , k" , and po depend, of course, on (m, r, m', m", r', r", 1). We do not denote this dependence explicitly because we are soon going to take sup norms where the dependence of the G, on p, , k" , and po drops out. We have omitted the caret and the superscript F on C since we shall only use momentum space in the following, and because we shall take the continuum limit soon. With all this, the component form of the RG equation is (9A G

A0 (k2,

km)

=

sm Q 7/;:rA,A0 ( k2,

km ) ,

(3.38)

with cglryt,A0 given by (3.31) and S given by (3.27). Contrary to what one might believe at a first glance, the details of this equation and of its derivation are quite simple to understand, and it has a simple and convenient structure, which we explain in some detail in the next section. Also, we relate it to the Feynman graph representations derived in Chap. 2, which, as mentioned, provide explicit solutions to this equation.

3.4 The Structure of the RG equation 3.4.1 The Graphical Representation Visualizing Gr niAr A° as in Fig. 3.1, every term in the sum over r', r", m', m", and 1 has the graphical representation shown in Fig. 3.3. In other words, it is the value of a graph with only two vertices, with m' and in" legs, such that the number of external legs is m. The graph has 1 > 0 loops. The sum over rre and m" starts at 1 because otherwise ( I '(p) in (3.23) would have had nothing to act on. Thus Go,, does not appear on the right hand side of the flow equation. The graph is connected for all 1 > 0 because there is always the line carrying 8ACA,A0(p0) connecting the two vertices. If 1 = 0 it is a tree graph. If 1 = 1 it is a one-loop graph, usually called a bubble graph. Retaining only this term in the sum over 1 and integrating the RG equation gives the so-called ladder approximation, done in Sect. 3.9. A brief description of the procedure used in Appendix A.4 to derive the right hand side of the RG equation is as follows. The functional Laplacian generates the tree graph shown in Fig. 3.2, where both vertices correspond to

3.4 The Structure of the RG equation

73

Fig. 3.3. The right hand side of the Wick ordered RG equation coefficients of Wick-ordered monomials. There are 7n1 legs of the first vertex on which the first 0—derivative can act, and m" legs of the second vertex on which the second derivative can act. By the permutational symmetry of the Gmr , we may permute and thus get a factor m'm" (this argument involving the permutational symmetry will be applied also in the following to derive the combinatorial factor by counting the number of graphs). We now have to reorder the product of two Wick ordered monomials; this gives a sum over Wick monomials, indexed by /. In every term, all possible self-contractions of the new vertex associated to the entire graph are subtracted, hence appear as soft lines. Since both vertices already correspond to Wick-ordered monomials, all soft lines must join vertex 1 to vertex 2. If / > 0 is the number of Wick lines, there are r ' ï 1 \) possibilities to choose a subset of / legs of vertex 1, and (mu -1 t ) possibilities to choose a subset of / legs of vertex 2, to form pairs. Given these legs, there are /! ways of pairing them together. This explains the combinatorial factor. The remaining m legs are the external legs of the graph. Counting legs and internal lines gives the constraint in' + 7n" = 771 + 2 1 + 2. In Fig. 3.4, we show how the flow equation looks for small values of m and even interactions.

A_0_ =_0÷0_

d_ dA

-

+

Fig. 3.4. The RG equation for m = 2 and m = 4

+

74

3. The Renormahzation Group

3.4.2 The Relation to the Feynman Graph Expansion The RG equation is a differential equation for the kernels of the effective action, for which we derived a representation in terms of values of connected Feynman graphs in Chap. 2. Thus the sum over values of connected Feynman graphs is a solution to the RG equation. It is a consequence of the Feynman graph representation that fh(r)
1, and all those with m" = 4 and r" > 1.

3.4.3 The Continuum Limit at Fixed Ao Thus, the functional form gets translated into a system of differential equations, with the following structure. The system is labelled by (m, r) where, as shown in the representation by Feynman graphs, m A so that p2+17no 2 < Note that (3.73) was crucial for this argument. The contribution to the m = 4 term is

f

q)dp.

COA(P)aAcAA 0 (p

(3.77)

At q = 0, this is, by supp K' C [1, 4),

f

1

COA(p)aAcAA0(p)dp1

f p2 M

2P21 p2 / 7/ 1 2 A3 I KI I. 1

1 p 2 1_ -r

Mo 2 dP

2A

2

1

f p dp (p2 A3 1/000 1

mo2) 2P

(3.78)

A 2A

22 A3 IIK'lloo f P = A3

iIK'

log2.

A

There is no log A term because the lower limit of the p-integral is A. In any case, a single logarithm does not create factorials. Similarly, as for m = 6, the r = 3,m = 8 tree graph requires two propagators, CA A. and is therefore 0(A -4 ). This leads us to the ansatz for the Green functions. Theorem 3.11. For Vt(A n r0) = -8m441 (Y 4 -theory) in two dimensions, there are rymr > 0 such that, for all A> 3 and for all q

A2 A, Ti ( GAmAr0 )
4, Ao

A

The triple (m', m", 1) contributes to OAG AmAr° only if m + 2 1 + 2 = m' + m". Inserting this, we have Ao

X

(m+21-1-2) (log s)1 < 7m' r' 7m" r" f ds 55— A Ao

=

/M I T.'

7m" r"

log s \I ds s3-m( s 2f ) •

(3.82)

A

Since A > 3 was assumed, we have for m > 6 Ao

X
4, Ao

X < f as,

(log s)t eymy7m,,,,,s-(m"-4) s3

(3.85)

82

3. The Renormalization Group

so the power of s that appears is -3 + 4 - (m + 2 1 ) = 1 - m - 2 1 , and

X


4, 2/ = m' + m" - 4 > 4, so / > 4 In case (i), we insert the inductive hypothesis into (3.80), to get Ao

X < f dS 3-3 727.'72r"

Ç

1

72r" A

2.

(3.87)

A

In case (ii), we may use (3.81) and replace m' + m" = 2 1 + 4, to get A0

Ao

X < f ds(log s)1.95- (21+4) = A Ao

f ds (log s) 2

ds(log

8)28 _3 1 10g S \1-2 s2

A

S —3
1 and r > 1. Proof. For d = 2, we use (3.59) and (3.56) to get Ao 1717AnAvr0n,,

f ds s -3 (log

A

A23 ,17 (G

(3.99)

)A28 , 71 (0A 0 G m s t„ )

and A2 A

,n (Q m A orA 0 ‘

)


2, and the statement be proven for all r' m, Ao Ao

f dA /AA° (0) ? Q 1 log T

(3.105)

Ai

and

Ao Ao

f dA II/AA01100 Q2 log Ai . A'

(3.106)

86

3. The Renormalization Group

Proof. Estimating OACAA„ by its sup norm (as usual), we get /AA() (k) < I ICOAII? IlaAcAA0 II.

K1 = KA (K2A) 2 2,7

-Al

for all k, so the upper bound in (3.104), and (3.106), hold with Q2 = The lower bound in (3.104) is proven in Appendix A.5.

(3.107)

K1 K3. •

Of course, this graph will appear as a subgraph in many others, and so its divergence will cause all Green functions to diverge - and we have not even looked at larger graphs yet. In a graphical analysis, one can postpone this problem by considering only graphs without two-legged insertions. We can do the same in the flow equation by truncating the sums on the right hand side so that mi = 2 and m" = 2 do not occur. It follows from the graphical representation of the flow equation that this removes two-legged subgraphs from the sum over graphs that solves this truncated flow. We shall derive bounds for the so obtained functions dAmAr° and turn back to the full problem later. One can apply other truncations, which provide a lot of insight, to the flow equation. One can, e.g., also remove the four-legged insertions from the right hand side, or remove all terms except for 1 = 0 from the sum over 1 this truncation of the flow produces tree graphs only. Keeping 1 = 1 only does not lead to one-loop graphs, but to sums over all graphs where every line is part of one loop only, the non-overlapping graphs. In these truncated systems, the flow of the coupling constants can often be calculated (see Sect. 3.9). -

3.7.1 Power Counting for the Truncated Equation We now denote the modification where the sums over m' and m" both start at 4 by f dk mr . By (3.54), 'rub < K2A for d = 3, so by (3.60),

(aiiA G AA.0 ) < _1 fA2,r‘ diz_ m

2K1

-Al r A3

AA° AA. A2A, n (G rn t r i )A2A m (G m li r li ).

(3.108)

Motivated by the results in d = 2, we try the ansatz A2A

07 (am AAro )
6, then 3 - m - / 2

(3.129)

where ( 8$R) ) r >1 is a given sequence which is independent of Ao. (the superscript R only indicates that this is a renormalization condition; R is not a variable). Then (3.128) implies finiteness of G2Ar' A0 (k) for all k. This boundary condition determines the mass counterterms at Ao uniquely in the following sense. Lemma 3.17. There exists a unique sequence (8/4-110) )r>2 such that the effective action GAmlrA0 at scale A 1 obtained from the initial interaction

{ —1 for r = 1 and m = 4 ( vo) 81. `1°) for m = 2 and r > 2 mr = for all other m and r 0

(3.130)

satisfies (3.129). Remark 3.18. In fact, a renormalization condition like (3.129) is necessary to fix the counterterms since the requirement that a term growing in Ao is cancelled leaves the freedom of adding finite constants.

A lrA° Proof. Suppose we have a theory with vertices given by (3.130). Then Gm is a sum over values of Feynman graphs with four legged and two-legged vertices, such that the orders of the vertices add up to r. This fixes in particular G2Ari A0 (0). For instance

3.7 AiAo (0 )

Three Dimensions

= 0,

G2A.A o (0)

91 (3.131)

f

Ao (pi ) CA, Ao (p2) CA, Ao ( —pl — p2)( 3 pid3 P2 + (5P(2A°) •

It is obvious that this equation can be solved for 8/.4110) if G A° (0) is given. We now proceed by induction in r. In the induction step, we may assume that the 6/4A,°) have already been fixed for all r' < r, in terms of the G 2A,r1,A0 (0) with r' < r. In order r, we have G2ArI A0 (0) = 8/ ( `10) + a sum over values of graphs with two legs and at least two vertices. In this sum, only 8/4j,i0) with • r' 1, a multiindex a is an m • dtuple a = (ai,... ,ûm ) E ( 4) m ,

ai =

• • • lai,d)

We denote

(3.132)

m d

I ûIIûlI+.+IûmI = and

ûj

z =1

m

a! = !

am ! =

(3.133)

j =1

d

HIT au! .

(3.134)

i.1 j.1 We write a' + a" = a if a'ii + aiiii = au for all 1, j. Note that a' + a" = a =

la'l +

= lal.

(3.135)

Leibniz's rule, Da (fi • • • fp)

a! =

a(1)!

.û(P)!

Da(l)fi ...D'(P)fp,

(3.136)

is proven by induction. AAr° are C0° in the external momenta We shall show inductively that all Gm if the same holds for the initial interaction vISIAr o . For this we note that CAA° E C"(R d ,R) and

3. The Renormalization Group

92

DctaACAAo(P)

-

(3.137)

Da (; Kt( A P22 ) p2 +P2m0 2 2

E

mo 2

P2

a!

A3 ce±crii =a aqa"!

Da' K i (

)Da (1

A2

p2 mo2 ).

By the support properties of K, D ' K'() = 0 for all a' unless IPI E [A, 2/1], Moreover 1 if a" = 0 (3.138) K (a" ) 1Da" ( 1 p27Y14)2 2 )1 < M if la"I 1 A 2+1a" {

SO

IDaaAcAA0(p)1‹

A K31+(a121 „

(3.139)

< Ipl < 2 A)

(A

with a combinatorial constant K1 (which depends on mo). We also use the following lemma to reduce the number of cases to go through as much as possible. Lemma 3.20. Let m,m 1 ,m" E N, / E No such that m" + m" = m + 21+2. Define

D(m) = min{4 — m, 0 } E(m) = max{ 4 — m, 0 } .

and

(3.140)

Let 6 E R. Then

D(m 1 )+D(m")+16-3 = D(m)-1—(2-6)1+E(m)—E(m 1 )—E(m"). (3.141) In particular, if m > 4,

D(mi ) + D(m") +16 D(m) + E(m) = 4

Proof.

—

—

3 < D(m)

—

1

—

(2

—

m, so

D(mi) + E(mi ) + D(m") + E(m") — 16 — 3 = 4 m' + 4 = 5 16 — (m' + m") = 5 16 — (m + 21+ 2) = 1 + 4 m (2 8)1 = D(m) — 1 — (2 6)1 + E(m) —

—

—

—

—

—

(3.142)

6)1.

—

m"

—

16 — 3

(3.143)

—

—

• Theorem 3.21. Let d = 3, 8pR,2 E R and 04 with a second-order mass counterterm be given by

V2°) = - 67-1 6m4 where 8,14i1° ) is fixed by the condition

G .1.21A0 (0) _

6r2 8

e

3.7 Three Dimensions Then all

alrinAr° with m odd G 71,1° (pi , p2,

93

are identically zero,

(3.146)

for all pi, A, m,

= —8m 4

the GAmAr0 for m even, r > 2 are Cc° functions of the external momenta (Pi, • • . ,pm—i) and for any multiindes a there are constants grnri al, depending all 77 > A1 , all m > 2, and all r> 1, on 614 , such that for all A E [AI,

A2A, n (D'

ilAro ) Gm

< I

9220 lOg A

g,-11 AD( m)— lai

m=r= 2 a= 0 otherwise

(3.147)

Remark 3.22. 1. More precisely, we prove that

m = r = 2, a = m = r = 2, lal > 1 m = 2, r> 2, a = m = 2,r > 2, lal > 1 or m = 4, r > 2 m > 6.

I log A

A-1 log A 01(G A rnAr° J
6 is important for these bounds, to ensure that A2A, n (GAmAr0 ) AD(m) — I'l for m > 6. It can be relaxed slightly to a condition 14,Ar° ) A0D(m) . This is useful for the so-called "improvement", i.e. speeding up the convergence of the GAmAr° to their limits Gr as Ao cx) by adjusting the boundary condition [30]. AAr. = Da Qm AAro By Leibniz's rule Proof. By the flow equation, DaaAGm

E

f dKmr f dwA (12)

a l +all +cell

Dcv"

a! aga"!am!

D ai GA ntlior°,

(p, k i )

(3.1 49)

k") Dam aAcAA0(p0)

so, by (3.139), A2 A,n( Da Q A mAr °)

f d,,mrilcoAg

E

a l +ce, ±am =a

A2A,7 1 (.13''

AA,roi , Gm

)

a! Ki (am ) aqa"!a"' ! A 3+1c," I

A2A, n (Dcv" GA mjilt°rif ).

(3.150)

This bound holds in any dimension. Specializing to d = 3, we have liCoA Iii < K2 A, so

94

3. The Renormalization Group

A2A,T) (Da

Qm AAro ) \

a!

E


1, we integrate the flow equation from A to Ao and use (3.152), to get A0

Ki(a)K 23-7 constfds 8 -1 ' 1-7

A2A,, i (D'GAnti° )

A

Ki (a)K23-7 const A — i al — 't1 +1

(3.154)

which satisfies (3.148). For r = 2, a = 0, we integrate from A1 to Ao, to get A

IG1212A0 (A i

16141 + f ds Ic4P(p) I A1 A

f 7ds

< 164 I + IC1 1q const

(3.155)

A,

A

< 16141+ KIM log — . AI Since A > 3, log A > 1, so we can bound this sum by g220 log A, with the same constant g220. Let r > 3, and assume (3.148) for all r' < r. We integrate the flow equation from A to Ao and use IdAr°) = 0 for all m (since r > 3), and the inductive hypothesis (3.148), to get A2A m (13'

G AAro) m < f clitmr

E

cx 1 -1-ce"-Fai" =a

a! aqa"!am!

K1 (a") .Kg m , eke I gm" r's I a" I XAAo

(3.156)

3.7 Three Dimensions

95

(a)

454'1

Fig. 3.6. Scaling bounds: (a) unrenormalized; (b) renormalized. The scaling behaviour is unchanged because it depends on the propagator. The boundary condition is changed under renormalization.

with XA Ao

= XAA0(m%m"7 1/ 7) )r",ial)ia"kia t"I) A0 . f ds 8 -3+1-1&" I Ornirs la' I (9 )

(3.157)

A

where log s s—a

onira(s)

s'a (log s) 6-0 3-1—a

=

84—m—a {

m = r = 2, a = 0 M = r = 2, laj > 1 m = 2, r > 2 m = 4, r > 2 M > 6.

