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Phase Transitions and Critical P h e n o m e n a

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Phase Transitions and Critical P h e n o m e n a Volume 19

Edited by C. Domb

Department of Physics, Bar-Ilan University, Ramat-Gan, Israel

and

J. L. Lebowitz

Department of Mathematics and Physics, Rutgers University, New Brunswick, New Jersey, USA

ACADEMIC PRESS A HarcourtScienceand TechnologyCompany San Diego San Francisco NewYork London Sydney Tokyo

Boston

This book is printed on acid-free paper. Copyright 9 2001 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press A Harcourt Science and Technology Company Harcourt Place, 32 Jamestown Road, London NWl 7BY, UK http://www.academicpress.com Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com ISBN 0-12-220319-4

A catalogue record for this book is available from the British Library

Typeset by Focal image Ltd, London Printed and bound in Great Britain by MPG Books Ltd, Cornwall, UK 01 02 03 04 05 MP 9 8 7 6 5 4 3 2 1

Contributors G. M. SCHUTZ, Institut fiir Fesstk6perforschung, Forschungszentrum Jiilich, 52425 Jiilich, Germany K. J. WIESE, Fachbereich Physik, Universitiit GH Essen, 45117 Essen, Germany


General Preface This series of publications was first planned by Domb and Green in 1970. During the previous decade the research literature on phase transitions and critical phenomena had grown rapidly and, because of the interdisciplinary nature of the field, it was scattered among physical, chemical, mathematical and other journals. Much of this literature was of ephemeral value, and was rapidly rendered obsolete. However, a body of established results had accumulated, and the aim was to produce articles that would present a coherent account of all that was definitely known about phase transitions and critical phenomena, and that could serve as a standard reference, particularly for graduate students. During the early 1970s the renormalization group burst dramatically into the field, accompanied by an unprecedented growth in the research literature. Volume 6 of the series, published in 1976, attempted to deal with this new literature, maintaining the same principles as had guided the publication of previous volumes. The number of research publications has continued to grow steadily, and because of the great progress in explaining the properties of simple models, it has been possible to tackle more sophisticated models which would previously have been considered intractable. The ideas and techniques of critical phenomena have found new areas of application. After a break of a few years following the death of Mel Green, the series continued under the editorship of Domb and Lebowitz, Volumes 7 and 8 appearing in 1983, Volume 9 in 1984, Volume 10 in 1986, Volume 11 in 1987, Volume 12 in 1988, Volume 13 in 1989 and Volume 14 in 1991. The new volumes differed from the old in two new features. The average number of articles per volume was smaller, and articles were published as they were received without worrying too much about the uniformity of content of a particular volume. Both of these steps were designed to reduce the time lag between the receipt of the author's manuscript and its appearance in print. The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It is not longer an area of specialist interest, but has moved into a central place in

viii

General Preface

condensed matter studies. The editors feel that there is ample scope for the series to continue, but the major aim will remain to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments. CYRIL DOMB JOEL L. LEBOWITZ

Preface to Volume 19 Statistical mechanics provides a framework for describing how well-defined higher level patterns of organized behavior may result from the activity of a multitude of interacting lower level individual entities. The subject was developed for, and has had its greatest success so far in, relating macroscopic thermal phenomena to the microscopic dynamics of atoms and molecules. While some of these phenomena can be understood as the additive effects of the actions of individual atoms, e.g. the pressure exerted by a gas on the walls of its container, others are paradigms of emergent cooperative behavior. The latter have no direct counterpart in the properties or dynamics of the microscopic constituents considered in isolation. A paradigm of such phenomena are phase transitions, such as occur in the boiling or freezing of a liquid, where dramatic, essentially discontinuous, changes in structure and behavior of a macroscopic system are brought about by very small changes in the control parameters. The methods of statistical mechanics used to understand and predict these phenomena owe their success to the fact that even very crude modeling of the microscopic structure and dynamics of the atoms and molecules yields many essential features of their collective behavior. This is well established for equilibrium phase translations, where not only qualitative features, such as the basic similarity of the phase diagrams of different substances, but also quantitative ones, such as critical exponents are "universal". Less well understood, are emergent cooperative phenomena in nonequilibrium and in intrinsically spatially inhomogeneous equilibrium systems such as membranes. These are the subjects of the two review articles in this volume. They will surely be central topics of the statistical mechanics in the new century. To put these in context, let us remind the reader very briefly of the statistical mechanical formalism describing first order phase transitions in homogeneous equilibrium systems. On the macroscopic level such transitions are encoded in the phase diagram of the system. These phase diagrams can be very complicated but their essence is already present in the familiar, simplified two dimensional diagram for a one

x

Preface to Volume 19

component system like water or argon. This has axis marked by the temperature T and pressure p, and gives the decomposition of this thermodynamic parameter space into different regions: the blank regions generally correspond to parameter values in which there is a unique pure phase, gas, liquid, or solid, while the lines between these regions represent values of the parameters at which two pure phases can exist. At the triple point, the system can exist in any of three pure phases. In general, a macroscopic system with a given Hamiltonian is said to undergo or be at a first-order phase transition when the temperature and pressure, or more generally the temperature and chemical potentials, do not uniquely specify its homogeneous equilibrium state. The different properties of the pure phases coexisting at such a transition manifest themselves as discontinuities in certain observables, e.g., a discontinuity in the density as a function of temperature. On the other hand, when one moves between two points in the thermodynamic parameter space along a path which does not intersect any coexistence line the properties of the system change smoothly. A beautiful part of the statistical mechanics developed in the past century is the analysis of this macroscopic behavior in terms of Gibbs ensembles specified by the microscopic Hamiltonians. While the use of ensembles was anticipated by Boltzmann and independently discovered by Einstein, it was Gibbs who, by his brilliant systematic treatment of statistical ensembles, i.e. probability measures on the phase space, developed a useful elegant tool for relating, not only typical but also fluctuating behavior in equilibrium systems, to microscopic Hamiltonians. In a really remarkable way the formalism has survived essentially intact the transition to quantum mechanics. The key ingredient in connecting ensemble properties to observable equilibrium behavior in individual macroscopic systems is that the functions on the phase space of the system, with energy in some specified interval, are of a particular form. They are sums of functions, each of which depend only on the coordinates and momenta of a few elementary constituents, e.g. atoms. The values taken by such sum functions are essentially constant on the energy surface when the size of the system is large on the molecular scale. Thus the relevant collective properties of a macroscopic system are typical of points on the energy surface, i.e. the fraction of microstates for which some property, say the kinetic energy of particles contained in the left half (or some other portion) of the container is significantly different from its microcanonical average goes to zero as the size of the system increases. This constancy of macroscopic variables and consequent equivalence of equilibrium ensembles carries over also to the suitably defined empirical fluctuations in these variables. These grow typically like the square root of the number of microscopic variables involved whenever the system is in a pure phase, away from a critical point. In the vicinity of critical points certain fluctuations increase, i.e. they grow with an exponent greater than 1/2. These exponents depend on

Preface to Volume 19

xi

the dimensionality and some symmetry properties of the relevant microscopic variables but are otherwise universal, i.e. independent of the precise Hamiltonian. Such properties have not been proven for non-equilibrium systems. It is therefore very interesting and helpful that there are some non-trivial, exactly solvable examples of such models, which correspond to integrable quantum systems. While this correspondence is currently known only for one dimensional systems--it still offers many insights and even physical predictions, as explained in the article by Gunter M. Schlitz. The second article, by Kay J6g Wiese, deals with cooperative phenomena in a very interesting class of systems, polymerized tethered membranes. These are membranes with a fixed internal connectivity-somewhat analogous to polymers. Both equilibrium and dynamical properties are studied. CYRIL DOMB JOEL L. LEBOWITZ


Contents Contributors . . . . . . . . General Preface . . . . . . Preface to Volume 19 . . . Contents of Volumes 1-18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

v vii ix xv

1 Exactly Solvable Models for Many-Body Systems Far from Equilibrium G. M. 1 2 3 4 5 6 7 8 9 10

SCHUTZ

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Q u a n t u m Hamiltonian formalism for the master equation . . . . . . . . 17 Integrable stochastic processes . . . . . . . . . . . . . . . . . . . . . . 30 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Equivalences of stochastic processes . . . . . . . . . . . . . . . . . . . 72 The symmetric exclusion process . . . . . . . . . . . . . . . . . . . . . 79 Driven lattice gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Reaction-diffusion processes . . . . . . . . . . . . . . . . . . . . . . . 162 Free-fermion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Experimental realizations of integrable reaction--diffusion systems . . . 215 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Appendix A: The two-dimensional vertex model . . . . . . . . . . . . . . . 225 Appendix B: Universality of interface fluctuations . . . . . . . . . . . . . . 230 Appendix C: Exact solution for empty-interval probabilities in the A S E P with open boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 A p p e n d i x D: Frequently used notation . . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

xiv

Contents

2 Polymerized Membranes, a Review K. J. WIESE 1 Introduction and outline . . . . . . . . . . . . . . . . . . . . . . . . . . 256 2 Basic properties of m e m b r a n e s . . . . . . . . . . . . . . . . . . . . . . 261 3 Field-theoretical treatment of tethered m e m b r a n e s . . . . . . . . . . . . 280 4 S o m e useful tools and relation to p o l y m e r theory . . . . . . . . . . . . 306 5 Proof of perturbative renormalizability . . . . . . . . . . . . . . . . . . 319 6 Calculations at two-loop order . . . . . . . . . . . . . . . . . . . . . . 341 7 Extracting the physical informations: extrapolations . . . . . . . . . . . 348 8 Other critical exponents, stability of the fixed point and boundaries . . . 358 9 The tricritical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 10 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 11 D y n a m i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 12 Disorder and n o n c o n s e r v e d forces . . . . . . . . . . . . . . . . . . . . 389 13 N - c o l o u r e d m e m b r a n e s . . . . . . . . . . . . . . . . . . . . . . . . . . 409 14 Large orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 15 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Appendix A: Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . 452 Appendix B: List of symbols and notations used in the main text . . . . . . 454 Appendix C: Longitudinal and transversal projectors . . . . . . . . . . . . 455 Appendix D: Derivation of the R G equations . . . . . . . . . . . . . . . . . 456 Appendix E: Reparametrization invariance . . . . . . . . . . . . . . . . . . 459 Appendix F: Useful formulas . . . . . . . . . . . . . . . . . . . . . . . . . 460 Appendix G: Derivation of the Green function . . . . . . . . . . . . . . . . 461 Appendix H: Exercises with solutions . . . . . . . . . . . . . . . . . . . . 462 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Contents of Volumes 1-18 Contents of Volume I (Exact Results) t

Introductory Note on Phase Transitions and critical phenomena. C. N. YANG. Rigorous Results and Theorems. R. B. GRIFFITHS. Dilute Quantum Systems. J. GINIBRE. C* Algebra Approach to Phase Transitions. G. Emch. One Dimensional Models--Short Range Forces. C. J. THOMPSON. Two Dimensional Ising Models. H. N. V. TEMPERLEY. Transformation of Ising Models. I. SYOZI. Two Dimensional Ferroelectric Models. E. H. LIEB and F. Y. Wu. Contents of Volume 2 t

Thermodynamics. M. J. BUCKINGHAM. Equilibrium Scaling in Fluids and Magnets. M. VICENTINI-MISSONI. Surface Tension of Fluids. B. WIDOM. Surface and Size Effects in Lattice Models. P. G. WATSON. Exact Calculations on a Random lsing System. B. McCoY. Percolation and Cluster Size. J. W. ESSAM. Melting and Statistical Geometry of Simple Liquids. R. COLLINS. Lattice Gas Theories of Melting L. K. RUNNELS. Closed Form Approximations for Lattice Systems. D. M. BURLEY. Critical Properties of the Spherical Model. G. S. JOYCE. Kinetics of Ising Models. K. KAWASAKI. Contents of Volume 3 (Series Expansions for Lattice Models) t

Graph Theory and Embeddings. C. DOMB. Computer Enumerations. J. L. MARTIN. Linked Cluster Expansions. M. WORTIS. t Out of print.

xvi

Contents of Volumes 1-18

Asymptotic Analysis of Coefficients. D. S. GAUNT and A. J. GUTTMAN. Ising Model. C. DOMB Heisenberg Model. G. A. BAKER, G. S. RUSHBROOKE and P. W. WOOD. Classical Vector Models. H. E. STANLEY. Ferroelectric Models. J. E NAGLE. X - Y Model. D. D. BETTS. Contents of Volume 4 t

Theory of Correlations in the Critical Region. M. E. FISHER and D. JASNOW. Contents of Volume 5a t

Scaling, Universality and Operator Algebras. Leo E KADANOFF. Generalized Landau Theories. Marshall Luban. Neutron Scattering and Spatial Correlation near the Critical Point. JENS ALS-NIELSEN. Mode Coupling and Critical Dynamics. KYOZI KAWASAKI. Contents of Volume 5b t

Monte Carlo Investigations of Phase Transitions and Critical Phenomena. K. BINDER. Systems with Weak Long-Range Potentials. E C. HEMMER and J. L. LEBOWITZ. Correlation Functions and Their Generating Functionals: General Relations with Applications to the Theory of Fluids. G. STELE. Heisenberg Ferromagnet in the Green's Function Approximation. R. A. TAHIR-KHELI. Thermal Measurements and Critical Phenomena in Liquids. A. V. VORONEE. Contents of Volume 6 (The Renormalization Group and its Applications) t

Introduction. K. G. WILSON. The Critical State, General Aspects. E J. WEGNER. Field Theoretical Approach. E. BREZIN, J. C. LE GUIELOU and J. ZINN-JIJSTIN. The l/n Expansion. S. M A. The e-Expansion and Equation of State in Isotropic Systems. D. J. WALLACE. Universal Critical Behaviour. A AHARONY. Renormalization: Ising-like Spin Systems. TH. NIEMEUER and J. M. J. VAN LEEUWEN Renormalization Group Approach. C. DI CASTRO and G. JONA-LASINIO.

Contents of Volumes 1-18

xvii

Contents of Volume 7 t

Defect-Mediated Phase Transitions. D. R. NELSON. Conformational Phase Transitions in a Macromolecule: Exactly Solvable Models. E W. WIEGEL. Dilute Magnetism. R. B. STINCHCOMBE. Contents of Volume 8

Critical Behaviour at Surfaces. K. BINDER. Finite-Size Scaling. M. N. BARBER. The Dynamics of First Order Phase Transitions. J. D. GUNTON, M. SAN MIGUEL and P. S. SAHNI. Contents of Volume 9 t

Theory of Tricritical Points. I. D. LAWRIE and S. SARBACH. Multicritical Points in Fluid Mixtures: Experimental Studies. C. M. KNOBLER and R. L. SCOTT. Critical Point Statistical Mechanics and Quantum Field Theory. G. A. BAKER, JR. Contents of Volume I0

Surface Structures and Phase Transitions--Exact Results. D. B. ABRAHAM. Field-Theoretic Approach to Critical Behaviour at Surfaces. H. W. DIEHL. Renormalization Group Theory of Interfaces. D. JASNOW. Contents of Volume II

Coulomb Gas Formulation of Two-Dimensional Phase Transitions. B. NIENHUIS. Conformal Invariance. J. L. C ARDY. Low-Temperature Properties of Classical Lattice Systems: Phase Transitions and Phase Diagrams. J. SLAWNY. Contents of Volume 12'

Wetting Phenomena. S. DIETRICH. The Domain Wall Theory of Two-Dimensional Commensurate-Incommensurate Phase Transitions. M. DEN NUS. The Growth of Fractal Aggregates and their Fractal Measures. P. MEAKIN.

XVlII o . .

Contents of Volumes 1 - 1 8

Contents of Volume 13

Asymptotic Analysis of Power-Series Expansions. A. J. GUTTMANN. Dimer Models on Anisotropic Lattices. J. F. NAGLE,. S. O. YOKOI and S. M. BHATTACHARJEE. Contents of Volume 14

Universal Critical-Point Amplitude Relations. V. PRIVMAN, P. C. HOHENBERG and A. AHARONY. The Behaviour of Interfaces in Ordered and Disordered Systems. G. FORGACS, R. LIPOWSKY and TH. M. NIEUWENHUIZEN. Contents of Volume 15

Spatially Modulated Structures in Systems with Competing Interactions. W. SELKE. The Large-n Limit in Statistical Mechanics and the Spectral Theory of Disordered Systems. A. M. KHORUNZHY, B. A. KHORUZHENKO, L. A. PASTUR and M. V. SHCHERBINA. Contents of Volume 16

Self-Assembling Amphiphilic Systems. G. GOMPPER and M. SCHICK. Contents of Volume 17

Statistical Mechanics of Driven Diffusive Systems. B. SCHMITTMANN and R. K. P. ZIA. Contents of Volume 18

The Random Geometry of Equlibrium Phase. H.-O. GEORGII, O. HAGGSTROM and C. MAES. Exact Combinatorial Algorithms: Ground States of Disordered Systems. M. J. ALAVA, P. M. DUXBURY, C. F. MOUKARZEL and H. RIEGER.

Exactly Solvable Models for Many-Body Systems Far from

Equilibrium G. M. SchQtz Institut ffir Festk6rperforschung, Forschungszentrum JiJlich, 52425 JiJlich, Germany 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1

Stochastic dynamics of interacting particle systems

1.2

Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

............

1.3

Polymers and traffic flow: some notes about modelling

1.4

Outline

..........

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Quantum Hamiltonian formalism for the master equation

............

3 3 4 6 14 17

2.1

The master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2

Expectation values

22

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Many-body systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.4

Nonstochastic generators

..........................

28

3 lntegrable stochastic processes

..........................

30

3.1

The lsing and Heisenberg spin models . . . . . . . . . . . . . . . . . . .

32

3.2

Bethe ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.3

Quantum systems in disguise: some stochastic processes

41

3.4

Algebraic properties of integrable models

.........

.................

4 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 53

4.1

The infinite-time limit . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

4.2

Late-time behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.3

Separation of time scales

69

..........................

5 Equivalences of stochastic processes 5.1

.......................

Similarity transformations revisited . . . . . . . . . . . . . . . . . . . . .

PHASE TRANSITIONS VOLUME 19 ISBN 0-12-220319-4

72 72

Copyright 9 2001 Academic Press Limited All rights of reproduction in any form reserved

G. M. SchOtz 5.2

Enantiodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

First-passage-time and persistence probabilities

74 ..............

75

6 The symmetric exclusion process . . . . . . . . . . . . . . . . . . . . . . . . .

79

6.1

S U ( 2 ) - s y m m e t r y and stationary states . . . . . . . . . . . . . . . . . . .