( 3. 15 8)

We can write this more compactly as

Ornra(s)

= S p(m)—a—rnir (log s )5m25.0

(3.159)

where rmr = 6m2(1 — 67.2) — 6m4(1 —

41 ) .

(3.160)

It is clear from (3.156) that if we can show that XAA0 has the A-behaviour of (3.148), then the Theorem is proven because 9mr 1,1 can be determined

96

3. The Renormalization Group

recursively. We verify this now by going through all cases of (m', m", r', r", 1) that can occur in f ditm.,.. Recall in particular that m' + m" = m + 2 1 + 2, and that r' + r" = r > 3, so one of r' and r" has to be larger than 1.

1. Let 2 fl {rre , m"}. By (3.159), the integrand for

XAA 0

is

7 = -3 + / - Ian + D(m 1 ) WI +D(m") — 3+1+ D(me) + D(rn") — lal — ./"' = and where r = rmy + F,,,,, ,,, > O. 1.1 m > 4: By (3.142) (with 6 = 1)), -

7 < D(m) — 1 — 1 — lal — r SO

-

.9 7 with lal -

r (3.161)

= 3 — m — lal — 1 — F

(3.162)

Ao X AA 0

< f ds 33- m- lal -l-r

(3.163)

A

For m > 6, we bound this using 1 > 0 and F > 0 by

A0

X A lt o < f d8 3 3- "H a l < A

1 A4-m- l a 1 4 l — a + l M

(3.164)

so (3.148) holds. Let m = 4. Then 2 1 = mid-mu-m-2 > 4+4-4-2 = 2, so 1 > 1, and

A0

XAA 0 < f d88 3-4-1- lal

(3.165)

- 1 + lal

A

also implies (3.148). 1.2 m = 2. By (3.141) (with E(m) = 2, D(m) = 0, and 6 = 1), the integrand is .9 7 with

7=1

-

1

-

lal - E(me) - E(m")

-

F.

(3.166)

In the present case, we have assumed m' > 4 and m" > 4, so E(mt) = E(m") = 0, and (3.167) 7 1 or where m" = 4 and r" > 1. This removes two- or four-legged proper (and nontrivial, i.e. consisting of more than one four-legged vertex) subgraphs from the graphical —AA0 expansion. Denoting the solution to the truncated equation by G mr 1 we make the ansatz that for all m > 6, il2A,17 (D a

(3.193)

GA mAr° ) < gmr A4- m- I'l.

This bound is obvious for r = 1 and m > 6, since GAmAi° = 0 for those m. For m = 4 and r = 1, it holds because of the initial condition V4(111°) = —1. Because of the truncation, other terms do not occur on the right hand side of the RG equation. We now proceed inductively in r. Since m' + m" = m + 2 1 + 2, A0 f

ds 8- 3+21- lam I A2A,ri

(Daiont sAior, , ) A2Am (Da"

Ontst„)

A

Ao

< gm/r' gm"r"

f ds 3-

3+2/+4-m 1 +4-m n -loci

A

Ao

= grai r, gm u ri,

f ds 3 5' 24' 1

(3.194)

A

A4- m - I'l m±la -4

grniri gm"r"

5 log A0 m = 4, a = 0 or A024' 1 m = 2, lal < 1. m+

la I

?

m = 2, lal = 2

We have thus proven the following lemma. Lemma 3.24. In unrenormalized, truncated (skeleton) irk 4 , i.e. with

V4Ar° ) (pi , p2,p3) = —6 ri 45m4

3.8 Four Dimensions

103

and with the truncation m' > 6 and m" > 6 in f dKrnr, there are constants grnr i cd such that

A2A,n(D

(:,,n1 AAr

m + lal > 5 log A0 m lal = 4 m = 2, lal < 1.

o

(3.195)

Remark 3.25. It is no contradiction to calculate the behaviour of alr and omit all these terms on the right side of the RG equation. This is clear operationally, since this simply corresponds to a particular modification of the RG equation. In terms of the expansion in Feynman graphs, air is the order r contribution to the four-point function from all those graphs that do not contain two— or four—legged subgraphs. Thus, leaving out these terms on the right hand side only removes the four-legged subgraphs that contain more than one vertex, and thus its feedback for the other Green functions; this truncation does not make the four-point function zero.

Remark 3.26. In contrast to d = 3, there is no extra decay 8— / in the integral A2 produces a 21 in the exponent instead of an in (3.194), because IAA Iii 1. In other words, the power counting becomes independent of 1. For d > 5, (d A d-2 1 so there remains a factor 8 -4) 1 in the integral. The theory I I COA is then perturbatively non-renormalizable.

3.8.3 The Boundary Conditions for Renormalization Recall that the original action of the model was 1

(V) 2sf(

—if)

02 + A04 )d4 s

(3.196)

2

and that we had put the quadratic terms into the Gaussian measure and (Ao ) the 04 term as the initial interaction Vmr = 6r 1 &nil (corresponding to e —xf 04 dx,. ) To incorporate the counterterms, we write an action Sb

= f

m(b 2 +74) 2 02 _ +45)004)d4x

(3.197)

where Z —1, Smo 2 and SA are the formal power series in A, starting at order A2 , i.e. , r > 2. In other words, we put GA m0rA0

= vr(i,Ar0) =

if m > 6 or m is odd

G A° (1)11P27P3) = V4(rA ° n P27 P3) =

qi

? A°

(p)

= V2(71.10)

= b '° p2

This means that as formal power series,

co

arA0

if r = 1 if r > 2 r > 2.

(3.198)

104

3. The Renormalization Group

smo 2

s

00 E brA° Ar

2

Z— 1 2

E

r=2

co

_Ao r r A

(3.199)

r=2 co

6A

E

CrA° Ar .

r=2

Although Z — 1 and 6m4, 2 multiply quadratic terms in 10, these terms are not put into the measure, but left as interaction terms. The formal series start at r = 2 because of Wick ordering. More explicitly, we are now expanding

f

diLC (0) e-AV(A°1(°41P)

(3.200)

in A, where V (A ° ) is the formal power series V(A0)

=

1 )(v0)2 + 57710202) + (A + 6)44)) nco , A0 (f clEy(Z — 1

(3.201) To fix the counterterm, we impose renormalization conditions at A = Al. They are

WI? A ° (0,0,0) =

—1

G141:A0 (0, 0, 0) = Cr qi! A ° (0) = ar

a2 apt ap

GA1A0)(0) =

24,143

Vr > 2 Vr > 2

(3.202)

Vr > 2.

The following lemma states that the boundary conditions (3.202) fix the counterterms uniquely to all orders in A, provided the form of the action is fixed by (3.198). Lemma 3.27. Let (arR b be a sequence in1;13 . There is exactly one (ar A o ,brAo C 0 r) > 2 such that the effective action Ç A ' A ° at scale A1 sequence obtained from the initial condition (3 .198) has expansion coefficients GA,T,17.A° obeying (3.202). ,

Proof. The proof is a trivial generalization of that of Lemma 3.17.

•

Remark 3.28. By Wick ordering,

Gi41? )10 (Pi7P2IP3) = —1 -. G 4111)11° (P17P2,P3) = —1.

(3.203)

3.8 Four Dimensions

105

The renormalization conditions can be thought of as being fixed by experiment. For instance, setting crR = 0 Vr > 2, i.e. , imposing 00

GpA0(0,0,0) = EArG4A,1^0(0,0,0) =

—A

(3.204)

r=1

means fixing the renormalized coupling constant A by the four-point function at (0, 0, 0) (which can be related to scattering amplitudes). The bare coupling constant 00 Ab = A + SA = A +

E crAo A

(3.205)

r=2

will be seen to diverge for Ao oo. As we shall discuss later, one nice point about the RG equation is that it allows us to write all quantities in terms of renormalized parameters, so that the bare parameters never appear explicitly. 3.8.4 Renormalized 0 4 Theory

We define renormalized 0 4 theory as the sequence of effective actions Gm AAro in d = 4, satisfying

GorA l 0 = vr(,,Ar 0 ) = o

for

m>6

Or

m

odd,

and satisfying the renormalization condition (3.202). By Lemma 3.27, the initial interaction V,S,Ar°) is fixed completely by this to be given by (3.198), and in particular, V,;,,Ar° ) is C' in all momentum variables, invariant under Euclidean rotations, and even in p. Consequently, the GA„,Ar° have the same properties for all A E [0, Ao]• Theorem 3.29. In renormalized 04 theory, for any multiindex a, all i> 1, all m > 1 and r > 1, there are constants gmficd > 0 and Ent,. E No such that for all A> A1 > 3,

A2A,n (D "

Gm AAro ) 0, since we want to have convergent integrals, i.e. the 04 term has the "right" sign). If we had imposed the condition (3.236) at a different scale A2 > A1, we would have got the same Green functions, but expressed in terms of a different renormalized coupling. Note that in (3.235),

3.9 The RG Flow in the Ladder Approximation

111

Ao is removed: the RG equation relates only the Green functions at scales A, without any reference to the cutoff scale. The semigroup property of the flow implies that 00

A2 = Al +

E /37-A1 r=2

(3.237)

where the Or are given as sums over Feynman graphs with propagators CA1)123 as discussed. In other words, if we define a scale-dependent coupling v (A) by defining v(A i ) by (3.236) and setting

u(A) = —G141 (A; 0,0,0),

(3.238)

and approximate the G`41 in (3.235) by v(A) (as suggested by the power counting behaviour of derivatives) then the relation between all these different choices of A is given by

aAv(A) = 132 (A) v (A) 2 .

(3.239)

where the coefficient 02(A) = M f

ddpC0A(p)

aAcoA(p)

(3.240)

is the lowest order term in the beta function of the RG. Since

02 (A) =

Mf

p2 p2 2p2 1 ddp (p2 + mo2)2 ( A3 ) KI (A2 )K ( A2))

(3.241)

we have (again for A> m) oA d-5 < 02 (A) 0 by the following lemma. Lemma 4.1. For all 0 > 0 and all y E R, ZA is nonzero, so ( ) is well-

defined. Moreover,

( o K 1 loll

(4.25)

where 11011 denotes the operator norm of 0, and for all linear operators B on Jr, (4.26) (B* B) > 0. Proof. Tr has an ONB of eigenvectors 77,„ of H — AN. The eigenvalues cc, of H — ILN are real because H and N are selfadjoint and p is real. Thus Z11 = E (i ia I e— (3(H - 01) lia) =

E e—o'fa 1 oc, I

zA(B*B)

= E e—'36" I (qc,

(4.27)

a

a

z111(0)1

E e-03E- > 0,

07/0)1

E e—acc. 11011 = zit 11011,

(4.28)

E e—'3f c" IIB%11 2 ? 0.

(4.29)

and

I B * B77)1 =

a

Corollary 4.2. For all p E R, A (N) > O.

a

•

4.2 Many-Fermion Systems on a Lattice

119

Proof.

l(N) ap

a2

=

ap2

= =

log ZA =

a ZA

Tr (e -fl (H- A N) N)

(V + ( N2 ) = ((N — (N))* (N - (N)))

(4.30)

(N)) 2 )

0.

In the second line we used that (11) = 1. Thus (N) is an increasing function of p. This is not unexpected because of the term AN that p contributes to the energy. Typical observables that we want to study are the correlation functions (I1 ax* 0. 11 ay,r ). In particular, we are interested in the behaviour of the twopoint function By spin symmetry, we have (c4,0.ay,T ) = (5„(a;,,,a y,,) and, by (4.18), (4.20), and the linearity of ( ),

*

1".3c,cr u'Y,a/

= c d E e ik(x-y)(4. ,ack,a )

(4.31)

kEA*

because, by translation invariance, (c 0.cie ,u ) = other bases, the relation reads

ox* ,cray,c )

L -d

0 for k k'.

In terms of the

E

(4.32)

kEA*

4.2.3

The Fermi Gas

Ultimately, we want to study the system with a weak interaction (A small) by means of a perturbation expansion in A. It is thus natural to look at the case of no interaction (A = 0) first. For A = 0, the fermions are independent ("free"), and calculating the partition function and (ce1) is easy: we evaluate the trace for Z A as a sum over the ONB (OA )A c r. given by A = n(x,0)EA cf2 (we fix an ordering on r* and take the product in the ordering induced on A). Recall that 1/0 = >k,0. 6(k)iik,o. with fik,o- = qck,a• The iikosr fulfil the

relations

=0

,cr 7 qc; , 0-1] = Skki cr CZ, 0- •

— tbk,o- 7

,

Thus

e -fl(Ho-t,N)

_ H e -0E(k)fik,ff

( 4.33)

(4.34)

(k,cr)Er*

where

E(k) =ê(k) For (lc,

(k', o'), fik os,

-

p.

and cZ, ,0., commute, and fik,012 = 0, so

(4.35)

120

4. The Fermi Surface Problem

ii

=

e - )3( Ho - AN4A

e--)3E(k)fik,Π4,13

(k,cr)EA

H e-f3E(k)fik,,

=

. ri

e—flE(k)fik'' f2

ez ,o. (2 (k:1 vA

(4.36)

(k,a)EA and all factors in the last product acting on f2 anticommute. An expansion of the exponential and (4.33) imply that e -SE(k)fik, ff = 1 + (e-r3E(k) - 1)iik, c,-.

(4.37)

e—flE(k)fik '' 4 ,012 = 4 ,,f2 + (e -flE(k) - 1)4,04,012.

(4.38)

Therefore

By (4.33), nk,0rc12 =

50

e -)3(Ho-piv)

'

A

= H

e -f3E(k

A•

(4.39)

(k,cr)EA

This equation has the simple interpretation that for independent electrons, the state ' Cl l which contains particles in the states (k, ci), for (k, a) E A, has total energy E (k,o.)EA E(k). By the binomial theorem,

Z

A

(c4 1 e--0(110--AN4A) E Acr.

=

= E H e—I3E(k) (k,o-)EA = il (1 ±

(4.40)

ACT'

(k,cr)Er*

The number density is calculated in the same way: 1 Tr (e -0(110-AN) n- 13 " ) (flp,T) = _

Z zA (7-ip,,-) = E (IA I e--0(110-AN)/ip,74A). , 7

A

SO

(4.41) (4.42)

ACT*

Since iipo-f2 = 0, hp,4■ A = 0 unless (p, 7) E A. If (p 7 T) E A, then iip,74A = A • Thus -f3E(k) . e (4.43) A(fip,r) =

Z

E II

ACr* (kMEA

(pr)EA

The constraint in the sum that (p, T) must be in A can be enforced by a factor 7 .., (ka) vA Arp,r (k, a.) in the summand, where Arp,T (k ,a)

={0 1

if k = p and cr = T otherwise.

4.2 Many-Fermion Systems on a Lattice

121

Again, by the binomial theorem, ZA(lip,T)

c II A437-(kIr) H e -'3E(k) (k,cr)VA z' H (ArP,T(k, Gr) + e -flE(k) )

= V

ACT' (k,o-)EA

.

(4.45)

(k, o-)Er

TT

e

=

(I- + e-f3E(k))

(k,n œ)Er* (k,a)*(13,1

)

Dividing out Z A l we get

e —OE(p)

(fip,T ) = 1 + e- OE(P) = f o(E (P))

(4.46)

1 fo(E) = 1 + 0E

(4.47)

where

is the Fermi-Dirac distribution. It has the zero-temperature limit 1

lim fo(E) = {

/3-+ao

if E < 0 if E = 0

(4.48)

0 ifE>0.

The interpretation of this is that in the Fermi gas at T = 0, all states with energy E < 0 are filled and all states with E > 0 are empty. The set of p for which E(p) = 0 is the surface of the occupied region, the Fermi surface

S= p {

E

B: E(p) = 0}.

(4.49)

Here B = Rd PaZd is the first Brillouin zone for the infinite lattice. On a finite E lattice, also A* is a finite set, so {E(p) : p E A* } is finite, and S n A* 0 can happen only for finitely many p. The discontinuity of the zero-temperature Fermi-Dirac distribution at the Fermi surface is the characteristic feature of the Fermi gas; it is a direct consequence of the Pauli principle: the energy of different particles simply adds up, no state can be occupied twice, and thus in the ground state, only the states with lowest possible energies are filled. On the other hand, (4.31) implies that the position space two-point function is simply the inverse Fourier transform of the number density, so, in the infinite volume limit L oo,

(a * a

) = f ddk eik(x- Y)fo(E(k)).

f (27r)d

(4.50)

13

The discontinuity of the Fermi distribution at T = 0 implies a slow decay of the two-point function in infinite volume - if the two-point function were

122

4. The Fermi Surface Problem

in LI, its Fourier transform would be continuous by the Riemann-Lebesgue lemma. As an explicit example, we calculate the two-point function for d = 3 in the continuum limit e -> 0, where e(k) = k2 /27n: in infinite volume and at zero temperature (T 4. 0), with kF = N/I' WIT

(4,,ay

, Π)

-

f

ddk ik oc 3,)

(27r) d e J pci 0 let C't be an invertible, antisymrnetric linear operator acting on functions defined on F, i.e. (Ct f)(X) = dXC t (X, X') f (X I ) with Let

Ct (X I , X) = —Ct (X, X').