79

6.2

Nonequilibrium behaviour

80

6.3

First-passage-time distributions . . . . . . . . . . . . . . . . . . . . . . .

6.4

Bethe ansatz solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

6.5

Algebraic formulation and solution . . . . . . . . . . . . . . . . . . . . .

88

.........................

83

7 Driven lattice gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

7.1

The asymmetric exclusion process . . . . . . . . . . . . . . . . . . . . .

104

7.2

TASEP with open boundaries . . . . . . . . . . . . . . . . . . . . . . . .

125

7.3

More on the origin of domain-wall physics . . . . . . . . . . . . . . . . .

136

7.4

Theory of boundary-induced phase transitions . . . . . . . . . . . . . . .

143

7.5

Traffic flow models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

154

8 Reaction--diffusion processes . . . . . . . . . . . . . . . . . . . . . . . . . . .

162

8.1

Enantiodromy relations . . . . . . . . . . . . . . . . . . . . . . . . . . .

162

8.2

Decoupling of the equations of motion . . . . . . . . . . . . . . . . . . .

164

8.3

Field-induced density oscillations

167

8.4

Field-driven phase transitions . . . . . . . . . . . . . . . . . . . . . . . .

.....................

169

9 Free-fermion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

9.1

The J o r d a n - W i g n e r transformation . . . . . . . . . . . . . . . . . . . . .

9.2

Stochasticity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .

178

9.3

Equivalences

181

177

................................

9.4

Stationary and spectral properties . . . . . . . . . . . . . . . . . . . . . .

191

9.5

Diffusion-limited pair annihilation . . . . . . . . . . . . . . . . . . . . .

193

9.6

Open boundaries

211

..............................

10 Experimental realizations of integrable reaction-diffusion systems

.......

215

10.1 Dynamics of entangled DNA . . . . . . . . . . . . . . . . . . . . . . . .

215

10.2 Kinetics of biopolymerization

220

.......................

10.3 Exciton dynamics on polymer chains . . . . . . . . . . . . . . . . . . . .

222

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223

Appendix A: The two-dimensional vertex model . . . . . . . . . . . . . . . . . . .

225

Appendix B: Universality of interface fluctuations . . . . . . . . . . . . . . . . . .

230

Appendix C: Exact solution for empty-interval probabilities in the A S E P with open boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237

Appendix D: Frequently used notation . . . . . . . . . . . . . . . . . . . . . . . .

239

D. 1 Single-site basis vectors and Pauli matrices D.2

Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D.3

Other notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

................

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 239 241 242

1 Exactly solvable models for many-body systems 1

Introduction

1.1 Stochastic dynamics of interacting particle systems Many complex systems of interacting particles that one encounters in nature behave on a phenomenological level in some random fashion. Therefore the theoretical treatment of these systems has to employ statistical approaches. Examples come from areas as diverse as the growth of surfaces (involving atoms or molecules) or the growth of biological systems (involving macroscopic cells), spin-relaxation dynamics, reaction-diffusion processes or, reaching into the sphere of sociological behaviour, the study of traffic flow. There is a general framework for the statistical description of equilibrium systems, but equally general concepts for systems far from thermal equilibrium are lacking. Hence one usually investigates specific model systems, hoping to gain insight into either the general behaviour of classes of systems or into the specific properties of the system under investigation. Indeed, compared to 'completed' classical theories such as electromagnetism or thermodynamics the current state of nonequilibrium statistical mechanics may be seen as a treasure of accumulated knowledge, but with relatively little profound understanding. Yet, some structure has emerged in the recent past, most notably in the concept of universality. It provides a theoretical framework for the observation that often quantities such as critical exponents or certain amplitude ratios do not depend on the specific details of the interactions between the basic constituents of the system. Intimately related is the idea of scaling, expressing the self-similarity of a system if observed on different length scales. These notions apply not only to equilibrium systems, but also to nonequilibrium behaviour of microscopically very different random processes. Naturally, the discovery of universality and other concepts has largely come from the study of specific systems and of simple models. These possess only those mechanisms that are deemed essential for the understanding of what one observes in real complex systems. Examples of nonequilibrium random processes include driven lattice gases (Spohn, 1991; Schmittmann and Zia, 1995) and reaction--diffusion mechanisms (Privman, 1997; Mattis and Glasser, 1998). They play an important role in the theoretical understanding not only of chemical systems and purely diffusive physical systems. Such models are able, through various mappings and different physical interpretations of the observables, to describe a wide variety of phenomena in physics and beyond. Thus they shed light on the mechanisms leading to universality in particle systems with short-ranged interactions, the emergence of simple collective behaviour which allows for a description of the many-body dynamics in terms of a few relevant variables, and other genetic features of systems far from thermal equilibrium.

4

G . M . SchLitz

Usually even simple models are not amenable to exact mathematical analysis. However, it has long been known that there are classes of nontrivial one- and two-dimensional equilibrium statistical mechanical models which can be solved exactly. These have considerably advanced our understanding of critical phenomena in general and of the physics of low-dimensional systems in particular (Baxter, 1982). Recent work has shown that by using a different interpretation of the variables such models may describe nonequilibrium behaviour as well. It is the aim of this work to provide an introduction into exactly solvable nonequilibrium models and to survey some of the insights that have been achieved through the detailed understanding that exact analysis has made possible.

1.2 Integrability Randomness - which may result from effectively stochastic forces or which may be intrinsic in the underlying microscopic t h e o r y - leads to the description of observables in terms of random variables and expectation values (Feller, 1950; van Kampen, 1981; Liggett, 1985; Spohn, 1991). The problem posed by the random behaviour of nonequilibrium systems is to develop tools beyond classical and quantum thermodynamics which allow for a theoretical investigation of these quantities. The oldest approach to the treatment of reaction-diffusion systems is to formulate rate equations for the reactants in a mean-field approximation. One ignores correlations between particles and often obtains reasonable results by invoking the old law of mass action: the rate of reaction of two species of particles is proportional to the product of their concentrations. However, particularly in lowdimensional systems, mean-field methods tend to be inadequate due to inefficient diffusive mixing. Moreover, in one dimension even short-ranged repulsive interactions represent obstacles seriously blocking the diffusive motion. As a result, large fluctuations persist and rate equation or other mean-field approaches fail (Schmittmann and Zia, 1995; Privman, 1997). This shortcoming was realized quite long ago and to some extent accounted for in Smoluchowski's theory of diffusion-limited reactions (von Smoluchowski, 1917). This correlationimproved mean-field theory is successful for many problems of interest, but both verification of the assumptions made in this theory and other still untractable problems involving fluctuations require the application of more sophisticated techniques. Progress may be achieved by adding a suitably chosen noise function to an otherwise deterministic differential equation as in the Langevin approach, or by a Fokker-Planck description, or through the formulation of the stochastic dynamics in terms of a master equation (see, e.g. van Kampen (1981) for these methods). In this type of modelling of a real system usually three approximations

1 Exactly solvable models for many-body systems

are made: the first approximation consists, as in the rate equation approach, in the identification of a few coarse-grained observables such as particle density, magnetization etc. with an effective interaction between these quantities. The second approximation concerns the mathematical prescription of the nature of the random forces which leads to the full dynamical equation describing the system. Solving these equations is a formidable task and therefore usually a third approximation is necessary for the solution of these equations. For instance, in recent years, Monte Carlo simulations on increasingly powerful computers have become a widely applicable numerical technique. Moreover, in the context of critical phenomena, the renormalization group has emerged as an extremely fruitful approach in the study of stochastic processes. Really e x a c t solutions of the dynamical equations for complex systems are comparatively rare. Besides some isolated exact results for various reaction--diffusion systems derived in the past, the only general framework which can produce exact and rigorous results has traditionally been the mathematical treatment using the tools of probability theory (Feller, 1950; van Kampen, 1981; Liggett, 1985, 1999; Spohn, 1991; Kipnis and Landim, 1999). The past few years have seen an exciting new development which has led to a series of remarkable exact solutions for the stochastic dynamics of interacting particle systems and also to an understanding of the mathematical structure underlying some of the already existing exact, numerical and renormalization group results. At the heart of this development is the close relationship between the Markov generator of the stochastic time evolution in the master equation approach on the one hand and the Hamiltonians for quantum spin systems (or the transfer matrices of statistical mechanics models respectively) on the other. The master equation for the probability distribution of a many-body system is a linear equation of a form similar to the quantum mechanical Schrrdinger equation. It can be written as a vector equation with a time-translation operator T (for discrete time evolution) or H (for continuous time evolution) acting on a many-particle Fock space (Kadanoff and Swift, 1968; Doi, 1976; Grassberger and Scheunert, 1980; Sandow and Trimper, 1993). The new insight is the somewhat surprising observation that for some of the most interesting interacting particle systems the time-translation operators, T or H respectively, turn out to be the transfer matrix or quantum Hamiltonian respectively of well-known equilibrium statistical mechanics models. Moreover, in some important cases, these models are integrable, i.e., have an infinite set of conserved charges like six- or eight-vertex models (Baxter, 1982). It was recognized in the early 1990s that in this way the toolbox of many-body quantum mechanics becomes available for the study of equilibrium and nonequilibrium stochastic processes. Typical results which one obtains using free-fermion techniques, the Bethe ansatz and related algebraic methods, or global symmetries and similarity transformations include firstly stationary properties of the process.

6

G . M . SchOtz

Thus one can study a variety of phenomena including phase transitions with divergent length scales in one-dimensional nonequilibrium systems. Going further, one may investigate spectral properties of the time evolution operator which give relaxation times and exact dynamical exponents. In some cases explicit expressions for time-dependent correlation functions can be found and one obtains detailed information on the collective behaviour of the particle system. This mapping to quantum spin systems applies to processes where each lattice site can be occupied by only a finite number of particles, i.e. where each lattice site can be found in a finite number n of distinct states. The physical origin of this restriction may be hard-core constraints or fast on-site annihilation processes. Examples describing lattice diffusion of particles with on-site interaction, combined with a chemical annihilation-creation reaction, include the Hamiltonians of the anisotropic transverse X Y model (Felderhof and Suzuki, 1971; Siggia, 1977), of the Heisenberg ferromagnet (Alexander and Holstein, 1978; Dieterich et al., 1980; Gwa and Spohn, 1992a) and higher-spin analogues (Alcaraz and Rittenberg, 1993), or the transfer matrices of vertex models (Kandel et al., 1990; Schiitz, 1993a). Integrability gives rise to what we want to call integrable stochastic processes. By exploiting this property this review attempts to expose the unified mathematical framework underlying the exact treatment of these systems and to provide insight into the role of inefficient diffusive mixing for the kinetics of diffusion-limited chemical reactions, in the dynamics of shocks and in other fundamental mechanisms which determine the behaviour of lowdimensional systems far from thermal equilibrium.

1.3

Polymers and traffic flow: some notes about modelling

While our main concern is generic nonequilibrium behaviour of interacting stochastic particle systems (as opposed to the specific properties of a given system) such a discussion must not rest on the study of abstract models alone, but has to retain a close relationship to actual physical systems encountered in nature. So first the general approach to modelling complex systems taken here should be made clear. Obviously, in an experimental investigation of a many-body system one does not wish to explore the individual motion of each particle. Such a huge amount of data can neither be gathered nor processed. As pointed out above, one tries instead to identify a few characteristic macroscopic quantities such as magnetization, density or current and measures how they depend on other quantities which can be controlled in an experiment. Implicit in this description of a system with many degrees of freedom in terms of an effective system characterized by just a few variables is the belief that there are some 'simple' basic mechanisms which determine the mutual dependence of these quantities. Inevitably such a reduction leads to inaccuracies in the


description. However, considerable progress has been made if with a simple model an order-of-magnitude agreement of predicted and experimental data can be achieved. Guided by the experimental results one may then build on this rough basic understanding to include other mechanisms in the model. Thus a prediction of experimental data with, say, a 10% accuracy might become possible. In this way one can continue to identify further subleading contributions to the dominant behaviour and proceed from qualitative to detailed quantitative understanding. If some modification of the model does not significantly change the theoretical prediction, then one has learned that the corresponding mechanism is irrelevant. Moreover, if some result like, e.g. a critical exponent does not change at all, then this observation is a hint at universality. On the other hand, if a supposedly small additional change in the model leads to a marked decrease in accuracy in the prediction, then this would point to a misconception of what had previously been identified as a leading mechanism. The ultimate goal of such modelling is the development of a picture of reality which provides an understanding of how the collective behaviour of the single components of a complex system leads to the emergence of simple macroscopic mechanisms. This onion-like picture of m o d e l l i n g - a core which provides a first basic understanding of the dominant behaviour, coated with subsequent layers of decreasing significance - represents the basic strategy adopted here. From the considerations above it should have become clear that any model can and should capture only certain aspects of a real system. For example, in the modelling of traffic flow it is entirely irrelevant how the intention of a driver to accelerate his or her car is technically transmitted to an increased number of revolutions per second of the wheels on the road. Surely this is not to say that this question is not of importance in general, but it is not of any relevance for the questions that one asks in traffic flow modelling. Models are always designed to answer specific questions asked by a specific set of people. If a model was not just a simplifying picture of reality but as complex as reality itself, then no progress of understanding would have been achieved. This strategy indeed suggests the use of integrable models as a starting point for understanding certain aspects of physical reality. As shall become increasingly clear in the treatment of integrable models below, the reduction of the collective many-body dynamics to a problem described by just a few effective variables may turn out to be not an uncontrolled approximation (which one usually has to hope to be justified) but the integrability and associated symmetries turn this reduction into an exact relationship and rigorous analysis of the many-body dynamics becomes possible. Moreover, in some instances integrability allows one to obtain results even where such a reduction completely fails and the full manybody problem has to be solved. Thus integrable models provide a test laboratory for investigating the general concepts outlined above.

8

G.M. SchQtz

A severe constraint to the application of integrable models to experiments seems to be the fact that they are all one-dimensional. However, it turns out that for many different problems (e.g. involving polymers or traffic flow), a one-dimensional description of the stochastic dynamics is appropriate. Furthermore, there are systems where only the projection of the system onto one space coordinate is of actual experimental interest. In these cases integrable reaction-diffusion processes can and do play an important role in the theoretical understanding of the underlying physical mechanisms. In this spirit we can now try to get some idea how simple models may capture essential features of very different physical systems.

1.3.1 Reptation A fundamental topic of soft matter physics concerns the motion of single polymer chains in a random environment of other polymers. If this environment is made up of long polymers and is sufficiently dense then they will form an entanglement network through which polymer chains may diffuse. Such a situation is, e.g. encountered in gel electrophoresis for the separation of a mixture of DNA strands of different length. A gel is a network-like structure of entangled polymers, and the DNA is a long polymer itself which wriggles through the pores of the network spanned by the gel strands. This motion can be understood by the concepts of the confining tube (Edwards, 1967) and of reptation (de Gennes, 1971) which derive from the topological constraints of the entanglement. The confining tube is a hypothetical object which may be considered to be the sequence of pores within the gel which the DNA occupies. Hence the shape of the tube is determined by the contour of the DNA, but coarse-grained on the level of the average pore size of the gel matrix. In the simplest approximation of the dynamics one assumes that the bulk of the DNA can only move within the tube. The topological constraints of the surrounding gel entanglement strongly suppresses any transverse motion out of the tube into neighbouring pores. Instead, if a DNA segment is not fully stretched in a given pore, then some of the stored length may move to a neighbouring pore along the tube. Only the end segments of the DNA can move freely to arbitrary neighbouring pores. As a result, on small time scales the tube can change its shape only at its ends. The bulk of the tube retains its present shape for a long time. In analogy to the motion of a snake, this mechanism is called reptation (Fig. 1). This picture is very simple and the diffusive motion of polymer segments (often called 'defects') within the tube can be modelled by a one-dimensional lattice gas in the following way: we consider the network as a disordered structure made up of distinct neighbouring pores. We discretize the DNA into L


Fig. 1 Reptation of an entangled polymer in the confining tube and mapping to the symmetric exclusion process. Polymer segments ('reptons') 1. . . . . L -- 10 with a size of the mean entanglement distance ~ move diffusively to neighbouring pores. Segments connecting two consecutive 'pores' of the surrounding network correspond to particles (full circles), segments fully contained in a pore correspond to vacancies. Diffusion within the tube amounts to particle-hole exchange. The motion of the end reptons in and out of the tube respectively correspond to annihilation and creation respectively of a particle.

consecutive unit segments of the mean pore size. Hence any such segment will typically be either fully contained within a pore (in which case it corresponds to a 'defect') or it extends from one pore to the next along the tube. We shall refer to such segments as 'particles', whereas we consider the defects as 'holes'. These consecutive particles and holes represent the conformation of the polymer chain, but may also be interpreted as a configuration of a one-dimensional lattice gas of L sites where each lattice site can be occupied by at most one particle. The total number N of particles gives the tube length in units of the mean pore size. Diffusion of defects corresponds to particle-hole exchange. We assume this motion to be a random process similar to lattice Brownian motion, but with an exclusion interaction. Particles on site k attempt to move to neighbouring lattice sites k + 1 after a random waiting time. If the site which has been chosen (randomly with equal probability) is empty, the move succeeds. Otherwise the move is rejected. As in lattice Brownian motion we assume the random time to have an exponential distribution with mean r0, corresponding to a hopping rate l/r0. Therefore, after a time r0 the probability that the particle (defect) is still in the same pore has reduced to l/e.

10

G.M. SchQtz

This lattice gas model, which is one-dimensional even though it describes a three dimensional system, is known in the probabilistic literature as the symmetric simple exclusion process (Spitzer, 1970) (SSEP). To account for the additional degrees of freedom at the boundary one must allow for annihilation and creation of particles with suitably chosen rates at the boundaries. Thus one obtains the SSEP with open boundaries where the lattice gas can exchange particles with imaginary exterior reservoirs of some fixed density. The generator which appears in the master equation for this process is the quantum Hamiltonian for a Heisenberg ferromagnet (Alexander and Holstein, 1978). Many experiments have been performed verifying the reptation picture by indirect means. However, only some years ago K~is et al. (1994) and Perkins et al. (1994) were able to directly monitor the motion of single entangled polymers (actin filaments and DNA strands respectively) by using fluorescence microscopy. Perkins et al. recorded the relaxation of an initially stretched DNA strand to its equilibrium conformation with a video camera. The experimentally accessible quantity is the time-dependent, relaxing tube length which allows for a quantitative comparison of experimental data with theoretical predictions from continuum reptation theory and from the lattice gas model. As will be shown in Section 10, by using the results of Section 6 on the particle number relaxation in the exclusion process the agreement is very good, thus confirming the simple picture of reptation summarized above. The dominant mechanism of motion is indeed the diffusion of stored length along the confining tube.