(4.83)

Let C't be continuously differentiable in t; denote 49,4*-' = Ô. Let clfict be the linear functional (Grassmann Gaussian measure) with characteristic function

f

dpc, (0) e (n,r = e l(n, ct n)r

(4.84)

The integrals of arbitrary monomials are obtained from this formula by taking derivatives with respect to 71. The measure is normalized: f clitc t (0) =1. Let V(7,b) E A be an element of the even subalgebra and have no constant = AV(0). The effective action at t> 0 is part, A E C, and

mop)

g (t,) = log f dp ct (x)

(4.85)

Because the measure is normalized, eg(t4) = 1 for A = O. By the nilpotency of the Grassmann variables, eg (t ' ll') is a polynomial in A (the degree of which grows with Thus 0,0) is analytic in A for I AI < Ao (F).

in.

4.3 The Renormalization Group Differential Equation

131

4.3.2 The RG Equation The RGDE is obtained by taking a derivative with respect to t of the defining equation (4.85) of the effective action. The Boltzmann factor eG (t M is given by a convolution of the initial Boltzmann factor eG(°,0) with a Gaussian measure, just as the solution to a heat equation is the convolution of the initial function with the heat kernel. And indeed, the differential equation satisfied by the Boltzmann factor is simply the associated heat equation. Proposition 4.3. Let

1

f dX fdX'

F Then

r

6

scx)

a

Ct(X, X')

(4.86)

e go "0 )• = , Ûte go, ,,b)

(4.87)

eg(t 'lb) = eAct eg(°M.

(4.88)

`9i

a

and If g(0,0) is an element of the even subalgebra, then for all t > 0, 0, 0) is an element of the even subalge bra, and it satisfies the renormalization group equation

—a ol o = .Act g (t, 0) + 21 at

coop) A sg(t op)) SO

7

Lit

SO

.

(4.89)

Proof. For any F(0) E A, define F(4) by replacing every factor 11)(X) by in the polynomial expression for F. Then F(0) -- [F(h) e (no,b)r int---_o T( R-T60 (the derivatives also generate a finite-dimensional Grassmann algebra, so the expansion for F terminates at some power). Since Grassmann integration is a continuous operation, and by (4.84),

pr,

e g(t ' 0)

= [eg(° '* )

f dpot (x) e(71 ')c±'b)11 n=o

= [eG(0,*) e i(n,co)r e (no,b)ri i n.o For any formal power series f (z) =

(4.90)

E Axk,

f (ict) e ( ") r = f ( 21 (Th Ct TI)r)

(4.91)

so e I (7) , C' n)r+("b)r = et e (no,b ) r. Since .Act is bilinear in the derivatives, it commutes with all factors that depend only on 77 and can be taken out in front in (4.90). This implies (4.88). Since Lic, also commutes with (4.87) follows.

132

4. The Fermi Surface Problem

If g(0, 0) is an element of the even subalgebra, the same holds for g(t, by (4.88), since every application of ,Act removes two fields. Thus performing the derivatives with respect to 0 gives (4.89). • Let the Grassmann polynomials

gr(t,o)

be defined by

00

= E Argr (t) 0).

g(t,

(4.92)

r=1

Assume that C = Ct exists and let Dt = C Ct so that Dt is the covariance of the unintegrated fields. Again, we expand in the Wick ordered monomials —

f2Dt (1,b(X1)

. . . (Xp )) =

'

t 11)(X 1 ) ...

(Xp ).

ôt = —ii. (4.93)

They form a basis of the Grassmann algebra because the relation (4.93) can simply be inverted by multiplying by eApt , and they inherit the antisymmetry from the ordinary monomials: f2D,( 1P(Xx(1)) . • •

ti)(X/r(p)))

= E(7 )flpt(IP(Xi) • • • 1,b(Xp)).

The polynomial Ç7. (t, 0) has the representation (here fh(r)

gr (t)

=E

f d;_v_ Ginf(t

f2D,

X = (XI).

(11 cx0) .

(4.94) , Xn,)) (4.95)

k=1

m=0

where G mr (t I X) is the connected, amputated, Wick-ordered m-point Green function. Using Wick ordered monomials provides even greater simplifications for the fermionic models than for the 04 case. The Gm ,.(t I X 1 , ... ,X„,,,) are assumed to be totally antisymmetric, that Xir ( m )) = E (71- )G mr (t I X1 • • • X m) is, for all 7r E Sm, Gmr (t I X ir (1) because any part of G that is not antisymmetric would cancel in (4.95). Application of to (4.95) gives a sum of two terms since two factors depend on t. By (4.93) and Ôt = —be, the derivative of the Wick ordered monomial is if2D tt t (11k 11) (X0) = ZiCt f2D,(Ilko(x0). This implies that the term linear in g drops out of (4.89) by Wick ordering with respect to Dt , and (4.89) now reads

g

m(r)

f

,opt (11

m=o r,n

k=1

E

a G mr (t _ at

I X) = — Q 7.(t,t1)) 2

(4.96)

where Q,.(t, 0) is defined by (sg(t,11))

(50

_o

,

00

6g(toP)) = E Ar a7.(t ) 1,b). À.1

(4.97)

4.3 The Renormalization Group Differential Equation

133

Being an element of the Grassmann algebra, Qr (t,ip) has the representation

f

771 ( f )

Qr(t, 0) = E

dX_ Qinr(t

Ix) f2Dt

(

m=0 Tn,

Because g is in the even subalgebra and Ot =

H:, 11)(xk))

.

(4.98)

-A,

Qr(t,o) = -2-1 [((t,x),Ottip- g(top))] x=11, = — 12

(4.99)

rfitg(t,x)g(t,o)] x=,,b

- L -

with Ht = ( A7, D t 4). Upon inserting the expansion of g, we have to reorder expressions d the form m n 1. (4.100) P = ---Htf)D, flipt (Htp(m)

(11x(xo) k=1

i=1

By definition (4.93) of the Wick ordered monomials, n _A(0) 1 ,_ r-r m P = - -2 lite Dt il X(Xk) e D t H ip(Y1).

(4.101)

/=1

k=1

Here the superscripts on the Laplacians indicate with respect to which variables the derivatives are taken. All differential operators are bilinear, hence commute with one another. Moreover, ,6 A ( ) commutes with all polynomials in x. Thus 7, =

1 —

m

n

A 00_,6 (0) . D t Ht 11 X(Xk) e Dt

HIP

k=1

Since

(

(4.102)

Yi ) .

/=1

— ,4') = — ,6 V')

(4.103)

where 6 — Vi ± Vri, 1 Dt(4 +

A(x0P) _ 1 ( 6 Di

we get 1 _(x') ( gt Ht) Dt - e

m

(4.104) n

11 x (x,)

H

k=1

1=1

V) (11) -

(4.105)

Because

II x(xo

kEK

Howoi tEL

= f2Dt (Hltock) H V) ( Yi) ) x=lb

kEK

tEL

(4.106) because e — D t is and a linear differential operator, it now suffices to calculate the action of ellt on the monomial of V) and x. A cx,Ab)

134

4. The Fermi Surface Problem

Lemma 4.4. m

n

ell' rr 11 X(Xk) k=1

H V; ( Yi ) =

E(_1)m8+ 3 4.E1)

1=1

s>0

E

v(A)v(B)

ACm,BCn

Itil=1B1=a

detAB(Do

x(X) pflA

II 0(yq).

(4.107 )

qflB

Here m = {1, ... ,m}, 8

det AB (Dt ) = E e(7,)

H Dt(Xak,n, r(k ) ),

(4.108)

k=1

irES3

where e(7r) is the sign of the permutation 7r, and v(0) = 1, and for A = { a i ,... ,a,} with a l < < a„

v(A) = (-1)Ek

(4.109)

ak.

Proof. We abbreviate xk = x(Xk), h = 0(Y/), and 15k/ = Dt (Xk, Y1 ). Then m n

Ht

=

EE a

a

(4.110)

—• 19 01

k=1 1=1 aXk

The expansion of the exponential gives e

t=

E

H X°a 7=1 —a •

SCmXn(k,l)ES

(4.111)

alP1

k

A set S with s > 1 elements contributing a nonzero term to this sum is of the form S = f(ak,b7r(k)) k E {1, , as } c m with s }} , where A = al, al < < as and B = < bs , and where 7r is a ,b8 } C n with b1 < permutation of {1, , s}. The map S (A, B, 7r) is bijective. Thus

=

E 8>1

a A H uXai.

01/4 1r(k)

ACm,BCn rES s k=1

(4.112)

IA1=1B1=3

and we need to calculate its action on xi • • • Xml,b1 • • • 1,bn. All sign factors now come from reordering the Grossmann monomials: 8

H

8

S

8

( 1)8(8-10

TT

eXak athir(k)

TT

8 eXak

k=1

1=1

k=1

8 "bw(t)

s

=

s a IA axa k II aiPbt '

( ..... 1) 8(8-1)/2, (70 H

k=1

1=1

8-1 = (-1) 8(8-1)12 6(70

11

k=0

s-1 A

a

as—k

IT .91148_1 a

1=0

(4.113)

4.3 The Renormalization Group Differential Equation

and

8

Xi • • • Xm = v(A)(-1)8(8+)/2

xk

xa.

135

(4.114)

kA

k-=1

(the factors in the last product are ordered according to increasing k), and similarly for the 0. Commuting the product of derivatives with respect to with x i xm gives a factor (-1) 8m. There are no additional sign factors. • Lemma 4.4 and a comparison of the coefficients give the component form of the RGE

1 a yiGmr(t I Xi,— Xm) = AtlIQmr(t I Xi, .. Xm) where Am is the antisymmetrization operator

(Amf)(Xi, • and

1 E ,xm)= m! irES,n

6(7) f (X70), • • • Xir(n)),

Qmr (t I X) is given by the following expression.

Proposition 4.5. Qmi (t I X)

= 0, and for r > 2,

= f dtz, f dV f dW (--aat det Di(I.) (V ,W))

Q mr (t I X)

Ti

Gm i ri (t

I

ri

X1, V) Gm2r2(t

I it, x2),

(4.117)

where f diz, stands for the sum

f dtzmr F =

r17r2> 1

E

mi,r2, m2, i)

(4.118)

E.A11.r 1 r 2 n

with positive weights li,m1tn2 i = (T) ( 77), so that dizm, is a positive > 1, 1 < m1 < measure. Mrir2m is the set of (m i ,m 2 ,i) such that i _ rri(ri), 1 < m2 < rri(r2), mi + m 2 = m + 2i, and m 1 and m2 are even. Xi = (X1, ... ,X mi —j), X2 = (Xmi—i-1-1, • • • , Xm), V = ( 171, • • • ,Vi), W = (W1, • • • , Wi), 1I = (Wi, • • • WO, and 130i) (V,W) is the i x i matrix

(12T) (V , W))

ki =

Dt (Vk

One important feature of (4.117) that distinguishes it from bosons is that the determinant of the propagators appears in this equation. The Gram bound for this determinant improves the combinatorics by a factorial (see [53, 541).

136

4.

The Fermi Surface Problem

4.3.3 The Component RGE in Fourier Space Let A= TxA and A* = M„, x A*, as given in Sect. 4.2.4. Let I' = A x {-1, 1} x {1, 2} and f* = A* x {-1, 1} x {1, 2}. The Fourier transforms of the Gmr (t I -) are m

dmr(t

I Pi, • • • , Pm)

= f

II dzk e-i' x 'c

Gmr (t

I Xi ,— ,Xm.).

(4.119)

Am k=1

For K = (k, a, j) E I' let - K = (-k, a, 3 - j), and define br,(K, IC) = 5A* (k, ki)(5„, (5ii, . The Fourier transform of the propagator Dt is of the form 15(K, IC) = 61-,..(K, , K') b t (K 1 ), where b(K) = (-1)iDt((-1)ik).

(4.120)

This combination of signs implies that Dt(X,Y) = -D t (Y, X). The propagator Dt for the many-fermion system was given in Sect. 4.2.5. The Fourier transform of Q mr (t I .) is Qmf(t i ri, • • • ,Pm) = i

f dkmr f cl.K 1 ...d1Ci (

81)t(Ki)) et f

Ti rI Di Ki ) (

(4.121)

j=-2 Omiri(t

I -P(1) , Ki, • • • ,Ki) Om2r2(t I^ ' Ki, • • • ,r- K1,P (2) )•

Here f diimr = f dismrii!, P (1) = (P1,- - ,Pmi-i), and P(2) = (Pmi-i+i, ... mi +m2). By translation invariance in space x and time T,

6 mr( t I P) = 6A. (Pi + • • • + Pm,0)Imr(t I f)

(4.122)

with a totally antisymmetric function /mr (t I P) of P E Frn that satisfies

(1,...,1) • Vimr (t I P) = 0. (4.123) A priori, (4.122) only implies the existence of a function inar (t I P), defined for those P = (P1, . . • , P„,,) for which pi + ... + pm = 0. However, since

H = {Pi, • • • 'Pm : Pi + • • • + Pm = 0}. (4.124) is a linear subspace of (A*)m, one can simply extend f to a function on all space by defining I = f o Hi with //H the projection to the subspace H. Since //1/ is symmetric in all its arguments, I is totally antisymmetric. The product of the two deltas in (4.121) can be combined to cancel the delta in the relation between d and I, and to remove the integration over kl . Thus k1 is fixed as

k1 = —(k2 + . . . + ki + pi + . . . + Pmi i+1 ) —

(4.125)

4.4 Power Counting for Skeletons

137

and the RGE in Fourier space is &I,,r (t I P) = lAmanr (t I P), with

il Am

,,,,,.(t I P) =

f dii,,,, f dK2 . • • cliCi 1,,i1(—D.t(Ki)) ,E

1:

II NIC 8 /min (t I P (1) , K1, • • •

(4.126)

)

s=2 Itn 2 r2 (t

I"' Ki, • • • , — Ki , P (2) )

with IC1 = (ki , cri,ii), where k1 is given by (4.125). 4.4 Power Counting for Skeletons

We now proceed to analyze the field theory by the RGE in perturbation theory. To this end, we take the thermodynamic limit. The existence of the thermodynamic limit, as well as that of the limit n7. —> oo, is shown in Appendix B.7. There is no problem with singularities as long as the temperature is positive; taking L oo simplifies some of the technical arguments because the discrete momentum sums of the finite volume are replaced by integrals. 4.4.1 Bounds for the Infinite Volume Propagator -

In the thermodynamic limit, the spatial part p of momentum becomes continuous, p E B, and the set of Matsubara frequencies becomes M = { (2n +1) : n E Z}. It is convenient to take (po, p) E Rx5 and put all the 0-dependence into the integrand. To this end, we define the step function cos : R ME by

7 (2n + 1) if po E (271- n , 2— 71- (n + 1)]. w(po ) = — 0 0 0 For any continuous and integrable function f,

f —d2P: Awfoo))

= T31

E

f (w)

(4.127)

(4.128)

•

wEm

R

Moreover, sup

lcdfl(p0)

7r — po I = —

and

inf P (R)) I

PoER

0

PoER

=E 18

(4.129) '

> 1, and ws (—pi) ) = —ws (po) holds Lebesgue-almost everywhere, so that in integrals like (4.128), wo can be treated as an antisymmetric function. With this, the propagator now reads Di (p) = Ct(cdp(po),E(P)) with ICds (A ) I

Ct(z, Y) =

1 xi ( fT.2 (z2 + y2 )) y

IX —

(4.130)

138

4. The Fermi Surface Problem

(and et = co e—t ). In infinite volume, the restriction on the spatial part p of momentum improves the bounds on the propagator over the finite-volume ones used in Appendix B.7. in t and Cko in p. If t > log g-92, then Dt(p) = 0 for all p ER x B. For all multiindices a ENg with lai 0 such that

Lemma 4.6. Di is bounded, C

l

(4.131)

ac' 154)1 < Baft—i—lal 1 (liw(P0) — E(P)i ft)

Baft—i—lal 1 (1w00)0)1

ft) 1 (1E( 3)1

ft) •

Bo =4. For t < log g?,

f talaczA(P)1—< BafTial .(1E(p)i
log . Since

L5(p) = —24-2 (iwo (po) + E(P)) Xi (ft 2 PO (PO

) 2

+ E(p) 2 ))

(4.135)

and Ilxill oo < 2 , 1 154)1 Clicdo(Pa) — E(P)I < et) . Thus Bo = 4. Derivatives with respect to p can act on E(p) or on x'1 , in which case they produce factors bounded by ft lIVE(P)1, so there is B a such that (4.131) holds. Inserting (4.131) into the integral in (4.132) gives

1'154) 1
0

limr(t)1 o 4 if m = 2.

(4.145)

4.4 Power Counting for Skeletons

141

Proof. Induction on r, with (4.145) as the inductive hypothesis. For r = 1, there is nothing to prove because 0, 4 1 = O. Let r > 2. The inductive hypothesis applies to both factors I/mk rk leI on the RHS of the RGE. Since mi = 2 and m2 = 2 are excluded from the sum in f dkmr , we can insert the upper line of (4.145) for both factors. Because m1 + m2 = in + 2i, the exponent becomes independent of i, mi , and m2, and we get iQmr(t)10 with

_< amr( 1 + t) r-20(m-4)

1 Ni amr = J1 ez f Cliçrnr (8 th C) O a rn 1 rl aM2 1.2 •

(4.146)

(4.147)

All that remains is to use Ifmr(t)lo < Ifmr ( 0 )10 + 1 fot ds 10mr(5)10 and to do the s-integral. For in > 6, in — 4 > 0, so t

t

fo ds I 0mr(s)lo

amr

f ds (1 + s)r_ 2 ei (m -4) o

2a mr (1 + or-2 e (m-4) . 771 - 4'


6 is thus completed by noting that 1 < (1 + t) r-2 < (1 + t) r - i and defining a, recursively as am,. = aZ. + a,/(m— 4). If m =-4,

t

t

fo

ds 104r(s)lo
i T2(si , s2).

(4.158)

By Lemma 4.6, and since B0 = 4,

(s)
2. The inductive hypothesis applies to /miri and 1m2r2 . Let 1 < s < m. By the inductive hypothesis and Theorem 4.8, and because mi < s and mi > 4 and m2 > 4,

(s, t)

j 1,0yb Ji1E0

Similarly, si < mi and s2

(s, t)

<
i .1+82=8

). max brainsi bm2r282 (1 -I- t r 2 e1("1-8-121

81,82>i

.1+.2=.

(4.165)

4.5 The Four-Point Function

145

Thus

l ti imr (oi s < el (m-8- ') (1 + t )'

2 f dkmr(8J1f0)iMmrs

(4.166)

max bmirisi 1'm2r282 b

(4.167)

with Mmrs

=

max{

—iris am2r2)

:48322>=13

and therefore

11 mr (t)l s 1-a > 0, so the integral over t' is bounded by 1a (1 + t) r_2 e (nI-8—a) . Since this quantity is at least one, the induction step is completed for in < s by defining

bmrs = ers + If

s = m, m - s - a f

f

= -a < 0, so the integral over

dt1(1 ty-2e-te

e2

(4.169)

dkmr (8theo) i M m,...

t'

is

dt (1 + e)r-2e-1(1-Ft1)

f

(4.170)

o 00

0


log . For = oo, convergence follows from (4.166) because for s = m, the right hand side of (4.166) vanishes as t oo. •

4.5 The Four-Point Function In this section, we do some explicit calculations for a truncation of the RGDE. This truncation corresponds to the ladder approximation to the four-point function. The justification of this truncation will be discussed in Sect. 4.6.

146

4. The Fermi Surface Problem

4.5.1 Motivation

We have seen in the last section that the four-point function produced factors tr
0 corresponds to a repulsive interaction, and Ao < 0 to an attractive one. The 6 symbols make f antisymmetric under exchange of i and 6, and 6 and 4 4. The factor 24 and the extra sign are due to 7 — 1-1,72 ET1T2E 7271 = —2. The solution is so simple because the momentum dependence of f (t16 , ... , ,i) is only via the sum of the ingoing momenta pi + /32 . Thus integrating this truncated equation amounts to a resummation of the particle-particle ladders. For oo, D(k) — > 0, so

t

B (-) (q) = lim B(q) = t-400

f

dk Do(k)D0(q - k).

(4.199)

For T > 0, B( —) (0) is finite, but proportional to log 0: up to a term coming from the ultraviolet cutoff that is bounded uniformly in the temperature, we have, by the symmetry E(—k) = E(k),

B (—) (0) =

h. E

f

dk

1 1 iw — E(k) —iw — E(—k)

(4)EmlE(k)15_c0 fO =

f

dE N(E) 113

_EC)

E wEm

1 2 ±E2

(4.200)

with the density of states N(E) as in (4.77). Because E w -2 converges we can put in a factor ei" and get , by (4.68), 1

1 V"`

7

to EM

w2 +E2

= lim 1 N--. 40 /3

=

•(--'d

wEM

eiEw LiJ2 + E2

1 ( ei" + 40 o E -2E ic,, E wEm 1 ( 1 —2E 1 + OE 1 b OE 2E tan " 2 '

fin, i

-

=

--=

—

icd

—E (4.201)

SO fo

B (—) (0) =

fE

dE

—

0

tanh

q N(E) =

qQ-

f

136 0 2

k il tanhz

NC1). -

(4.202)

4.5 The Four-Point Function

153

As shown in Sect. 4.2.6, N(E) is Ck0 -1 around zero, so a Taylor expansion gives for large 0

1 0E0 B (-) (0) = - N(0) log( — ) + 0(1) 2 2

(4.203)

(if ko > 3 then one can use Taylor expansion to second order; the first order term then cancels because the integrand is odd, and one gets 0(A) instead of 0(1)). N(0) is the density of states at the Fermi surface. For a repulsive interaction (Ao > 0), the function f remains bounded and vanishes for 0 -> co at pi +p2 = 0 ("asymptotic freedom": a running coupling constant defined by

A(t) =

A0

(4.204)

1 + 24A0B -) (0)

decreases to 0 as t -> oc. For an attractive interaction (A0 < 0), A(t) has a singularity at a finite value of t (the BCS case: the effective coupling constant grows as the energy scale is lowered), and f becomes singular if 0 gets large. However, f remains finite and analytic in Ao as long as IA0 IN(0) log(0f0) is small enough (this statement was proved in the last section for the skeleton four-point function, not just the ladder resummation). If one fixes Ao and increases 0, a singularity indicating the onset of superconductivity occurs at a temperature To = 1/00 of order To cc foe-241AIN(0). (4.205) It is natural that c o sets the scale in this equation: the above considerations are valid only when the particle-particle ladders dominate the RG flow. If the Fermi surface has positive curvature, it is possible to choose fo such that the ladders dominate the flow for energies below fo. The role of the curvature for this will become clear when the volume improvement from overlapping loops is shown; it will be discussed further in Sect. 4.8. The coupling constant A0 is not the original coupling constant, but the effective four-point coupling at scale eo. It is a regular, in fact, analytic, function of the original coupling constant A, if I AI is small enough (dependent on co ) but it is of course not true in general that this effective interaction is local, i.e. a constant in momentum space. This can be taken into account in the following way. Let

00-(t1Q)k)i) = f (tI( - k + ?) al)) (k + L, (72)2 +

as), ( -1 +

0'4)) , (4.206)

then

04,

at- (t I Q k , 1) =

12 f dr [4(Dt(-r + ?)Dt(r + ?))]

E 00Ti ,T2

3. 0. 2n.7-2( t

I Q5 kf r)

727134 (t

(4.207) 1).

4. The Fermi Surface Problem

154

Note that the sum of the ingoing momenta Q now plays the role of a parameter in this function. Because Q = 0 is the singular case, we now restrict to Q = O. The flow for 0, has the initial condition 0,0 I 0, k,1) = v,(k,l)

(4.208)

where y is a C2 function obeying t-independent bounds, since it is the initial condition. The particle-particle bubble in (4.207) is independent of the momenta k and 1. Thus derivatives with respect to k and 1 do not act on the propagators Di and therefore do not deteriorate the scaling behaviour. Writing the integration momentum as r = (w,r(p, 0)), we may thus do a Taylor expansion in p without influencing the t-dependence. The terms proportional to p have power counting that is a factor e-t better. Denoting

w(t I 0i,u29 0

(4.209)

cf4,01 ) = Off(t I 0 , 1r( 0 3 0)2 1r( 0 5 01 )),

we thus have, up to terms that are multiplied by an extra factor e - t,

aw

at (t

cr i , (72 , ; cr3 , (74 , Of )

=

(w2 P2))2 f (xi(fT2 12 f dw dpgt (10J(0,0") co 2 p2

(4.210)

"

E w(t I ul,o2,0; 715 7-2; °H )W(t

7-2 2 71 5

0" ; (73, (74,

01 ).

Ti T2

We can now interpret [44] the function w as the kernel of an operator acting on the weighted L2 space L2 ({-1, 1} 2 x Sd-1 ) where the weight is given by the local density of states J(0,0). An explicit construction of an ONB for spaces associated to Fermi surfaces (also called Fermi surface harmonics ) was given by Allen [49]. An expansion in an ONB of eigenfunctions of w, with t-dependent coefficients g, (t), 00 w(t

Ial , cr 2 ,0 ; 0.35 0'4) Of)

=

gk(t)Wk(cr i ,a2 ,0)Wk(a3,a4,00,

(4.211)

k=1

then gives, because of orthogonality, the uncoupled system of equations

agh(t) = 12-)(0)gk(t)2.

at

(4.212)

Thus, in the ladder approximation, the coupling constants associated to the various symmetry channels (which generalize angular momentum channels) flow independently, and all according to the law gk(t) =

gk(0) 1 + 12gk(0).131 -) (0) .

(4.213)

4.5 The Four-Point Function

155

Any coupling that starts out negative (gk (0) < 0) will eventually grow, but one can expect that the one that has largest absolute value will determine the symmetry of the superconducting order parameter. The full flow does not lead to a decoupling of the different angular momentum sectors (even here, we had to do a Taylor expansion and omit some terms to get this decoupling property). Kohn and Luttinger observed, by a second-order calculation [50], that even if the initial interaction is purely repulsive, the effective action at scale co, which determines the initial condition (4.208), must have gk(0) < 0 for some (in general very large) k. Feldman, Kniirrer, Sinclair and Trubowitz [51] have shown by a very careful analysis that although in d = 2 dimensions, the Kohn—Luttinger effect is not there in second order, there is a similar third order effect that drives the full RG flow to an 1 = 1 superconducting state [51]. The analysis of [51, 52] goes far beyond the original proposal [50]. In particular, the RG flow, and not just the Bethe-Salpeter kernel, is studied in [52].

4.5.5 The Particle—Hole Flow The particle-hole flow is obtained by leaving out the pp term on the right hand side of (4.178). Similarly to the particle-particle case, this now resums the particle-hole bubbles

13 +) (q) = f dk (Do (k)D 0 (q + k) — D t (k)D t (q + k)) .

(4.214)

In contrast to the particle particle case, this function is bounded uniformly co. This follows for NI away from in the temperature even in the limit t zero from Corollary 4.11 and for II near zero from the following theorem, which is based on an interpolation formula for Fermi surfaces. Theorem 4.12. Let q E B, qo E .2#Z, and B (+) (q)

=

f ([,214

1 iwn 1E(p) z(cdn+qc)_E(p+q) .

(4.215)

nEZ There is r > 0 and K > 0 such that for all II co. Let q 0. Using a Taylor expansion, B (+) (q) can be rewritten as

Proof. —

fR

156

4. The Fermi Surface Problem

f ddp 1 (27r) 1 iqo + E(p) — E(p +

B(+) (q) =

1

1

iw E (P)

( /1 3E nEZ

f

•

iWn E( P

))

dd p Xf3(E(P)) — X0 (E(p + q))

(27r)d iq o + E(p) — E(p +

(4.217)

ddp +E( p)-E(p+q) f dt 60(E(p, q, t)) (2 iod igo where

E(p, q, t) = (1 — t)E(p) + tE(p + q).

(4.218)

VE(p, q, t) = VE(p) + tE n (p + Vq) • q

(4.219)

Because with some '0 E [0,1] (here E" denotes the matrix of second partial derivatives of E), and because I VE(p)I > go on the compact set S and the second derivative of E is bounded, there is r > 0 such that for all II < r, VE(p, q, > go/ 2 . Thus the interpolating Fermi surface St = { p : E(p, q, t) = 0} is still a submanifold for all t E [0,1] (it is, in our case, even positively curved). We can therefore introduce a radial coordinate p = E(p, q, t) and a tangential coordinate 0 E Sd-1 in the standard way [45], with a bounded Jacobian and get

B(q) < f dt f dp d0

p, 0)60(p) co,

f dû J(0,0) = —N(0)

(4.222)

is just the density of states at the Fermi energy, which is finite by our assumptions on E. If q = 0, but qo 0 0, a similar calculation shows that • B (+) (q0, 0) = O. Equation (4.217) also implies that at qo = 0 the particle-hole bubble is a Cko function of q for II < r if the dispersion relation E is Ck°.

4.6 Improved Power Counting

157

In Theorem 4.12, we have left out the ultraviolet cutoff on Do . This is justified because in the integrand for the difference, at least one of the propagators is cut off away from the Fermi surface, hence bounded, and thus the integral converges because f dpiDo (p)1 is finite. In summary, the finiteness of the particle-hole ladders holds under two conditions: there is no van Hove singularity on the Fermi surface, and the Fermi surface has positive curvature. If one of these conditions fails, the particle-hole ladders can have singularities. A relevant example for that is the half-filled Hubbard model, where singularities occur (in d = 2) on the two lines p i = p2 and pi = —p2 (here p = (pi , p2 )). Near to, but not exactly at, half-filling, these singularities are replaced by large values of the function, which may be important in the RG flow, although they do not introduce true instabilities at very low energy scales for small values of the coupling constant.

4.5.6 The Combined Flow An iteration of the full parquet equation (4.178) also generates terms that mix the particle-particle and particle-hole bubbles. However, these terms all contain overlapping loops (defined in Sect. 4.6), and therefore they affect the leading low-energy behaviour, which is given by the particle-particle ladders, only by exponentially small terms. The particle-particle singularity is more generic in that it occurs at any value of the filling factor, whereas the particle-hole singularity occurs only for specific filling values (e.g. half-filling in the Hubbard model).

4.6 Improved Power Counting In the last section, we have seen that the ladder approximation to the flow of the four-point function has a logarithmic growth when the sum of the ingoing momenta equals zero. This by itself does not prove the occurrence of such a singularity in the full four-point function because there may be other contributions that cancel the singularity in the ladders. In this section we shall prove, among other things, that if the ladder term is left out, the fourpoint function is bounded uniformly in the temperature. Thus, although in the full flow, the ladder term and the others are coupled, such cancellations do not occur.

4.6.1 Overlapping Loops We now proceed to discuss the bounds that are at the heart of the analysis of perturbation theory to all orders. They apply to contributions from graphs with overlapping loops. Examples for such graphs are given in Fig. 4.5. The

158

4. The Fermi Surface Problem

first one is a contribution to the four-point function, the second one to the selfenergy. The third one is a graph that occurs when the integral equation corresponding to the parquet RGE is iterated, and when the particle-particle and particle-hole flows mix. All these graphs have at least two loops, and two different loops have one line in common. This line is drawn heavy in the figure.

:CΠ-a-K)Fig. 4.5. Examples for overlapping graphs To motivate the bounds that follow, we discuss the two-legged graph in detail. If associate a vertex function that is identical to one to each of the two vertices, we get the second order contribution to the skeleton selfenergy in the Hubbard model with on-site interaction, given by the integral

È2(q) = f dkidk2 Dt(ki)Dt(k2)1 5 t(q + viki + v2k2)

(4.223)

with vi E { 1, 1} . The property that a linear combination of two integration momenta, like the q + viki + v2k2 in (4.223), appears in the argument of a propagator, is a defining property of graphs with overlapping loops. A purely graphical definition of the property of a graph to be overlapping can be given [45]. If a spanning tree T for a graph G is fixed, the loop generated by a line l i fl T consists of ii and the closed nonselfintersecting path connecting one endpoint of / 1 to the other one. If there are two lines 1 1 ,12 ¢ T, the graph is called overlapping if the loops generated by l i and 1 2 contain a common line of T. Whether a graph is overlapping or not does not depend on which spanning tree one chooses to check this property [45]. Although the graphical definition and interpretation just given will not be necessary for understanding how the bounds work in the RG differential equation, they allow to decide rather easily if there are gains in power counting also when one does the analysis with discrete renormalization steps. One can give a complete characterization of nonoverlapping Feynman graphs with two and four external legs [45]. The power counting of Sect. 4.4 gave —

( .4)±2(q)1

iia a btlioollptiii

130,(8.11)2 E.