1.3.2

Kinetics o f biopolymerization

Protein synthesis is a rather complex process involving a complicated interplay of many different agents. In order to get an understanding of this important process many simplifying models have been developed, usually focusing on certain aspects of the whole process (von Heijne et al., 1987). (MacDonald et al., 1968; MacDonald and Gibbs, 1969) studied the kinetics of biopolymerization on nucleic acid templates with a lattice gas model. The mechanism they try to describe is (in a very simplified manner) the following: ribosomes attach to the beginning of a messenger-RNA chain and 'read' the genetic information which is encoded in triplets of base pairs, called codons, by moving along the m-RNA, t Each time a unit of information is being read, a monomer (some amino acid) determined by the genetic information is added to a part of a biopolymer (e.g. haemoglobin) which is attached to the ribosome. After having added the monomer the ribosome moves one triplet further and reads again. So in each reading step the biopolymer tThe m-RNA is a long molecule made up of such consecutive triplets.


11

grows in length by one monomer and is thus synthesized. The ribosomes are much bigger than the triplets on the m-RNA; they cover 20--30 of such triplets. Therefore neighbouring ribosomes sitting at the same time on the m-RNA cannot simultaneously read the same information. More importantly, they cannot pass each other: if a ribosome is currently located at a particular codon and does not (temporarily) proceed further, then an oncoming ribosome from behind will stop until the first eventually moves on. Finally, when a ribosome has reached the end of the m-RNA the polymer is fully synthesized and the ribosome is released (Fig. 2).

[]

Codons ~

~)

[]

-~

O

m - RNA

[]

Amino acids

~

Ribosome

i

initiation

o

[]

~

Peptide chain

translation

oI

[]

[]

O

[]

[]

[]

2

3

3~

[] [] I

polyribosome

I

[]

release

Fig. 2 Kinetics of biopolymerization on an m-RNA template. In order to describe the kinetics of this process MacDonald et al. introduced the following simple model. The m-RNA is represented by a one-dimensional lattice of L sites where each lattice site represents one codon. The ribosome is a big particle covering r neighbouring sites which moves randomly by one lattice site with a constant rate p from site 1 until it reaches site L of the lattice. These particles interact via hard-core repulsion, i.e. there is no long-range interaction, but there is also no overlap of ribosomes. At the beginning of the chain, particles are added with rate u p (the initialization) and at the end of the chain they are removed with rate ~p (release). In the idealized case r = 1 this model is the asymmetric simple exclusion process (ASEP) with open boundary conditions. Its generator is related to the anisotropic Heisenberg quantum chain (Gwa and Spohn, 1992b).

12

G . M . SchQtz

The experimentally accessible quantity is the specific activity, giving the relative rate of synthesis of an individual chain as it moves along the template. In terms of the exclusion process the specific activity is proportional to the stationary average number of particles on the chain between site 1 up to some site k. By taking the space derivative of the specific activity curve one finds then the stationary density profile. Therefore the main quantity of theoretical interest is the stationary density profile of the exclusion process. Experimental data suggest two regions of constant, but different densities, corresponding to a slowing-down of ribosomes. Originally it was speculated that this change in specific activity might be due to some control point on the m-RNA at which the rate of chain growth changes (Winslow and Ingram, 1966). The exclusion model, however, suggests that this change of growth is not due to such a control point, but arises from a 'traffic jam' of ribosomes (MacDonald and Gibbs, 1969): after a region of low density of ribosomes at the beginning of the RNA a sharp increase in density occurs and the following ribosomes are in a congested high-density state with equal current, but low velocity. The presence of this 'shock' in the average ribosome density has later been confirmed in other models (von Heijne et al., 1987). This 'traffic jam' and some of its implications for the interpretation of the experimental results is discussed in Sections 7 and 10. We note that unlike in our first example here one is not interested in a time-dependent property, but in the stationary state of the system. Nevertheless one deals with a nonequilibrium situation since there is always a stationary current of particles, i.e. ribosomes. 1.3.3

Traffic flow

The one-dimensional exclusion processes, symmetric or asymmetric, and some of their variants serve as a model not only for the systems described above, but also, e.g. for diffusion in thin channels (Kukla et al., 1996), ionic conductors (Katz et al., 1984), spin relaxation dynamics (Kawasaki, 1966), interface growth (Meakin et al., 1986; Plischke et al., 1987) (see Appendix B), or traffic flow (Schreckenberg et al., 1995; Nagel, 1996). Since the most interesting lattice models for traffic flow are not integrable, we shall not study such systems in great detail. Nevertheless in the spirit of the onion way of modelling some insights may be gained from the simplest possible models which are integrable. While in many respects very unrealistic for traffic flow, the ASEP has a stationary current-density relation with a single maximum of the current at some intermediate density as is known from the flow diagram of real freeway traffic (Hall et al., 1986). Moreover, we have seen above that the ASEP exhibits shocks, unfortunately an important feature of real traffic. Within the exclusion model one can study in a detailed manner how shocks move. Notice that unlike in the previous examples the motion of shocks involves a problem which is both timedependent and where one does not approach an equilibrium state.


1.3.4

13

Exciton dynamics on polymer chains

As a final introductory example we consider the modelling of reaction-diffusion mechanisms by lattice gases. An experimentally relevant class of physical (rather than chemical) reactions comprises diffusion-limited annihilation processes. These may be used to describe the dynamics of laser-induced excitons on polymers and similar processes (Privman, 1997). The excitons hop along a polymer chain (a process symbolically represented by A0 ~- 0A), may decay spontaneously after some typical lifetime (,4 --~ 0), but can also annihilate either in pairs (,4,4 --~ 0~) or undergo fusion (A,4 --~ AO, ~,4). If the excitons have a lifetime that is orders of magnitude larger than the hopping time, then one may neglect spontaneous decay and is left with the pure pair annihilation (or fusion) process. This can be observed in exciton annihilation on TMMC chains ((CH3)4NMnC13) (Kroon and Sprik, 1997; Kopelman and Lin, 1997) where excitons of the Mn 2§ ion are initially created by laser excitations. The excitons then move along the widely separated MnCI3 chains and coalesce (undergo fusion) when they meet, leading to the emission of light. The intensity of light, which is proportional to the exciton density, can be measured. A minimal lattice gas model that captures the essential physics of diffusionlimited annihilation in one dimension is the symmetric simple exclusion process augmented by a pair annihilation reaction. If a particle attempts to move on an occupied lattice site then the move is not rejected, but both particles annihilate, thus modelling an instantaneous annihilation which is suggested by further experimental evidence. This lattice gas model is related to the integrable sevenvertex model in a submanifold that maps to a free-fermion problem. In a different mapping the same process describes Glauber spin relaxation dynamics of the onedimensional Ising model. This process, discussed in detail in Section 9 indeed predicts the correct asymptotic decay of the luminosity and allows for a detailed analysis of problems not readily accessible within the renormalization group approach and Smoluchowski theory. It will also become clear why pair-annihilation (,4,4 --~ ~0) and coalescence (fusion A,4 --~ 0,4, AO) have the same universal power law decay. Notice that here the stationary state is completely trivial: stationarity is reached when all particles are annihilated and the process stops. However, trivial equilibrium behaviour does not imply trivial dynamics.

1.4

Outline

The treatment of transfer matrices for the description of discrete-time processes is conceptually not much different from the discussion of continuous-time processes in terms of quantum spin chains. In many cases the quantum spin Hamiltonian

14

G . M . SchOtz

can be obtained directly from a transfer matrix by taking an appropriate and physically harmless limit. Therefore the presentation given here will focus on continuous-time problems. Only one subsection is concerned with a discrete-time model for reasons which will become apparent below. In order to avoid irrelevant technical and notational difficulties we shall consider mainly finite systems. The thermodynamic limit is conceptually straightforward on the level of the equations of motion for expectation values unless ergodicity breaking is involved. This is a topic in its own right and is not treated here. The general structure of the bulk of this article is a division into three parts comprising the relationship between stochastic dynamics and quantum spin chains (Sections 2-5), the properties of lattice gas systems with hard-core repulsion (Sections 6 and 7), and of reaction-diffusion systems (Sections 8 and 9). In the concluding Section 10 we return to the three polymer systems introduced above and briefly review theoretical and further experimental results. An overview over the full range of exact results is impossible and therefore the selection of material has a strong personal note. To partially make up for this shortcoming, most sections end with comments on related topics and references to the relevant literature. The following short summary may give the uninitiated reader some idea what to expect. In the first part we introduce some of the basic tools and notions used later and thus provide a 'dictionary' for the correspondence between probabilistic quantities and quantum spin language. For this purpose the tensor basis for finite particle systems is introduced and the master equation is written in terms of a many-body operator acting on a suitably chosen tensor space. As the reader will soon realize, this Fock space technique is very easy to implement, but it may be necessary to take some time to get used to the language. Even though this formalism is used in much of the existing literature, there is no systematic and pedagogical introduction. Here we try to fill this gap. The following brief review of quantum spin systems is meant to acquaint an unexperienced reader with some elementary properties of these systems and to familiarize him with some basic techniques of their treatment. For an advanced understanding we refer to the book by Baxter (1982). Applications to stochastic processes are given in the context of specific questions in subsequent sections. Finally some notions pertaining to the late-time behaviour of stochastic processes, their equivalences and other specific properties are introduced in general terms. This will, hopefully, provide a frame of reference and some motivation for the study of specific systems that is to follow. The second part (Sections 6 and 7) is concerned with the theoretical treatment of purely diffusive systems, beginning with a treatment of the symmetric exclusion process (Section 6). This model has been much studied in the past (see, e.g. the book by Liggett, 1985), yet many new results with interesting applications have been obtained using its integrability. Lattice gases which are driven by an external force (Section 7) were introduced in a systematic way not


15

very long ago by Katz et al. (1984). Some of the most interesting applications are in one dimension and we focus our attention on this case. Again a considerable wealth of insights has been gained in the probabilistic literature (Liggett, 1985; Spohn, 1991), by numerical means (Janowsky and Lebowitz, 1997), renormalization group methods (Schmittmann and Zia, 1995) and by a matrix product description of stationary states (Derrida and Evans, 1997; Derrida, 1998) which are well documented in the existing literature. Further important exact results have been obtained using integrability and other alternative approaches. These are reviewed and placed into context here. The most far-reaching result discussed in this second part is the exact solution of stationary properties of the ASEP with open boundary conditions (Schlitz and Domany, 1993; Derrida et al., 1993a) where the system is coupled to particle reservoirs of constant density (Krug, 1991) (Section 7.2). Much of the physics of the asymmetric exclusion process can be understood in terms of shocks (domain walls separating regions of low and high density) propagating through the system and from the motion of local perturbations. These mechanisms provide an essentially complete understanding of how the open system selects its bulk steady state and allow us to predict the phase diagram of quite genetic one-dimensional driven lattice gases (Section 7.4). The third main part of this work (Sections 8 and 9) deals with diffusive systems of a single species of reacting particles. More by way of illustration of quantum mechanical methods rather than with the serious intention of investigating a particular real system we first describe the unusual phenomenon of microscopic field-induced density oscillations. This demonstrates the possibility of interesting dynamical behaviour in these rather simple toy models of diffusion-limited chemical reactions (Sections 8.3 and 8.4). It should be made clear that Section 8 is not intended to map out the vast territory of nonequilibrium phenomena in reactiondiffusion systems. Such an undertaking would be far too ambitious. Instead we just provide some further pieces in the puzzle, hoping that at least a rough outline of the class of systems discussed here may be drawn in the foreseeable future. Section 9 discusses a class of exactly solvable models comprising superficially very different systems, ranging from Glauber dynamics for the spin relaxation of the one-dimensional Ising model (Glauber, 1963) to branching and coalescing random walks which describe diffusion-limited pair reactions. Various relations between these and other models have been known for some time and can be found scattered in the literature cited below. Sections 9.1-9.3 are devoted to bringing some order into this web of relations. In one dimension the quantum Hamiltonian for the stochastic dynamics describes an exactly solvable system of free fermions (Siggia, 1977). This property becomes manifest by some suitably chosen similarity transformation and constitutes the common mathematical ground on which these models stand. Our main message is that all known equivalences can be generated by two families of similarity transformations. Our derivation and the

16

G . M . Sch0tz

form of these transformations lead us to conjecture that the models described here are all equivalent single-species free-fermion processes with pair-interaction between sites. Various dynamical properties of diffusion-limited annihilation are worked out in detail. This work ends with some appendices. Appendix A explains the relationship between the discrete-time exclusion process of Section 7.3 and a two-dimensional vertex model. In Appendix B we discuss universality of interface fluctuations. In Appendix C we present the exact solution of recursion relations for the asymmetric exclusion process with open boundaries. In Appendix D frequently used conventions (including the definition of tensor products) and notation are summarized. All the topics discussed in this work represent active areas of research. There are still plenty of open questions some of which are pointed out as we go along. An important class of systems which may appear underrepresented in our survey are two-species lattice gases. In one dimension such models are not only of experimental relevance in tracer diffusion in narrow channels (Kukla et al., 1996) or for the description of gel electrophoresis (Barkema and Schiitz, 1996) (Section 10.1), but also exhibit novel and interesting phenomena, e.g. spontaneous symmetry breaking (Evans et al., 1995, 1998; Godr~che et al., 1995; Amdt et al., 1998b). Moreover, with such processes one can study the behaviour of single tagged particles in a lattice gas rather than only the collective behaviour treated in this work. However, the consequences of the quantum Hamiltonian formulation for these processes are largely unexplored and thus represent an area of future research.

1 Exactlysolvable models for many-body systems 2

2.1

17

Quantum Hamiltonian formalism for the master equation

The master equation

Our real concern is not the modelling of a physical system by some stochastic equation (the first approximation discussed in the introduction), but the analytical treatment of such an equation once it has been obtained through experimental observation and physical intuition. For our purposes the only important issue in the modelling is the general philosophy of adopting a coarse-grained point of view: rather than considering a continuum of possible states of the system (defined by positions and momenta) we assume the particles to be located on some lattice on which they can move and interact with each other. At any instant of time one thinks of the system as being in a configuration r/ ~ X, defined by the positions of the particles. X is the set of all states in which the system may be found. Unlike in classical many-body physics, where the state of a system at a given time is specified by a set of positions and conjugate momenta, here a complete description is provided by the probability Po(t) of finding the system in the state r/at time t. The system dynamics has to be described accordingly. Instead of being defined by Newton's deterministic equations of motion one assumes the time evolution to proceed according to certain stochastic rules. These rules are encoded in transition probabilities P,7~,7' for elementary moves from a state r/to a state r/'. An elementary move is a transition which takes place instantaneously after some time interval At, e.g. the spontaneous decay of a particle, or the hopping of a particle from one lattice site to another (see Fig. 3 for modelling Brownian motion by a random walk on a lattice). These probabilities do not depend on how the system got into the state r/in a previous move: the dynamics of the system has no explicit memory of its own history. Thus the stochastic processes which are the subject of our investigation are so-called Markov processes. By following the time evolution of a specific system one realizes that starting from a given initial state, the system can take many different paths in state space, each with a probability which is the product of all transition probabilities between the elementary moves that constitute the path. This is the basic difference to a deterministic system where the equations of motion uniquely determine the motion of a many-body system in phase space from a given initial state. Of course, not all random processes which one observes in nature are Markovian, but the lack of memory for the history of a stochastic system seems to be a reasonable approximation for many systems of interest. In modelling the dynamics of a real system one may use either discrete time steps and/or continuous time. In both cases the heuristic description of the time evolution given above can be properly defined in terms of a master equation for the probability distribution Po(t). Assuming time to proceed in discrete time

G. M. SchQtz

18

t + At

t

(~

(~,

(~

t

t

t

t

I

I

I

I

I

Px~x~Px~x+l t

I

I

~

x-I

x

x+l

Fig. 3 A random walk on the integer lattice with nearest-neighbour moves. The state r/ of the system, i.e. the position of the random walker is given by an integer x e Z. Each possible elementary move, indicated by the arrows, takes place with some given probability. The sojourn probability of not moving is given by 1 - Px ~ x - 1 - Px ~ x + 1. These probabilities define the stochastic dynamics of the random walk. steps At the master equation relates the probability distribution Po(t + At) at time t + At to the distribution at time t

Po(t + A t ) =

Z

Po'~oPo'(t)"

(2.1)

o'~X

This equation represents the action of a linear time evolution operator, the generator of the Markov process, on the probability distribution: the probability Po(t + At) of finding the system in a certain state 17 at some time t + At is given by the product of the transition probabilities with the probabilities of finding it in any of the possible states before the move into state r/took place. Here the sojourn probability ps(O) = Po--*o which is included in the equation (2.1) is not an independent quantity, but given by conservation of probability as ps(rl) = I - )--~'~0'~,7P0~ 0" In such a discrete-time description the random lifetime of a configuration r/is geometrically distributed: the probability that the system is still in state r/after n steps is given by ps(rl) n. The mean lifetime (sometimes called mean sojourn time)is r, 7 = ps(rl)/(1 - ps(rl)). One may pass to a continuous-time description by defining the process in terms of rates wo__,o, = po__,o,/At (for 17 # r/') which are the transition probabilities per time unit. In the limit of infinitesimal time steps At ~ 0, (2.1) turns through the Taylor expansion into the continuous-time master equation d dt P~

= Z

[wo'--*oPo'(t) - wo~o' P'7(t)] "

(2.2)

OtC:rl rlt EX

The rates satisfy 0 < w(o, 0') < c~. In this continuous-time description the probability that the system does not change its state to some other state decays


19

exponentially in time with decay constant r o given by the mean sojourn time r0-1 ~ Y~r/'#rt w(rf, 11). These lines of thought can be well illustrated in terms of Brownian motion of a single particle which may be studied by describing the actual path of the panicle in space and time in terms of a stochastic differential equation m.~ = F (x, ./:)+ 17(x), i.e. Newton' s equation of motion with a deterministic force F plus a noise term 17representing the random forces acting on the particle. According to the philosophy outlined above we proceed in a different way by representing the stochastic dynamics in terms of a particle hopping on a lattice with certain rates (Fig. 3). To obtain the continuous-time master equation for the hopping process we assume the particle to hop with rates Wx-~x+l = DR ( W x - - - ~ x - I = DL) to the fight (left) which, for simplicity's sake, we assume to be space-independent. The spatial asymmetry in the hopping rates represents the effect of a constant driving force acting on the particle.* Then the master equation for the probability of finding the particle at site x reads d

dtPx(t) = DRPx-I(t) + D L P x + I ( t ) - (DL + DR)Px(t).