(4.224)

For a = 0, this leaves a decay Et oc e—t, so that E2 (q) = f(!,, ds È2 is finite. For a > 1, however, the integral grows with 3 and diverges for zero temperature,

4.6 Improved Power Counting

159

thus indicating that already the first derivative of the selfenergy is divergent at zero temperature, if the momentum is put on the Fermi surface. This is the true behaviour in one dimension, but not for curved Fermi surfaces for d > 2: Proposition 4.13. [45, 46] For the class of models obeying the assumptions in Sect. 4.2.6, there is a constant (independent of temperature) such that t led

()c' ±2(q) 1
3.

Thus the first derivative of E2 is bounded uniformly in derivative grows at most as a power of log O.

(4.225) and the second

Here and in the following, derivatives with respect to po are understood as a difference operation (f(po + — f (po)). The RGE actually defines the Green functions for (almost all) real values of po, not just the discrete set of Matsubara frequencies, so the effect of such a difference can be bounded in terms of a derivative by Taylor expansion. This changes at most constants, so we shall not write this out explicitly in the proofs. The proof of this proposition will use the volume improvement bound from overlapping loops. It is a generalization of the one-loop volume bound given in Lemma 4.10, but it has much stronger consequences because it is uniform in the external momentum. It applies to all contributions to the right hand side of the RGE with number of internal lines i at least three. Using this bound we shall generalize Proposition 4.13 to the full skeleton selfenergy. Proof. We write k • = ki) for the integration variables, use again (4.187), (4.131), and the inequality 1(1 qo + vi col + v26)21 _< et ) < 1, and do the frequency sums over co l and 6)2 . This gives the bound log(0€0) Oq

±2(q)
t and t2 > t, V2(q)t) ti 5 t2)

QT€t1ft2 et {

1+t

if d = 2

(4.228)

The constant QT depends on the minimal curvature of the Fermi surface. The naive bound for V2 (q, t, t 1 , t2) , obtained by dropping the indicator function in (4.227), is V2

(q, t, ti, t2) < const

Et 1 Et 2)

(4.229)

so, up to the factor t in two dimensions, the Lemma implies an improvement by a full power Et. We call Et cc e —t the volume improvement factor. The most important feature of this bound is that the improvement is uniform in the external momentum q. It does therefore not matter if q is near to the Fermi surface (as would be the case in the selfenergy) or if q is any linear combination of momenta, as is the case for the four-point function and the higher m-point functions. Volume improvement from overlapping loops was first proven in [45]. The analogue of Theorem 4.14 proven in [45] gave an improvement factor e-6t with some 6 > 0 under much weaker assumptions on the Fermi surface (it need not be convex, not even connected, and the curvature may vanish on subsets of positive codimension of S, e.g., in d = 3 on a curve on S). On the other hand, the t-behaviour stated in Theorem 4.14 is optimal, that is, one cannot gain more than one power of ft from an overlap of two loops. Theorem 4.14 was first proven in [46] in the form stated in Appendix B.8.2. We prove it for spherical Fermi surfaces in Appendix B.8.2. The proof for the nonspherical case, including surfaces that are not invariant under p —> —p, is in Appendix B of [46]. It does not use Lemma 4.10: in fact, in absence of the symmetry p —> —p, VI+) and V(—) i behave differently: an analogue of Lemma 4.10 holds for 14+) , but only a weaker bound holds for 14—) (see Appendix C of [46]). The existence of a volume improvement by a factor e—t/ 2 , as well as its uniformity in q, can be seen in a very simple way by the following argument

[48] using Lemma 4.10. By definition (4.227), V2

(q, t, t i , t2 ) =

f clici ,Vi(v2) max (q + viki , t, t2). vi,v2=±1 f (27r)a

(4.230)

By Lemma 4.10, VI(±) (P2 t, t2) < Qoft2

Ii / .N

if 1106 otherwise.

(4.231)

4.6 Improved Power Counting

161

Inserting this bound into (4.230), we get V2 (q, t, t i , t2) < Qo Et2(N/

+

vol R(Et i )

vol (R,(E t ,)

n

NR)

(4.232)

with B(q,r) = {13 e B iiP IL < r}. We have vol R.(eti ) < const et l . The set B(-v i q, VEt-)nS is either empty or a piece of the Fermi surface with diameter 2.%/Fri". Thus R(Et 1 ) n B( -v1cl7VEi) is a subset of the cylinder of height 2et 1 above that region, and therefore -

V01(1?(Et 1

n B(—viq,

)

< const et i et(d-1)/2

(4.233)

const ft 1 Et2

(4.234)

Thus, for all d > 2, V2 (CI, t7 tl, t2 )
3, We also want to bound derivatives. The derivative applied to Qm, gives

vrnr (t I P) = f

ac

i!X(t I P),

(4.235)

where, after a change of variables ki

Xct,i(t I /2)

=E

a! E EEtokti!a2!

crE{-1,1}i /e{ 1,2}i

f

dk2

Rx13

f

dki a'°./it(ki)

Rx /3

13 ctlImlr1 (t

a

H D(k)

(4.236)

j=2

p(1) K) aCt217n2r2 (t

K, p(2))

with Kj = (ki, g , 1 j), the sum over a running over all triples (c o , ai, a2) with ao + al + a2 = a, and ki = - Eii =2( -1 ) ilki - (PI + - • • +Pytt,-t-Ei)• X„,i depends also on (m i ,ri , m2 , r2 , r). Application of I k gives < 4i

E

iaal imiri (010 la a21m2r2 ( t )10 Yeto,i( t)

(4.237)

with

ya,i (t) =

sup CIEMx5

„,EI-1,1)

f

TT dki IDt (k)I pa i=2

(E viki + Q)I. j=2

(4.238)

162

4. The Fermi Surface Problem

Lemma 4.15. y(t) = o for t > log

, and

Ya,i(t) < (8.I i ) i-1 Ba E i -2- II .

(4.239)

Y(t) < ic(1)--1 (1+t) cr i- l al.

(4.240)

If i > 3,

with K (1) = BI3Ba(8M i-3 QT. Proof. By (4.131), Yo,,i(t) < B aft 1 lai llpt(k)11i i-1 , so (4.239) holds by (4.133). For i > 3, the integral (4.238) for y„,i (t) contains the subintegral S(Eii .4 viki + Q), with

S(q) = f dk2 f dk3IDt(k2)Dt(k3)it(v2k2 + v3k3 + q)I

(4.241)

which we recognize as the function A (see (4.223)). Thus by ProPosition 4.13, IS(q)I < BI,BaQTE t2 Inserting this bound, the remaining i -3 integrations are independent and contribute a factor IlDt(k) Ill i 3 . Thus (4.240) holds. • ' al .

Lemma 4.16. For all i,

IX«,i(t)10

4i (8J1) i-1

al

E aolaila21 1 aal irni ri (t) 10 1 a a2 /m2 r2 (0 10 a

B„Eit -2-la°1 .

(4.242)

For i > 3 IX0,,i(t)10
3.

(4. 24 3 )

In the next sections we use Lemma 4.16 to prove bounds on the non-ladder skeleton functions and on the derivatives of the selfenergy.

4.6.4 Bounds on the Non-Ladder Skeletons

We define the non-ladder skeleton functions as the solutions to the skeleton RGE where the ladder contributions to the four-point function are left out. That is, on the right hand side of the equation for in = 4, m i = 4 and m2 = 4 are left out.

4.6 Improved Power Counting

163

Definition 4.17. The functions A,N7), obtained from the truncation of the skeleton RGE, in which the equation for the four-point function is given by

_a pm (t ) = A4 Q 4 r,>3(t) at 2

(4.244)

4'r \

(i.e., where m 1 = 4 and m2 = 4 are left out in the sum non ladder skeleton functions.

f dk 4,r) are the

-

The next theorem [53] states that the non-ladder skeleton four-point function is bounded and that the non-ladder skeleton selfenergy is C I uniformly in Theorem 4.18. For all r > 1, the non-ladder r th order skeleton selfenergy -uso 12 r (t) converges for t —> oc to a C2 function. There are constants Lr and L., independent of 0, such that

l

1

if lal 1 Lr aa I,Nr ) (t)o < (log Ofo) 2 if la I = 2 and d = 2 log 0E0 if lal = 2 and d > 3.

and

(4.245)

1

if lal =0 log Oeo ifla' = 1


6-point functions with bounds in the L I norm. The following theorem implies Theorem 4.18. Note that all bounds in this Theorem are independent of O. Theorem 4.19. Let a be a multiindex with la 1 0 such that parmr (o)i < K, with K.Z. = 0 for m > 2r + 2, and 41 = Jm 4v , where

> O. Let a) (t) be the Green functions generated by the non-ladder skeleton RGE, as given in Definition 4.17, with initial values Imr (0). Let Km i = 4°1 and for r > 2, let Km ,. = K(°) 1 Mlf ditmri i!(3 2 Ji) i-1 KmiriKM2r2 mr+rn

(4.247)

where Al l = 240B K (1) , with B = maxe, B. Then Km,. = 0 if m> 2r + 2, and for all t > 0 and all m,r 2— rt—Ick:

laa (2(N) mr

< 2K

0—

Et Inr

El-1°,1 1+1 t

2 (j ) äd,2 k 2

if m > 6 if m =4 if m

2.

(4.248)

164

4. The Fermi Surface Problem

Moreover, for

m > 6,

l aa irr) (t)

Hai

(4.249)

< K4r

1 if = 0 (L2±1)2 if ict i 1 f T i p if _ 2,

(4.250)

1

if

lal 1

if

lal = 2

0< K

Mr Et2

for m = 4 l aa4zNr)(t) 1 ° and for

m=2 (L+1‘2

K2r

2) I-+ 2

if la'

=

and d = 2 2 and d = 3.

(4.251)

Proof. Induction in r, with the statement of the Theorem as the inductive hypothesis. r = 1 is trivial because iel) = 0 and because the statement holds for inzi (0). Let r > 2 and the statement hold for all r' < r. The inductive hypothesis applies to both factors 47,N)rk in 0;n - Nr) . For mk = 4, it implies that

1 for all

.-/Zk l o < Kink rk E T:al = Krnkrk EcT lal etaal+

(4.252)

t > O. Recall (4.242) and (4.243), and

la ' Arn -Q.Z.)(t)L =
6. By (4.242) and m1 + m2 = m + 2i, the t-dependent factors in X,i(t) are

et( "41- —2 -Ficti

e t( 2?-2+Icy2D e t(2-i+Ictol)

= et(*1 -2-Flal) .

(4.254)

Since ffz - 2 + lal > ¶ - 2? 1, integrating the RGE gives

actirr) (o)

aairr)(t)lo
oc of 4Nr) (t) exists and is a C I function of (Pi, p2 ). All constants are uniform in /3. The second derivative of Ce.) is 0(t) in d = 2 and 0(1) in d = 3. Since Orr) = 0 for t> log /3e0 , the integral for ir r ) over t runs only up to log /3€o, which gives the stated dependence on log /3€0. A look back at the proof shows that no factorials from powers of t were generated in the recursion for the coefficients (because there are no powers of t that grow with r).

4.6.5 The Derivatives of the Skeleton Selfenergy If the ladder contribution is kept in the RGE, powers of log )3E0 appear in the Green functions. We now prove [53] that if I Al log 13e0 is bounded by a constant, then the derivatives of the selfenergy are bounded independently of 13.

12,,. and 140. converge for t -> There are constants Lm,,. such that for all 0,

Theorem 4.20. The skeleton functions

(9Im ,,.(t)L < tm ,r (logOco)r

00.

(4.259)

for m = 2, lal 0 such that Imr(0)1 < Km, with Ifrç rC12 = 0 for m> 2r + 2, and 44)), = 6„7,4v, where > 0. Let Inzr (t) be the Green functions generated by the skeleton RGE, as given in Definition 4.7, with initial values Im ,.(0). Let Km1 KZ and for r > 2, let Km ,. be given by (4.247). Then for m > 4

(t)1 0 < 2Kmr (log OE° ) and

inzr(t)L

Km ,. (log Seori

(4.260) f7 Hal

(4.261)

For m = 2,

< 2K2, (log 0€0 )r -2 +5,1,24 — lal l aV2r(t) 10 —

(4.262)

and aq2r I

10

K2 r (log Ofo)' 2 Ii

‘ (log 0€0) 1+5d,2

lal < lal = 2.

(4.263)

Proof. The proof is by induction in r, with the statement of the theorem as the inductive hypothesis. It is similar to the proof of Theorem 4.19, with only a few changes. Note that m 1 = 2 and m2 = 2 never appear on the right hand side of the RGE because of the skeleton truncation in Definition 4.7. For m > 4, use (4.242); this gives aa(?Trzr(t)10

9vmr (log /36 0 )r -2

(4.264)

For m > 6, the scale integral is as in the proof of Theorem 4.19. For m = 4, the scale integral is now .1: ds t < log0E0 . This produces the powers of log 0€0

upon iteration. For m > 2, use (4.243); this gives K(net2- l a l (1(1 + t)) 6a.2 instead of Et 2 in (4.264). The theorem now follows by integration over t, recalling that the upper integration limit is at most log /3e0 . • Theorem 4.21 implies Theorem 4.20 as follows. Convergence of the selfenergy follows for lal < 1 as in the proof of Theorem 4.18. For m = 4, the function is bounded uniformly in t. For lal = 2, convergence at > 0 holds because iihr(t) = 0 for t > log 0E0. Corollary 4.22. For R> 1, let Ir) =limt,c„) Eril 12,r (Or . Then for all R > 1, there are constants -yR, independent of 0, such that for all I a I < 2, and all À with 1 Al log 0E0 < const , the skeleton selfenergy satisfies

a ct e) 1 0 < 'YR.

(4.265)

4.7 Renormalization

Subtractions

167

It is expected that 7R < const r, so that the skeleton selfenergy is analytic in A as long as the above temperature-dependent restriction holds. The present treatment of the combinatorics is far from showing anything of the kind, but a proof should be possible by combining the method of [55] with the bounds from overlapping loops [45, 46, 47, 53].

4.7 Renormalization Subtractions We now discuss what happens when the restriction to skeletons is removed, that is, when two-legged insertions are allowed.

4.7.1 Motivation; the Counterterm The two-point function in momentum space, 12,r(t I 1311 P2 (Pk ,Crk ik) of the form 12 7,-(t I Pi, P2) = Eii.12410.2 E2,r(t1( - 1) iPi)

is for Pk = (4.266)

(this holds by antisymmetry and because ./1H/2, r =1-2,r , where 17H projects to the subset pl = —p2). Accordingly, we write the right hand side of the RGE for m = 2 as 1 (4.267) 112 7 r (t = — Q2,r (t 2 with a function Q2 7r that is defined in terms of Q2 7 r in the same way as 1 2,r is determined by Also, in the neighbourhood of the Fermi surface where ,7r(p, 0)), with p and 0 the radial and the flow takes place, p = (co , p) = tangential coordinates around S. The skeleton m-point functions had the simple power counting behaviour ei (m -4); this bound iterated through the RGE in an almost trivial way because on the right hand side of the skeleton RGE, the m = 2 contributions were left out. For m = 2, e ( m-4) = e—t , but we have already seen that the skeleton two-point function is not 0(e —t ), but 0(1). Thus putting in the two-legged functions leads to a failure of the bounds. On the other hand, the derivative gii2,2 = Q2,2 was of order e—t , so one can once again attempt to solve the problem by a change of boundary conditions. The goal will be to choose an initial condition for the flow such that lim 112 r (tI0, ir(0, 0)) = 0

for all 0 E Sd-1 .

t-400 '

with the idea that, if RGE as

Q2, r

(4.268)

remains of order e—t , one can then integrate the

1[2AtI(0,7r(0,0))) =

00

f ds 02 7r(810,7r(0,6))

(4.269)

168

4. The Fermi Surface Problem

which implies that 1[2,7- (t I (0 ) Ir (0 ) 0))) is of order e —t , and hence gives hope that the bounds would not be upset when the skeleton equation is generalized to the full RGE. This strategy works [45, 46, 47], but this is not at all trivial. From a technical point of view, one has to be careful because in order to get from the Fermi surface (0, 7r(0, 0) to general values of p, one has to do Taylor expansion, and taking derivatives potentially generates singularities. From a conceptual point of view, one has to justify the change of boundary conditions. This is an important point, which itself generates further technical complications. One might think of bypassing the above discussion about Taylor expansion 0 as t co for all pEMx B. Indeed, there is by requiring that E2,r (tip) no unique way of imposing the boundary condition, so one has some freedom, However, both from a conceptual and from a technical point of view, there are significant differences between different boundary conditions. The boundary oc is really achieved by an appropriate initial condition condition for t E2,r ( 0 1P) = ]I()r)(P)- This initial condition changes the original model to one ( where the quadratic part of the original action, f dp E(P))0(P) •is changed to

IRA

f

dP /P(P)(iw

E(P) Irr (6) , P))09)*

(4.270)

This change has to be interpreted correctly (and justified, see below). For instance, it makes a difference whether the extra term really depends on w or not. If it does, then the quadratic part of the initial action will in general not correspond to a time-independent Hamiltonian, because the w-dependence of the action is fixed as ico in the functional integral we derived. We shall therefore require that the initial condition is given by a function K that depends only on p, and, to fix K uniquely, we furthermore require that K(p) = K (ir (p, 0)) depends only on the angular variable 0.