(2.3)

This equation can be obtained from the discrete-time description by taking the limit of continuous time, but is intuitively easy to understand directly from the definition of the process: the probability of finding the particle at site x increases through right (left) jumps from site x - 1 (x + 1) with rate Dn (DL). This assumes that the particle has not been at site x before the jump. On the other hand, if it was on site x, then the probability of finding it there decreases with rate DR + DL because it can hop in either direction away from site x. This gives the negative contribution in the master equation from which one reads off the mean sojourn time r = I/(DR + DL). The full solution of the master equation (see Section 4.2) then gives the time evolution of the probability of finding the particle at some lattice site x. The attentive reader may observe some formal analogies between this stochastic description of a classical system and the quantum mechanical formalism of the Schr6dinger equation. Like in quantum mechanics the description is probabilistic, and the time evolution is given by a linear equation involving the probability distribution. Indeed, a convenient presentation of the master equation and of quantities such as expectation values or probability distributions is in terms of a Dirac-Hilbert space notation as used in quantum mechanics. In this way the probability distribution maps to a time-dependent vector I P ( t ) ) in a suitably chosen vector space and the generator of the process is represented by a matrix acting as generator of the time translations of the distribution. *Ways of determining rates appropriate to a given physical situation are discussed later, but as explained above, this part of the modellingof a real system is not a central issue of this work.

20

G . M . SchQtz

To explore this - as we want to stress - purely formal analogy, consider instead of the random walk of Fig. 3 a system which can be found in one of m + 1 distinct states. Such a system could be an atom with a spin that takes values up or down (m = 1, X = { 1 , - 1 } ) or a single lattice site which is either occupied by a radioactive particle or empty once it has decayed (m = 1, X = {0, 1}). To each state 17 e X one assigns a canonical basis vector 117 ) of the vector space X - C m. Together with their transposed vectors ( r/I which form a basis of the dual space one defines a scalar product ( 17117' ) = 6o,o,. The probability vector is defined by I P(t) ) - Y]o Po (t)117 ) and the master equation (2.1) may now be written

[P(t + A t ) ) = TI P(t))

(2.4)

with the transfer matrix or transition matrix T defined by its matrix elements ( 17 IT117' ) = To, o, = Po'~ ~" This construction is easy to visualize for simple twostate spin-flip dynamics. With the choice of basis shown in Fig. 4 the probability vector is given by

'P(t))=Pt(t)'O)+P~(t)'l)=(P'(t)

)P~(t)

"

(2.5)

We assume spin up to flip with probability p and spin down to flip with probability q. This leads to the master equation Pt(t+At)

=

(1-P)Pt(t)+qP+(t),

(2.6)

P~(t + At)

=

q P t ( t ) + (1 - q ) P + ( t ) .

(2.7)

To derive the transfer matrix we define the ladder operators s + -- (orx 4- itrY)/2, and, with a view on application on particle systems, the number operators n = (1 - crz)/2, v = 1 - n = (1 + crz)/2 where crx'y'z are the usual Pauli matrices (see Appendix D). In the basis used here

s+ (01) 0

0

, s---

(00) 1

0

, n=

(00) 0

1

(2.8)

and one finds the transfer matrix

T = p s - + (l - p)v + qs+ + (l - q)n = ( l -

l -

) "

(2.9)

Interpreting a down-spin as a particle and an up-spin as a vacancy, this simple process describes decay of a radioactive particle with probability q per time unit and production of such a particle with probability p. If the transition probabilities are time-independent (as we assume throughout this work), then the solution to the master equation (2.4) with a given initial distribution [ P (0)) can formally be written I P ( t ) ) = Tnl P(O))

(2.10)


10/-(0),

l

'-

I~)-( ~, )

I

, q

t

21

t+At

Fig. 4 Vector representation of a simple stochastic two-state spin system with flip probabilities p and q. where t = n a t . The action of the transfer matrix has a simple interpretation in terms of the history of a given realization of the random process: in any given realization the system starts at some initial state 770 and proceeds through a series of n states to a final state r/n at time t - n At. This particular realization of the stochastic time evolution happens with the product of probabilities Poo--,o~ P r l l ~ , 1 2 " ' " Po,-~--,o,,. The matrix element ( r/' IT"I r/0 ) is just the sum of all probabilities of histories which lead from r/0 to some r/' = On in n steps. According to the definition of continuous-time dynamics one may write T = 1 - H A t . The off-diagonal matrix elements of H are the (negative) transition rates, H~, o, = - w o , ~ ~. The diagonal elements H0, 0 are the (positive) sum of all outgoing rates w o ~ ~,, i.e. the inverse lifetimes r (r/). For instance, setting for the two-state spin-flip process p -- c~At, q = y At yields H = ct(v - s - ) + y ( n s+). In particle language y has the interpretation of the inverse lifetime of the radioactive particle. This limit gives rise to the vector form of the continuoustime master equation d d~l P ( t ) ) - - H I e ( t ) ) (2.1 1) with the formal solution of the initial value problem I P(t)) - e-HtlP(O)).

(2.12)

From (2.1 1) one recovers the usual form (2.2) of the master equation by taking the scalar product with ( r/I and using the definition ( r/[HI rl') = Ho, o, - - w o , ~ o of the matrix elements of H. The master equation (2.1 1) has the form of a quantum mechanical Schr6dinger equation in imaginary time. This observation has given this treatment of the master equation the name 'quantum Hamiltonian formalism'. This notion is somewhat misleading, as quantum mechanical expectations for observables are calculated differently from the expectation values for the stochastic

22

G . M . SchQtz

variables. Also, the eigenvalues of the quantum Hamiltonian have nothing to do with energy levels of the classical particle system which usually determine the transition rates in the stochastic time evolution. Indeed, H may be, and in most cases of interest is, non-Hermitian and hence may have complex eigenvalues. But the term 'quantum Hamiltonian formalism' has become fairly standard and will be used here. For later reference we note a general property of stochastic time evolution operators. We introduce the row vector ( s I = Y~0 ( 171 with all components equal to one. Conservation of probability, i.e. ( s I P(t) ) = Y-~o Po (t) = 1 for all times, implies (s IT - (s I.

(2.13)

This is because in each column 17of T all matrix elements, i.e., transition probabilities Po~ 7' add up to one and this is tantamount to expressing completeness of the set X: the system always moves to some state 17 e X. A matrix with the property (2.13) and in which all matrix elements are real and satisfy 0 < To. o, < 1 is called a stochastic transfer matrix. For continuous-time dynamics, conservation of probability implies (s le - n t = (s I. Taking the time-derivative yields the eigenvalue equation

(sin -0.

(2.14)

Hence in each column of a stochastic Hamiltonian H all matrix elements add up to zero.

2.2

Expectation values

The most basic quantities which are usually measured in experiments are expectation values ( F ) = ~--~oF(o)Po (t) of an observable F. Here F(O) is some function of the state variables r/, e.g. the spin F(I') = 1, F(,I,) = - 1, or the position x = F(k) -- ka of a particle at site k in a lattice with lattice constant a. In a series of measurements the system may be found in states 17of the system with probabilities Po(t). Hence the expression ( F ) is the average value of what one measures in a series of many identical experiments, using the same initial state. If the initial states are not always the same fixed state, but some collection of different states, given by an initial distribution P0 = Po (0), then the expression ( F ) involves not only averaging over many realizations of the same process, but also averaging over the initial states. If we want to specify both time and initial condition, we write ( F ( t ) )Po" In the quantum Hamiltonian formalism the observable F is represented by the diagonal matrix F -- Y~-0 F(r/)[ 17)( r/I since ( F ) = Y~,7 F(rl)Po (t) = (s IFI P(t) ). For the continuous-time description we introduce the nondiagonal


23

time-dependent operator in the Heisenberg representation F(t) = e n t F e - n t

(2.15)

( F ( t ) ) Po = ( s IFe-H'l e(0) ) = ( s IF(t)l P(0) ).

(2.16)

and find, using (2.14),

Expectation values satisfy the equations of motion d d--~-( F ) = ([ H , F ] ) - - ( F H )

(2.17)

where the first equality follows from (2.15) and the second from conservation of probability (2.14). In discrete-time systems the expectation value (2.16) is given by the expression ( F(t))po = ( s l F T n l P(0)) (2.18) with t = n At. A special expectation value is the conditional probability P(r/; tit/; 0) of finding the system in state r/' at time t if at time t = 0 it was in state r/. The conditional probability is the solution to the initial value problem Po' (0) = ~0',,7 of the master equation. It is given by the matrix element e(r/'; tit/; 0) = ( r/' le-n/I r/).

(2.19)

For the random walk defined above where r / = x and the conditional probability defines the spatial probability distribution of the random walk at time t. The basic quantities in the study of Brownian motion are the moments (x n ) of the position distribution. For the random walk on the integer lattice the function x is represented by the operator Y~x x lx )( x I. Of particular interest are the drift velocity v and the diffusion coefficient D which we define by v

d lim ~ ( x )

--

t---~~ D

=

-

lira

(2.20)

dt --

2 t--->oodt

( x ) - ( x

.

(2.21)

Inserting the master equation (2.3) yields v = DR -- DL and D = (DR + DL)/2.