4.7.2 Full Amputation The singular behaviour of derivatives makes it necessary to transform from E2 to another function. The best way of getting optimal bounds is to transform to the one-particle irreducible functions. For the purposes of the present discussion, we shall, however, not need to Legendre transform, but just transform from 12 = Er›1 x71 2 , r to an analogue of the full selfenergy and amputate full propagator—s from the m-point functions for m > 4, as follows. Let (t1/9)

/(t1P) = 1+ Ct (P) 112(t1P)

(4.271)

This equation, as all following ones, is to be read as a formal power series in A, i.e. E = E E2(C 11 2)P. Since 112 is 0(A), E is a well-defined formal power series in A. The inverse relation is

4.7 Renormalization Subtractions

12(4)

E(t1p) - 1- Ct(p)E(tIP)

169 (4.272)

•

This transformation is motivated by the relation 112 (CO) = C-1 (G2 —

(4.273)

relating 12 to the full unamputated two-point function G2. In this equation, C gets subtracted because 112 contains only terms of order Ar with r > 1, whereas G2 contains also the free term, and the factors C-1 are there because 12 is an amputated function. The derivative with respect to t gives

È(t) = (1- CtE(t)) 2 012 (t )

— li2 (t)ÛE2 (0) .

(4.274)

The term 12 (t)CtE2(t) is the j = 1 term in the RGE for 12, so we see that this term drops out of the RGE for E. Similarly, we define the fully amputated m-point functions for m > 4 by setting (4.275) A(tilD) = (1- Ct ((-1)ip)E(t1(-1)ip)) 1 and defining

Pm) 11 A(tIPa). a=1

Wrnr(I P1 2 • 2 Pm) =

(4.276)

When inserting this into the RGE, on the internal lines, Dt(p) gets replaced by Dt (P)( 1 Ct(P)E(tin)) -2 , and A (p) gets replaced by tot (p)(1 Ct(p)E(t1p)) -2 , and the external amputations cancel out. The i = 1 term /m et h gets replaced by W m Ct È(t)(1 - C t E(t)) -1 . Thus the RGE for E and the Wm reads -

gEr (t ip) = L? )r ( t ip) Wmr

(4.277)

+ 02,r (tip)

( t i P ) = 411,)r (t1P) aVr(tIP) + mr( t IE)

with

1,2 ),r tIP) (

1 )Ct(( [Wm (t1 2

—1 )3m Pm)È(t1 —1)3m linz) A( t iPm)] r

f dk Wm+2(tIPI .. 2 Pm-2 (k, 1, 2), (—k, 1,1))

1472z)r(tIE)

i3(k)D(k)

Qmr( t 112)

[

riclkg L e ,j q=2

E(tik) (1 - CtE)3(k)jr Dt(kq)

[

(4.278) 15

t(k1)

(1— CtE(t)) 2 (kg ) (1 — CtE(t)) 2 (ki)

wmi (tip (1) ,K)wm2

170

4. The Fermi Surface Problem

0,,,,r

Apart from the replacements on the internal lines, is of the same form as the right hand side of the skeleton RGE. The notation [ - ],. means that the coefficient of Ar of the formal power series is taken.

4.7.3 Bounds for a Truncation In the following, we consider the truncation of the equations (4.277), where for m = 2 and m = 4, the "generalized tadpole" terms 14.V r are left out. The only reason for this is to keep the following proofs and theorems as simple as possible; it will be discussed at the end how one can take these terms into account. Again, we also distinguish between the cases where the ladder contribution i = 2 to Ch,r is removed from (non-ladder) or left in the RGE. As before, the difference in the result is that powers of log Ofo occur when the ladder term is taken into account. The recursion for the constants will be similar to the ones we have done up to now. We therefore do not keep track of these constants any more but use the following shorthand. We say that

Wrnr(tIE) = (9 (fa, [3))

(4.279)

if there is a constant cm ,., independent of 0 and t, such that

1Wmr(t1/2)1 4 and r > 1 let

WZ- be bounded Ck° functions on (R x B)m satisfying ./THWTT, )r = WZ-, and for p = (co, ir(p, 0)) E Mi x B let

1 7 (p) = (0, r(0, 0)).

(4.281)

The following theorem is an analogue of Theorem 4.8 for the full Green functions. It implies finiteness of the two- and four-point function. An analogue of Theorem 4.9, which implies finiteness of all the m-point functions with m> 6 as well, can be proven similarly, by using the norms I - I,. Theorem 4.23. For all r > 1, there are functions Kr E Cko (B, R) satisfying

. a — x-r (Tr(p,o))= 0 ap

(4.282)

such that the solutions Er and Wmr of the non-ladder equations with the initial condition Er(OIP) = Kr(P) Wmr (O1P) =

W(P)

are Ck° in p and satisfy for all a with lal 1, the selfenergy E r (t) and the non-ladder fourpoint function W4, r (t) converge to functions that are uniformly bounded in

0 Proof. The proof is by induction on r, with (4.282), (1) r-(8) r , and the hypothesis E(t I — Po , P) = E( tip° 2 P)

(4.284)

wmf(tIR12) = wmr(tIE),

where RP changes all (p)0 to -(pi) 0 , as the inductive hypothesis. We do the case r = 2 explicitly, to show the procedure for construction of Kr before going into the more complicated argument for general r. Note that the right hand side of the equation for Er and W,,,r contains only Er, and W„,,r, with I.' < r, thus the inductive hypothesis applies to all factors in there. Let r = 1. Set Ki (p) = 0 for all p. Then E1 (tip) = 0 for all t > O. Thus (1)1-(6)1 hold trivially. Since Orni = 0 for all m, (7)1 and (8)1 hold by the properties of Wri . Let r = 2. The Er, (t) occurring on the right hand side of (4.277) must have r' = 1, so all of them vanish. Thus E2(tip) is the second order skeleton selfenergy, and hence the results of Sect. 4.6 imply that (1)2 holds for &E2(tip) = a a 022(4)- In the integral for È2 (t1 — po, p), change variables from ko to -kc, for every loop momentum k. Because Dt( -PolP) = Dt(Po113 )1 and because the same holds for the potential f, by the hypotheses stated in Sect. 4.2.6, it follows that

±2(tl

—

po,p) = ±2 (tip° , p)

(4.285)

so (4.284) holds by integration over t. Set log )3f0 K2 (p) — —

f

dt 422 (417 (p)) ,

(4.286)

o

.ti

then K2(p) E R by (4.285), and (4.282) holds for K2 because H(p) = ir(0, 0) = O. Also, (1)2 implies (2)2 because dt 0ckQ22(t) converges for lal < 1. Integration of the RGE gives

g

r

172

4. The Fermi Surface Problem

t E2(t1p) = K2 (p) + f ds Q22 (.9 119).

(4.287)

0

Thus, by definition of K2, 00

E(t1//(p)) = — f ds Q22 (1(p)) t

(4.288)

which, together with (1)2, implies (3) 2 . (4.287), (1) 2 , and (2) 2 imply (4) 2 . Now (5)2 follows by Taylor expansion in p and co, and by (3)2 with a = 1. On supp Dt2 IP — H(P)I < const e - t, and I I Ct I 100 < const et, so (6) 2 holds. In second order, L( 1 ) and L(2 ) are zero because El = 0, so (7)2 and (8)2 follow from the results of Sect. 4.4, and (4.284) follows again by the properties of Di and f). Let r > 3. The main difference to the r = 2 case is that alz ),,- occurs for m > 4, L2 ), r occurs for m> 6, and that 1 — CL(t) is now 1 because the 0(r'), r' < r selfenergy will in general be nonvanishing. Whenever a factor (1— Ct E(t)) -1 appears, we expand it in a formal power series En (ct z(o)lL. As mentioned, since E is 0(A), all terms on the right hand side are of order 1, there is an open ball BR around E such that (R) (BR) C BR, and we can iterate the map in this ball. We have not proven the convergence of the iteration; to do this, one needs more detailed bounds, obtained in [46, 47] by an analysis involving a discrete RG flow (such bounds can, with a good deal more work, be given also using the continuous RG flow developed here). Uniqueness of the solution to (4.296) holds under weaker assumptions [45]. Once one can solve (4.296), one has a clear procedure for doing renormalization. Given the interaction, the dispersion relation e of the model, e.g. the tight-binding dispersion relation, and given any R > 1, one solves for E = E(R) = (45 (R) ,) — 1 (e). Then E + K(R) = e by construction of E, and one can put E into the propagator by the Gaussian shift formula

>L

,

PP dpce (0) = det(1 — CeK(R)) dpcE ( 0)e— f dP I7(P)K(R)()0() (4.297) (which follows directly from the definition of the Gaussian measure; see Appendix B). This automatically puts the counterterm K( R) into the action,

4.8

Conclusion

175

and therefore the expansion in A up to order R is now the renormalized expansion. One can interpret the procedure of solving (4.296) as calculating the interaction effects in two steps: first, one determines K( R), which is the part of the selfenergy that changes the Fermi surface. Going from e to E means changing from the Fermi surface of the free system to that of the interacting system. Once this is done, all other effects of the interaction can be calculated at the fixed Fermi surface { p : E(p) = 0}. It should be clear from the description of the solution of (4.296) that with the bounds we have, one can only show that the solution E = 45-1 (e) is C2 in A and in p. Thus a solution of (4.296) does not make sense in a formal power series setting, and even if the power series in A for K converges, the solution of (4.296) cannot be analytic in A (if it were, the infrared divergent unrenormalized expansion would converge, by the identity theorem).

4.8 Conclusion At the end of this long, and partly technical, discussion we review some of the arguments and results, to put them into perspective. The discussion in this chapter consisted of some rigorous bounds for the perturbative Green functions, together with an analysis of truncations of the flow that provide heuristic guidelines of how to understand the bounds. Needless to say, this leaves many open problems, a few of which we shall discuss briefly now.

4.8.1 Summary We have analyzed perturbation theory in the coupling constant A to all orders for the -skeleton Green functions, and also for the ones with properly subtracted two-legged insertions (i.e. with the appropriate boundary condition). The results show that in d > 2 spatial dimensions, and for Fermi surfaces that have strictly positive curvature, for all R > 1, the order < R renormalized selfenergy E(R) = limt_H„,E,R 1 Er (t)Ar exists and satisfies

'YR

(4.298)

for all lal < 2, and the order < R renormalized four-point function ir) = limt,,„ E rR_ I 14,7.(t)Ar exists and satisfies i 0 11 < Yi1:1)

(4.299)

with ,yR and YR independent of fl, provided the condition IA I log Ofo < const

(4.300)

176

4. The Fermi Surface Problem

holds. The higher Green functions are bounded in an L' norm. The detailed bounds also imply that if a single term in the RGE for the four-point function, the ladder term (i = 2), is left out, then the statements for the four-point function still hold, but without the restriction in (4.300), i.e. for Pti < const, uniformly in the temperature. This conclusion comes from the overlapping loop bounds; it applies to rather general Fermi surfaces because it holds whenever any kind of volume improvement is there (see [45] for more precise statements). One can get a heuristic insight in the behaviour of the full solution by the opposite truncation of the RGE where only the ladder term i = 2 is kept in the four-point function. The solution of the RG differential equation then describes a flow of a coupling constant that reproduces the familiar BCS formulas, e.g. the form of the critical temperature (4.205). Renormalization was done by introducing a counterterm function K that, in general, is momentum dependent. The justification for this requires the solution of a functional equation that relates the free and the interacting Fermi surface . This is a mathematically nontrivial problem even in finiteorder perturbation theory, and we have given only a hint at its solution

[46, 47]. We have not given any nonperturbative bounds here. Doing so is possible, at least in d = 2 spatial dimensions [55]. The nonperturbative analysis of these systems is the subject of ongoing research [56, 61] and goes beyond the scope of this introductory text. On the whole, it is expected that the above perturbative analysis is confirmed in the nonperturbative setting (but a proof of this is difficult). Also, some technical parts of it, in particular the overlapping loop method, are important for nonperturbative studies as well. The continuous RG method also works nonperturbatively for fermionic quantum field theory if the RGE is integrated [57, 63]; the continuous method may work without explicit integration as well [54, 53], but whether it does is at the moment an open problem. 4.8.2 A Fermi Liquid Criterion The results lead us to formulate a criterion for Fermi liquid behaviour at positive temperatures [53 ] . This is necessary because Fermi liquid (or non-Fermi liquid) behaviour is observed only above the critical temperature for superconductivity. Because of the Kohn-Luttinger effect, one may expect that every system satisfying the assumptions of Sect. 4.2.6 has such a superconducting transition (if the reflection symmetry is broken, the Cooper instability is suppressed and Fermi liquid behaviour can occur at zero temperature, as shown in [56]). It is not completely straightforward to formulate such a criterion that involves only equilibrium observables and that can distinguish between Fermi liquids and other states of many fermion systems (such as the Luttinger liquid behaviour that occurs in d = 1 [59, 601) because at positive temperatures, there is no discontinuity in the occupation number density which could serve

4.8 Conclusion

177

as a distinction (at T = 0 there isn't either, because the energy gap in the superconducting state removes the discontinuity). A criterion for Fermi liquid behaviour at positive temperature is as follows. 1. The renormalized perturbation expansion in A for the full Green functions and the counterterm function K converges in a region R. = {(A, 0) : I Al log 0 < const }. 2. The selfenergy is C2 in p, and its second derivative is uniformly bounded on compact subsets of R. 3. The counterterm function K is Ck° in p, with ko > d (where d is the spatial dimension), and there is a unique solution to (4.296). A slightly different, but equivalent, definition of Fermi liquid behaviour at positive temperature (referring to the skeleton functions instead of the full Green functions) was given in [53]. We conjecture that the above conditions can be verified in d = 2 dimensions for the jellium band relation and the Hubbard band relation, in the latter case for a filling that is sufficiently far away from half-filling, by a combination of existing techniques [45, 46, 47, 55]. The jellium model is much easier than the Hubbard model because in that case K is a constant, hence Condition 3 is trivially fulfilled. Condition 3 was proven in [45, 46, 47 1 in perturbation theory by a rather detailed singularity analysis [46] and an extension of the overlapping loop method [47]. Such an analysis is also possible using the RGDE, but not included here. Condition 3 is needed to invert the map i and to have certain summability properties of the propagator. While a proof that the above version of Fermi liquid behaviour occurs is, even for the jellium model, beyond what can be explained here, it is easy to explain why it is sensible to call systems fulfilling Conditions 1-3 Fermi liquids, and in particular, why the above definition allows to distinguish between Fermi and Luttinger liquids. This distinction is in the selfenergy. By Condition 2, one can do a first-order Taylor expansion in p 11(p), to get (writing 11(p) = (0, P(p))) —

EV)

= Po (ao E) (0 ,P (P)) + (P — P(P)) • VE( 0, P (P)) + E(p)

(4.30 1 )

from which one obtains a finite wave function renormalization

Z(p) = 1 + i(a0 E)(0,P(p))

(4.302)

and a finite correction to the Fermi velocity VE(p), and the Taylor remainder È(p) vanishes quadratically in the distance of the momentum p to its projection 11(p) on the Fermi surface. This property distinguishes Fermi liquids from other possible states of the many-fermion system, such as Luttinger liquids: In one dimension (where `Luttinger liquid behaviour' has been proven [58, 59, 601), already the first derivative of the selfenergy does not obey Condition 2. The geometric reason for this is that in one dimension, the Fermi "surface" consists only of two

178

4. The Fermi Surface Problem

points. There is no transversal direction and hence, of course, no curvature, and all the volume improvement effects are absent. Higher-dimensional systems with a flat Fermi surface also have singularities in the first derivative of the selfenergy, again because of absence of any volume improvement. Note that this distinction between the behaviours of the selfenergy in one and more than one dimension can only be made if )3 is allowed to vary; at fixed 0, the requirement that something is bounded independently of is trivial. This is the reason why a whole range of values of fi and À is included in the above definition. As discussed, the restriction (4.300) comes from the four-point function, in particular the ladder terms analyzed in Sect. 4.5.4. Thus this restriction is there to avoid the BCS instability: the temperature is above the critical temperature for superconductivity. Under the above definition, one can justify using renormalized perturbation theory: given a model with dispersion relation e, interaction 0, and coupling constant A, and at temperature T such that I Al logfl is small enough, solve the inversion equation (4.296) for E and change to E, whose zero set is the interacting Fermi surface, as described in Sect. 4.6. Because À is fixed, E is now fixed. By condition 1, and because À is fixed with log0 small enough, the renormalized expansion in À converges.

4.8.3 How the Curvature Sets a Scale Finally, we revisit the heuristic discussion about the validity of the ladder flow near to half-filling in the Hubbard model in the light of the above volume and power counting bounds. It was mentioned that an important role in the perturbative justification of the ladder approximation by the overlapping loop bound was played by the energy scale at which the flow is started, the 'initial' energy scale co. The reason for this is that the volume improvement is curvature dependent, and will lead to effects only below a certain scale, as follows. For the band relation E(p) = —tE cos ki — p, the curvature of the Fermi surface vanishes for p = O (half-filling). For small p, the size ro of the neighbourhood will therefore depend on p and vanish with p. The initial energy scale c o for the flow must be chosen < ro . The analysis of Sect. 4.6 showed that the i > 3 terms have an extra scaling factor QTE t , where the constant QT is proportional to 1/tc. Since tc p, we now see that the chemical potential sets a scale co , below which the ladders dominate, in a natural way: if cap > 1 is large, then the "improvement factor" QTct > 1, so there is no improvement at all, and thus no justification for restricting to the ladder flow. If co is chosen such that co /p < 1, the improvement holds for all t > 0, and then the ladders dominate the flow. The flow of the coupling constants in the various angular momentum sectors is then essentially independent, and one that started out negative will eventually diverge.

4.8 Conclusion

179

Thus the geometrical picture given by the volume bounds is simply that when Et /p > 1, the neighbourhood is so thick that curvature effects do not lead to any improvement; the scaling behaviour is similar to that of a onedimensional system. Only below scale p, the curvature becomes visible and its effects single out the ladders in the flow of the four-point function (see Fig. 4.6).

Fig. 4.6. A shell around the Fermi surface intersecting its translate, for the tightbinding dispersion relation near half-filling p ::---, O. The energy scale of the shell is above the scale set by p, therefore the shell is so thick that curvature effects do not improve power counting

There are a number of other effects that become important near to halffilling. The effective one-dimensionality suggested by the above considerations also implies that the selfenergy behaves rather differently above and below scale p, so that in particular a careful analysis of the wave-function renormalization is important for scales above p. It should be noted that above scale p, the parquet flow is also not justified as the leading contribution by the overlapping loop bounds. Finding out which approximation is tractable and gives the correct behaviour requires different arguments.

A. Appendix to Chapters 1-3

A.1 A Topology on the Ring of Formal Power Series In this appendix, we recall briefly some basic facts about formal power series. The ring of formal power series is the set F = 010 of sequences S = ( sk)k>o with addition defined in the obvious way and multiplication as k

(ssi)k =Est sik _i .

(A.1)

1= 0

F is a domain of integrity with unit element (1, 0, 0, ...) and the map 1: C F, a E-+ (a, 0, 0, ...) is an injection. The rule for the product is remembered easiest if one associates a formal expression S(z) = k>0 ZkS k to a formal variable z, to the sequence S. The lower degree of S E F is defined as w(S) = min{ k > 0 : sk 0 0} for S 0 0, and w(0) = oo. Then w(S + S') > ininlw(S), w(S 1 )}, and we can introduce a metric on F by the definition cl(S, S) = 0 and for S 0 S',

E

,

d(S,S') = 2 -w (s-s' ) .

(A.2)

With this, F becomes a complete metric space. We call a family (Sa ) OEE N of formal power series summable if the limit

n

lim

n—).co

E SI,

(A.3)

v=1

exists in F. Then a sequence (S„,),, E N is summable if and only if w(S) —i oc' as a oc. In particular, the substitution of one formal power series in another, S(S'(z)), is a well-defined formal power series if and only if cd(S') > 1.

A.2 Fourier Transformation Recall that

. N, so M =

t-- E 2N. The dual lattice to

r = eZ d I LZ d = { x E EZd

L

:

—

—< 2

xk


0 such that minf I K i (x) I

E 4) 2 , (751 ) 2 ll Ko•

(A.62)

(A.63)

190

A. Appendix to Chapters 1-3

Thus, for A > m and q E A(1A, aAcAA0(q)

(1.4) 2 2K0 K0 1 > A3 (0) 2 + mo 2> 3 A3.

Moreover, in the support of the integrand for

J(A),

— III > A— = A,

1g — A

5 lq

A

iPi < 4 A +

(A.65)

3A 2

(A.66)

> 0,

so, with L0 = miniK(x) : (i) 2 < x < COA

(A.64)

1 = (q — p) 2 + m 2 K( m 2

)

1,0 (A.67) A2'

4°

3 2+ mo 2 (0)

Inserting these bounds, we get A

AA)

>f

d3p L 0 K 0 p2 mo2 A2 3A3

4

f

d3q = k 1

f IrciP

A2 p2 + m02

tA)

A(4„1-)

9

(A.68)

8

with

= (-423r-)2KoL o 4) 3 — ( -9§- ) 3 ).

(A.69)

In the last integral, we use m 7 2 < A2 < (8p) 2 , i.e. , mo2 + p2 1

p2 + mo2 fig

A 8

Thus the lower bound in (3.104) holds with Qi = 5k2o •

(A.70)

B. Appendix to Chapter 4

BA Fermionic Fock Space We just give a brief summary here; for details, see [64]. If 7-t is the oneparticle Hilbert space (assumed as separable), the configuration space for an n-particle system is lin =1-1 0 ... 0 7-1 (tensor products are defined, e.g., in [65 ] ). The Fock space .F over 74 is defined as

00

T = 0 1-1 12

(B.1)

n=0

with 7-1 ° = C. elements of .F are sequences V) = (0(0) , 0(1) , ...) with 0°) E C, 0 1 ) E 71, etc. The infinite direct sum is a Hilbert space with the natural inner product (B.2) ((p I oF = 70°) + E((p(n) i on))-k.• n>i The set Y0 of finite vectors, which have only a finite number of 1/,(11 ) nonzero, is dense in T. Let P± be the linear operator defined by

1

P±(fi 0 • • • 0 fn) = 7 E (±1)hr ( 1 ) 0 • • • ® fw(n) n

(B.3)

• 71-Esn

where (+1) 7 = 1 and (-1) 7 = E(R)is the sign of the permutation r. The P± are projectors. The Bose Fock space is I.+ = RE T, the Fermi Fock space is .F_ = P__ T. For f E 7-1, the creation and annihilation operators a* (f) and a(f) are defined on .F0 as the linear operators satisfying a(f)0 (°) = 0, a* (f)(°) = f, and a(f)(fi 0 • • • 0 fn) =

AFL

(f

I

h)7.1 f2 0 • • • 0 fn

a*Cf)(flo•••ofn) = Vn+ifofio•••ofn.

(B.4)

The map f P,4 a* (f) is linear and the map f I-4 a(f) is antilinear. An elementary calculation shows that on 7tn, Ila(f )11 < AFL Ilf II and Ila*(f )1I

192

B. Appendix to Chapter 4

r\ri7— F 1 11f II. The bosonic and fermionic creation and annihilation operators are defined by

a(:) (f) = P±a( *) (f)P±.

(B.5)

From now on we focus on the fermionic case. The fermionic operators obey the canonical anticommutation relations (CAR) {a _ ( f), a_ (g)}

= { al (f), a!_(g)} = 0 {a_(f),a(g)} = 1 (f I g)7.1.

(B.6)

A consequence of the CAR is that the fermionic creation and annihilation operators are bounded,

11a(f)II= Ila*(f)11= If II -k ,

(B.7)

and thus extend uniquely to bounded operators defined on T and that, if (fcx)crEA is an ONB of 7-t, a ONB of T is given by the vectors 9 = (1, 0, 0, .. .) and HaEs a(f)Q, where S runs over all finite subsets of A. The physical interpretation is that fl is the vacuum, the state without any particles, and that the n-particle state n

H a(f)fl = -VW!P_(f al ®... 0

fa n )

(B.8)

kr=1

has n fermions occupying the states fai , ... , fan . The CAR imply in particular that a* (f) 2 = 0, i.e. that no state can be occupied twice. In the text, we only consider the Fermionic Fock space, so we drop the subscript and write a(f) = a_(f).

B.2 Calculus on Grassmann Algebras Let N be a finite set, without loss of generality Ar = {1, ... , N} (this is no loss of generality because the Grassmann Gaussian measure introduced below changes at most by a sign if the elements of Ar are permuted). The formal algebraic objects 01 , ... ON are called Grassma,nn variables if they obey

:= lkilAi + tAilPi = 0

(B.9)

for all i, j E A. The Grassmann algebra C { i, - - -, ON} generated by 0 1 , . ON consists of polynomials

g = b(g) + E E ci i ,...,in (g)Oi l .. . 0in n>i ii,...,inEA

(B.10)

where b,ci1 ,...,in (g) E C, with addition and multiplication defined as usual for polynomials and by taking account of the anticommutation rules for the generators 01 , ... , ON, namely, for any permutation ir,

B.2 Calculus on Grassmann Algebras • • '. Ikr(n) =

E(7)1141 " . Viin .

193

(B.11)

The coefficients c 1 ,...,j (g) in (B.10) are not uniquely fixed by g because the set of all monomials in the generators is not linearly independent by (B.11). However, the set B = {1} U { lPi, ...tpin I n > 1,i 1 , • • • , in E A, ii < i2 < . • . < in}

(B.12)

is a basis of the vector space C{0 1 , . .. , OAT}, so any element g can be written as

g

=

b(g)

+E

E

sii ,...,in (g)0i1 • • • iPin

n>1 ii0

since the sum on the right side involves only a finite number of terms. In the case b(g) 0, f (g) can be defined similarly by Taylor expansion around z = b(g) if the first N derivatives of f at this point exist. In analogy to the bosonic, that is, usual, differential and integral calculus, derivatives and integrals with respect to the Grassmann variables are given by special linear mappings. The derivative with respect to Ok, acting to the right, is the map

a

A

(B.19)

E(-1) , 16k,ifoil•••ipir_loil + 3.•• .1)b_n

(B.20)

490k

:

A

-4

determined by its action on elements of B,

490k

• • • Oin

1=1

and the requirement that it be linear. Similarly, the derivative with respect to IN, acting to the left, is the linear map 4--

a

alPk 473— IPT:1 Win

We

A A

E(_ir —i6k,i

0.21 • • •

(B.21)

For all i, k E A

(B.22) tNi 490k

aOk a'çbi

themselves generate another Grassmann algebra, and that the 4-a same holds for the —. (90 We have seen already when introducing Fock space that changes of basis in the one-particle space 1-1 correspond to linear transformations of the annihilation operators. Since the algebra of operators exists independently of the basis chosen in 1-1, such transformations do not change the Grassmann algebra generated by them. Similarly, the Grassmann algebra generated by so that the

195

B.2 Calculus on Grassmann Algebras

V = A(01 , ... opN)T is the same as that generated by = ON, • • • if A E GL(N, C). In particular, if 01, • • • ,I,bN and 771, • • • ,riN are Grassmann variables, then /Pi +7,1, ••• ON +riN generate the same Grassmann algebra as /Pi , • • • , IPN and 711,...,7w (This is a change of generators (0) 71 ) T 1- A(0) n) T (1 1) \ with A = 0 1) )•

Under a linear change of generators, = (01, • • •

with A

E

,ON) T 1-+ V = A(tPi, • • • ,ON) T ,

(B.23)

GL(N,C), the derivatives transform as 0 a = (A-1Y — IM' ,Mt

that is, for all a

E

(B.24)

A

a ,a ath

N

1--a.

= k=1

a'çbk

(B.25)

Contrary to the bosonic case, the integral with respect to the variable is defined to be identical to the derivative,

a

f do k g = —g 8114

Ok

Vg E A.

Multiple integrals are denoted by

f

d, .. . dOir, =

f dOi l f d 2 ... f (17,bir, ,

the most important case being the integral with respect to all generators (B.28)

which maps A-4 C. Lemma B.1. For all a E C,

f

diP1 F(01 — anl ) 02, • • . , N) — f di F(IPI, 02, • • • 2 ON) •

Proof. In every polynomial of the Grassmann algebra, such as F, the vari-

able 0 1 can appear only linearly by unchanged by addition of aiii .

nilpotency.

Thus the coefficient of

Vii

is Ill

From now on we assume that we have an even number N = 2L of generators which we label as /Pi, • • • '0/, and 6, • • • ,'Ç-b-L• Corollary B.2. Let A,B E MN(C). Then L

f 11

k=1

dtPkdOk f

L

OP + All, V) + BO

(B.30) k=1

196

B. Appendix to Chapter 4

B.3 Grassmann Gaussian Integrals For Q E ML(C) let (7,b

=i,j=1 E

I op)

(B.31)

and let DODO = d1 d1 • • • chbLd'IPL•

Lemma B.3.

f DODO

(B.32)

e - O-P-1 Qt1') = det Q.

Proof. Expanding the exponential, we have

I = f DI-PDO e-(;-61 Q1P) =

(1)P Eœ — f Dopo (7i p=0

PI

op)P.

(B.33)

(IT) I OOP is a monomial of degree p in and p in IT). Thus only p = L can contribute to the integral, and

f(o goLD,,T)Do

E 11 Qikim f DtpDo II t

Pikik .

(B.34)

k=1

k=1

The anticommutation rules for the Grassmann variables give

H 1,-/).0p20. =

L

L

k=1

k=1

( - 1) 1(.+1)12

k=1

=

H

-1) (L +1) /2Eil , , iL

Oik

, ...,3L

H

k=1

111Pk k=1

(B.35)

HI,T4Ok• k=1

Since

f chAr d0 r 7.1)r,1.1)r

15

1

L!

the Grassmann integral gives a factor (-1) L , and

E

Qikik

il,•...iL k=1

H Q w(k)(7(k) D 7r,crESL, 2 ME(47) k=1 -d

E E(p) H pESL,

(B.36)

E

Qk,p(k) .T! Le 1 = k=1 71. ES L

det Q. 1111

B.3 Grassmann Gaussian Integrals

197

Definition B.4. Let Q E GL(N,C). The Grassmann Gaussian measure with covariance C = Q-1 is the linear functional f i--+ (f)Q on the Grossmann algebra given by

,

L

1

TT (1

(f ) c2 = det Q f : LI

1-P dtPic e

f (IP 5 0).

(B.37)

k

We also write this as

(f)Q =

f

clitic(0)0f(0)0)-

(B.38)

If f depends on no other Grassmann variables, (f)c2 E C. But f may also depend on other Grassmann variables I?, in which case (f)Q is an element of the 'remaining' subalgebra generated by the 9. Remark B.5. Since f 1.-+ (f)Q is a linear map on finite-dimensional spaces, it is continuous in f. If f depends differentiably on a parameter A, derivatives with respect to A can be taken under the integral sign. Lemma B.6.

f clize (

5 0)e( 4 11,1))+0,T177) = e (771c77) .

(B.39)

Proof. Completing the square and using Corollary B.2, we get

f =

DOD/pe - (17'1Q0 )+01,,b)+(17, 177)

e Olen) f D,1-pDoe -(1:6-c T filc2(7,b-07))

(B.40)

= e (41c77) f NDIPe - OPIQO = eN cn) det Q.

• Lemma B.7. If fN -' (f)l-fil lfri , then = L_,I,Jco,...,L1401.,

(n cj =

E

(p 1 j (f)(_1)1 / 1(1 / 1 -1 ) 1 2 det Fj/ (C)

(B.41)

I ,J C{1,...,L }

1 1 1=1.1 1

where I'Ll(C) is the III x I/I matrix with elements (r.,(c)),,, = c i,, (and I = fi l , ... y ia J = 1:715 • • • f iLl, with i l < ... < iL and il < ... < iL). Remark B.8. The çoij(f) may be Grassmann-algebra valued, as long as they are independent of i/, and tp .

198

B. Appendix to Chapter 4

Remark B.9. Setting j = 0 can be done by putting Ilk = akxk, ak E R and Xk Grassmann variables, and letting ak -4 O. Since all anticommutators of Grassmann generators vanish, the ijk form a Grassmann algebra. Proof. By linearity of ( • )Q, it suffices to calculate (17/0.1 )Q . By expansion of IJI because only monomials the exponential we see that (i i tirl )c2 = 0 if I/I of the same degree L in and have a nonzero integral. Let I/I = I JI = m.

We write

in V-' 'V

Ti *e oltp)+(,--pi,) [H nt (— ,...,„i k )

=

(B.42)

77=o Exchanging the differentiation with respect to q at ii = 0 and the integral with respect to 0 is allowed since all operations are continuous and they commute. By Lemma B.6, k=1

H

(0 I 1P" c2 {

o 77.9, )

k=1

TT a e771)] eqjk

k=1 k=1

rm

77=0 L

M

HH

(— _

1=1 r=1

=

m

L

E

17=0

in

H(

II cirri

[k=1

ri,..•,r,n=i i=1

in

a7) .,.

in

=

II 7int] 77=0

1 =1

in

M

E H ci,„ [H ( ani ) H n E k=1 1.1 1 0, II • Il q is a norm on the Grassmann algebra oh}, and for all f and g E A, A = C{th,

Ilfg11,

(B.45)

11fIl q 11g11, •

Proof. It is obvious that II• 11 g is a norm on A. The product inequality holds because

E ih jgic

i lfg li q =

(B.46)

InK =0 J

and

E

IlfIl q 11911 g =

(B.47)

I ,J,K,L

•

Lemma B.13. (Gram's inequality) Let 1-1 be a separable Hilbert space, and for f = (f11 • • • fn) E lin and g = g71) E na let

(B.48)

(f, g) = det (((fi I gi))1 0, so all eigenvalues of 0 are nonnegative, anddet 0 = (F I F).F > O. If fi , ... , fn are linearly dependent, then, by (B.53), Q(u) = 0 for a nonzero u, so 0 has a zero eigenvalue. Then det 0 = 0 and (B.50) holds trivially. If fi , ... fn are linearly independent, Q(u) > 0 for all u 0, and the Gaussian integral

n f TT ......., du (det 0) -1 = 11

dft-ie _ ,1. Q(u)

27r

(B.54)

converges (here for u = a + ib, dudft = dadb). Since fi 0, we have P = (fi I fl) > 0 2 and (B.55) Q(u) = (p lui l 2 + ( du l + a g i ) + 0- (ii) where Q.-- (ii) -- E 1,i > 2 %O&M and a = a(E) = Enj=20 1 .7AL.7'- Thus (

Q(u)

4 + (ft)

= (Plul —
2

(-mot) n2_p __EBk,pop 1 P.

(B.120)

p>2

then

(B.121)

1 bpq111 iiBk,pli q < -1 with bp = aP/p!. Let Ar = {0, ... , n - 1}. Rn

=

E S U T=A( T#0

H (1 — t Ak)

H

kES

kET

Bk

=EE S u T=.1■1"

H(4,Ak) ri E Bk,r,„3". kET rk >2 ucskE,

(B.122)

TOO

Collecting powers of 0, we have AhL II (- n )

)

R71,1) =

E1

kEU

11

D. k,rk -I-)

(B.123)

kET

where E' denotes a sum over SUT =IV, with T 00, over U c S, and over sequences (rk)kET, rk > 2 for all k E T, such that

Erk +

in =P.

(B.124)

kET

We now take the norm and use the product inequality, and the estimates for iiAkiiq and 11Bk, r,",. With the notation Py = I r 1, we then have

B.5 Grassmann Integrals for Fock Space Traces E,

11Rn,plign
WI Tr qn b E/f1 kn- agn ) n 7.1. 2 rk '

G -

211

(B.125)

!CET -

By (B.124), lUi< p, so arl (lagn l iU1 < qn " (-n.

(B.126)

Inserting this, we can resum the rk and U and get, with b = Er>2 b r
0 is C(r, E(p)), where

5

E) =

-e-TE (1 — f p(E)) e _ T E f 0(E)

if T if T

>0 < 0.

(B.147)

Note that this obeys the antiperiodicity

C (r + 0, E) = -C(rE)

(B.148)

so that C is periodic with period 20. By (B.148) and because eiwn 0 = -1, the Fourier coefficients of C are O

2 en (E) = — f (IT C 20 -0 The Fourier series

converges for T

ei wnr

1 1 0 icon -E •

1 e -iWnT Cer ± 0, = 1---d icon - E 8 nEZ

OZ. Thus (4.68) follows by taking the limit T t 0 of (B.150).

B.6 Feynman Graph Expansions It is a natural question if there are any changes in the proofs of the connectedness theorems in Chap. 2 if we start from a fermionic theory instead of a bosonic one. In this section, we briefly discuss how the proofs given for the bosonic theory extend to fermions. We shall see that all essential formulas carry over unchanged, only that the variables are now Grossmann variables, and that the few formulas that do get changed get only trivial changes. In particular, we show that the proof that only connected graphs contribute to the effective action is unchanged even though the Grossmann algebra is noncommutative. In a nutshell, the reason for this is that the polymer activities

B.6 Feynman Graph Expansions

215

are elements of the even subalgebra, which is commutative, so that all results of Sect. 2.4 apply. The main problem in generalizing the previous representations to the fermionic case is that the fermionic algebra is noncommutative whereas we have assumed, e.g. in the polymer expansions, that the activities are in a commutative algebra over C. We now show that one can arrange everything such that only elements of the even subalgebra A2 [0, n1 occur, so that the assumption that everything commutes still holds. The only place where the fermion signs appear is at the very end, when the derivatives are taken and evaluated at 7./ = O. The definition of the fermionic effective action,

e a eff (0, c,

-

A v) .

f dttc,(0) e-w(d)+) ,

(B.151)

is formally identical to the one for bosonic fields, only that now the Grassmann integral is implied in (B.151). Similarly, (2.117) carries over, only that the v(mk) are now antisymmetric with respect to permutations, because no other partwould contribute, by the permutation antisymmetry of the Wick ordered polynomials. Thus we have CO

Geff(p,c,v)= EG(eifi(p,c,v)

Ai-

(B.152)

r=1

with Ge ff OA

C, V) = ET ( V,

...,

V, C, I,b)

(B.153)

and (see (2.121))

(B.154) P

f 11 d, 11 vrk)(xo

E Mi,...Mp

bEBO

k=1

a [ ll — E(q, C, CIAO] • am bE Bo n=0

In the product over derivatives, the order is from small to large k, and for every k, ik from 1 to mk, and

f

= L aA1 aP aA log ZFI

(B.155)

...

with ZF

=

f

P

CliiCi (0)

II (I. ±

(B.156)

k=1

(the A's are still ordinary complex variables, of course). Because of the simple algebraic nature of the Grassmann integral, there is not even any convergence question for this integral.

216

B. Appendix to Chapter 4

At this point we see that all factors are bilinear in the fermions, so all activities are in a commuting algebra. Expanding as in (2.126) and doing the Gaussian integrals, we get exactly the same formula as (2.127), only with ia replaced by ri. Thus all activitis commute because they are in the even subalgebra, and Theorem 2.17 applies, hence (2.128)42.139) hold, with ia replaced by ri. Thus the logarithm removes all disconnected graphs, exactly as in the bosonic case. The only notable difference to the bosonic case arises when one now does the derivatives. While Lemma 2.13 would carry over to fermions unchanged if the coefficients in an expansion in ri were C-valued, the extra 0-variables imply that the coefficients are actually in the Grassmann algebra (and in particular, they may have odd degree). The correct replacement of Lemma 2.13 is Lemma B.25. Let A and B be elements of the Grassmann algebra C{n},

that is, A=

E ani ,

B=

E bin/

(B.157)

where ai,bi E C, and let C(x) be the formal power series 00

C(x)

. E cdtr.

(B.158)

r=0

Then

a an

al, an

al, C((—,1,0) = [A( n) B(—)

[A(—) B(7 1) c((71,0))1

ail

71=0

]

(B.159)

where 21, an- is the derivative with respect to li, acting to the left. Moreover

a

a =-- [B(— [A(- ) A(71)]

071 ) B(7ù] 71=0

ail

.

(B.160)

71=0

Proof. The proof is an easy exercise in Grossmann algebra and is left to the • reader. The derivation of the graph rules is now easy. The place where the fermion signs arise is in the derivatives. Commuting them properly gives the usual fermionic sign factors. We leave these details to the reader. We finally note that for the case of QED-type or Yukawa-type or phonon theories, where one has boson and fermion fields with interaction term 0,-*, our results imply that all symmetry factors are equal to one.

B.7 The Thermodynamic Limit in Perturbation Theory

217

B.7 The Thermodynamic Limit in Perturbation Theory The propagator for the many-fermion system without cutoff is given in (4.64). Thus, for k = (w,k) E A* let 1

D(k) = i6) —

(B.161)

E(k) Xi ( €t 2 1 k") — E(k)12)

where i-.4) is defined in (4.65), xi is given in (4.71), and define C(k) similarly, with xi replaced by x2 = 1 — xi . Then C(k) + D(k) = (0 E(k)) -1 is independent of t. The functions Ct and Dt define operators lit (K,IC) and 6-`t (IC, IC) on the functions on F* in the same way as 6 defines C by (4.62). Denote iit = 4Dt, and let 11(A) = 1 if the event A is true and 0 otherwise. —

Proposition B.26. Dt is a C' function of t that vanishes identically t > log ".. If t IRe Cji > L the stated properties of Di follow. The t-derivative gives I Dt (k)1 = 2€T 2 go' — E(k) 1 lxi (€T2 )0 - E(k)I 2 ) i . Since xli (x) = 0 for s 41 (1,1), bt(k) 0 0 implies IQ < liiD E(k)I < Et which implies (B.163) and (B.162). (B.164) follows from these inequalities by • Lemma B.24, by (4.73), and by Dt = 1i°g('3 "12) ds 1)8 . —

The bounds (B.164) are crude because the restriction IE(k)I < et was replaced by IE(k)I 2r + 2, lim(n1.,L)-400 ar' L) (0) = /,„ (0) exists and is a bounded function on and 1,ç,12rr'L) (0) < KZ.. Let (41; 114 (t))m,r,n,,L be the solution to (B.165). Then, for all m and r, the function 14.1- ' 14 (t) = 0 if m> 2r + 2, and _a i(n,,L)/.‘ kt) = 0 Ot mr

for all t > log 2'

(B.168)

there are bounded functions Imr (t) : Pc:pm -4 C such that urn (nr ,L)--*oo

(t)

= imr (t).

(B.169)

-

Let Pmr be the polynomials defined recursively as

Pmf(x) = 6 1-r limr (0)1 ± EcT 1

f dkniri i!

(x)Pm2r2(x)

(B.170)

(in particular, the coefficients of Pm,. are independent of fl), then for all

nr ,L,O,t

1

4Infr'L)(t)10 < (€0,3)r -i Pmr

(LIVi log g;°) .

(B.171)

B.7 The Thermodynamic Limit in Perturbation Theory

219

Proof. Induction in r, with the statement of the Lemma as the inductive hypothesis. Let r = 1. Since ri > 1 and r2 > 1, the right hand side of the (t, = jr(nnir equation is zero, so " o .1-j) for all t. Thus the statement (

,

follows from the hypotheses on Let r > 2, and the statement hold for all r' < r. Let m > 2r + 2. Then ,L) (0) r 1 , m2, r2, i) contributes to the right hand = O. The five-tuple side only if r i + r2 = r, m 1 + m2 = m + 2i, i > 1, and by the inductive hypothesis, only if m i < 2r i + 2 and m2 < 2r2 + 2. Thus 4"4 (t) can be nonzero only if m = m i + m2 - 2i < m i +m2-2 < 2r i +2 + 2r2 + 2-2 =-- 2r+ 2. The integral appearing on the right hand side of (B.165) is a Riemann sum approximation to an integral over the (t, n„, L)-independent region s E [0, oc), kj E M x B. By the inductive hypothesis, the factors 4„r,:,L) have a limit (B.169) satisfying (B.171). Since for all t, .bt is a bounded C2 function, the same holds for the propagators (boundedness holds because f3 < oc). Thus the integrand converges pointwise, and it suffices to show that it is bounded by an integrable function to get (B.169) and (B.171) by an application of the dominated convergence theorem. Let a -= 41)1 log and let g be the function on [0, oc) x (M x B) 1-1 given by

2 (s < log 2?) f foe - s 1.2-1 Pmiri(a)07.1-1 Pm2r2(a)0

g(s,li) =

11 Ùf ds,

i i! (B.172)

8

< leoe-8 3 ) 1 (IE(kil < 260 e-83)

j=2 0

The integrand on the right hand side of (B.165) is bounded by g by Proposition B.26, Lemma B.24, and the inductive hypothesis (B.171). Because g vanishes identically for s > log 2?, it is integrable. By (B.163) and (B.164),

f

ds f

g(k)dk 2 .. . dki
6). Since the right hand side of (B.165) vanishes for t > log (B.168) holds. • As discussed, 47:1;'1') (0) is not the initial interaction because at t = 0, the (nr ,L) propagator Co 0 O. The /m r (0) are obtained from the original interaction by Wick ordering with respect to C and integrating over all fields with covariance Co. The existence of the limit (n„, L) -4 oc of the 47:fr'L) (0) can be shown by a similar inductive proof (it is not trivial because in the limit n, oc, the absolute value of the propagator is not summable). An alternative proof is in Appendix D of [46].

B. Appendix to Chapter 4

220

B.8 Volume Improvement Bounds In this Appendix, we prove the one-loop and two-loop volume bounds for the case E(p) = p2 1, that is, for the spherical Fermi surface. -

B.8.1 The One Loop Volume Bound -

Let

(19

T(E, q) = f (270d 11 (1E(7r(0, /9) + q)I e)

(B.174)

sd-I

Lemma B.28. Let E(p) = p2 - 1, q E Rd and q = 14. Then for all d> 2

and all

E
1. We change variables from 2vV1 - v2 , to get

T(E,


E and the integrand is identically zero. If lq 21 < 2E, then 14y2 + q 21 < e implies 4v2 2E, then 14v2 + q 21 < E implies 12.-Vri. < 21v1 < 1-Vrr-E. For Isl < x 11 < V2- x, so -

21y 1

-


and thus ((,o) Re, 0 can occur only in Case 2. In Case 2, ((,c)) 2 also occurs, and the proof in Cases 3 and 4 is similar (in fact easier). Thus we shall do only Case 2 in detail. We consider the contributions from the various regions given in the definition of separately by inserting the corresponding indicator function in the integral. The integral with 11 E (-14-, 2 is bounded by K(M)E as in Case 1. Let

k,

r

b))

Op)

ci

f

=

dçoisin (pl d-2

6

J1(2€ < (ço) < -L A, )

(ti9 )

(B.190)

. Since p > 3/4, this The restriction (cicr) < implies 2p(1 - cosy) < means that cp is near to zero. Changing variables to y = sin , we have OP) = X(v) = (( 1 - /3) 2 + 4Pv2 ) 1/2 2.Nrpv, and 1

cl < 2d

f

dv vd -2 (1 — v2 )

0 c1-2

For d > 3, we use 7 ; 7- _
0, -Va(u + a) -1 /2 < 1, so the boundary term is at most one. The same argument implies that the remaining integral is bounded by 1

1-1-cr

f du _ f du < log 2 + 21a1 fu±a f u E

(B.206)

e/2

V

Finally, we bound C4. If a v2 , so

< 4d+IEclia

dy V d-2


0 to get 8E? 1(v - -1)(y +

2 JIv -

2 4vIv - -Val,

(B.208)

that is, y must be in the interval iv - Val