2.3 Many-body systems 2.3.1

The tensor basis

In the two-state spin model discussed above we had just a single spin flipping up and down. In many-body physics one is interested in the behaviour of many

24

G . M . SchQtz

coupled spins sitting on some lattice. Such lattice systems are equivalent to particle systems: by identifying a spin up with a vacancy and a spin down with the presence of a particle on the lattice site, spin models can be seen as particle systems where each lattice site may be occupied by at most one particle. This correspondence can be generalized: allowing for different species of particles, or site occupation by more than one particle, one obtains models where each lattice site can be in one of m + 1 distinct states. Such a model can be viewed as spin( m / 2 ) system. Hence, on a technical level, spin systems and particle systems can be treated in the same way. We shall from now on use mainly particle language rather than spin language. In order to describe lattice systems we have to introduce some conventions and some new notation. On a lattice of L sites the states 17 are denoted by a set of occupation numbers r/ = {r/(1) . . . . . r/(L)}. Usually we shall use labels of the form ki, li, mi for lattice sites. In the context of the Bethe ansatz and the freefermion approach discussed below we shall also use the symbols xi, Yi. A sum over lattice sites Y~k~..... kn is understood as a sum over all distinct sets of n sites of the lattice. The natural extension of the vector description of a system with a single site to a lattice system is by taking a tensor basis as basis of the state space. The manyparticle configurations r / a r e represented by the basis vectors 117) = 117(1) ) | . . . | I r/(L) ) which form a basis of the tensor space (cm) | (Fig. 5). Since we are dealing with many-body systems we shall denote a state It/) with N particles located on sites k i . . . . . k,v by the vector Ikl . . . . . k N ). The empty lattice is always represented by the vector 10). The summation vector ( S I for a manyparticle system is a tensor product ( S I = (s I| of the single-site summation vectors. In any system it is the constant row vector (1, 1. . . . . 1), but for clarity we shall use a bra vector with a capital S if we specifically refer to a tensor product and a lower-case s otherwise.

+-

+

0

1 0 IO)|174

=

0

|

1

|

0

I ~ I

Fig. 5 Vector representation of a two-state spin (particle) configuration on three lattice sites. Annihilation of the particle on site 2 is represented by the matrix s~- = 1 | s + | 1. One of the main quantities of interest in the study of stochastic many-body systems is the expectation value of the local density Pk = ( n k ) . In the tensor


25

basis of a two-state system the operator nk is given by the projection operator nk -- (1 - cr[)/2 acting nontrivially only on site k of the lattice (see D.7). The total, space-averaged density expectation value p - ( N )/L is then given by the number operator N = Z nk. (2.22) k

The construction of nk for multi-species systems and for systems which allow for more than one particle on each site is analogous: nk is a tensor operator acting nontrivially only on site k. It is diagonal in the basis spanned by the state vectors 117) and gives as the eigenvalue the number of particles on this site. For two-state models the local particle occupation numbers nk take values nk = o(k) = 0, 1. Physically, this classical exclusion principle may result from hard-core repulsion of particles. In the correspondence to spin systems we shall use the convention of considering spin down as a particle and spin up as a vacancy. With this convention the summation vector has the useful representation (Sl = ((01 + (1 l) |

(2.23)

= (0[e s+

with the total spin-lowering operator S + - ~ k e s s~-. Here the row vector ( 01 = (1,0) represents a single empty site and ( 1 I = (0, 1) corresponds to an occupied site. So far we have introduced basis vectors representing all the configurations the system may take. A probability distribution is a normalized linear combination of the basis vectors. For interacting particle systems an important class of probability distributions are so-called product measures. These are distributions where the probability of finding a given state at site k is independent of the state of the system at other sites. In a many-body system such a factorized distribution is represented by a tensor product of single-site distributions I P ) = I Pl) | | IPL). It is easy to verify that there are no spatial correlations between local observables. Since the scalar product of two tensor vectors factorizes into the (ordinary) product of tensor products one has, e.g. (nknt) = (Slnknll P) = (slP1)(slPz)...(slnklPk)...(sJntlPt)...(slPL) = (nk)(nl) since (slPi)= 1 for all i. A homogeneous product measure for a two-state system has the form IP) = [(1 - p)lO) + p [ l )]|

--

1-p P

)|

(2.24)

with the single-site column vectors 10), I1 ) for empty and occupied sites respectively. In this distribution each lattice site is occupied by a particle with probability p. Any configuration with a fixed total number of particles appears

26

G.M. SchOtz

with equal probability. This distribution is important, e.g. when averaging over random initial states with density p. It also arises as a stationary distribution of some of the processes considered later. More detailed information on a system can be obtained from the m-point correlation functions (nk~(tl)...nkm(tm)) at different times ti > ti+l. These are the joint probabilities of finding particles at sites ki at times ti. They are given by the expression

(nkl (tl)...nkm(tm)) = (s Inkle-H(tt-t2)nk2 . . .nkme-ntml P(O) ).

(2.25)

Here the time-dependent operator nk(t) is defined by (2.16). Multi-time correlation functions are measured in multidimensional nuclear magnetic resonance experiments and yield information on the mechanisms of magnetization transfer from which one can infer knowledge about the structural and dynamical properties of the material under investigation. Other quantities of interest are discussed later in their respective context.

2.3.2

Construction of the quantum Hamiltonian

Having introduced a basis for the state space we are now in a position to formulate a recipe for the construction of the quantum Hamiltonian for a given process. The tensor basis makes the construction almost trivial. However, if one is not familiar with the tensor notation of many-body spin systems one needs to get used to the strategy. For purposes of illustration a beginner in the field may find it helpful to go directly to Section 3 for illustration after studying the general formalism presented here. In the example of the single-site two-state model discussed earlier we have seen that elementary stochastic moves are represented by off-diagonal matrices. These matrices generate the change in the probability distribution as time proceeds; see (2.4) and (2.11). According to our convention, in a two-state lattice system the matrix s~- annihilates a particle at site k and s k represents a creation event (Fig. 5). These matrices are given by a tensor product in the same way as the number operator nk. More precisely, the matrices s~: represent attempts rather than actual events: acting on the r.h.s, of the master equation on an already occupied site with s - yields zero, i.e. there is no change in the probability vector on the l.h.s of the master equation. This reflects the rejection of any attempt at creating a second particle on a given site. The exclusion of double occupancy is encoded in the properties of the Pauli matrices. Simultaneous events are represented by products of Pauli matrices, e.g. hopping of a particle from site k to site l is equivalent to annihilating a particle at site k and at the same time creating one at site I. Thus it is given by the matrix s-~s~. The hopping attempt is successful only if site k is occupied and site l is


27

empty. Otherwise acting with s~-s t- on the state gives zero and hence no change. An attempted annihilation of a pair of particles is given by the matrix s~-sl+. The rate of hopping (or of any other possible stochastic event) is the numerical prefactor of each hopping matrix (or other attempt matrix). Of course, in principle the rate may depend on the configuration of the complete system. Suppose the hopping rate is given by a function w(r/) where r/ is the configuration prior to hopping, as, e.g. in thermally activated processes. In this case the hopping matrix is given by s-~s~-w with the diagonal matrix w -- Y~,7w(r/)[ 17)( 17[. This matrix is obtained from the function w ( o ) by replacing all r/(k) by the projector nk = (1 cry)/2 on states with a particle at site k. This is easy to illustrate in an example. If for some reason hopping from site k to site l should occur with rate p if a third site m is empty but with a rate q if this site is occupied, then w(r/) = p(l - o ( m ) ) + q o ( m ) . The corresponding hopping matrix is given by s-~s~[p(1 - n m ) -k- qnm]. All elementary moves may be represented in this way. The (negative) sum of all attempt matrices form the off-diagonal part of H. The diagonal part is determined by conservation of probability (2.14). For two-state models one notes the useful identities (Sls~---(Slnk,

(Sls~- = ( S l ( 1 - n k )

(2.26)

which follow from the factorized form (2.23) of the state ( S I. Equations (2.26) provide a simple recipe for the construction of the diagonal part of the quantum Hamiltonian: to each off-diagonal attempt matrix one constructs a diagonal matrix by replacing all s-~ --~ nk and by replacing all s k --~ Ok = 1 -- nk; e.g. to hopping from k to l with rate w(rl) represented by - s ~ s ~ w one adds nk vtw. Conservation of probability (2.14) is then automatically satisfied for each elementary move. The (negative) sum of all attempt matrices minus their diagonal counterparts is then the full quantum Hamiltonian. For interacting particle systems involving more than one kind of particle or where one allows for multiple occupancy of particles of the same species, one proceeds analogously. The Hilbert space on which H acts is then (CP) | where p is the total number of different states an individual site may take. Stochastic moves from one configuration to another configuration are given by products of pry matrices E k acting locally on sites k of the lattice and changing the configuration on this site from state tr to state p. The p • p matrix E p~ has matrix element 1 in column p, row cr and zero elsewhere. The diagonal part of H corresponding to such a product is the product built p(7 pp from the matrices E~ p according to the replacement rule E k --~ E k . This construction is not restricted to a finite set of lattice sites, but may be generalized to arbitrary countable sets. The derivation of a transfer matrix from rules defined in discrete time proceeds in a completely analogous way.

G. M. Schfitz

28

2.4

Nonstochastic generators

For a variety of reasons also nonstochastic quantum Hamiltonians (or transfer matrices) play an important role in the investigation of stochastic dynamics. Since the definition of a stochastic matrix is basis-dependent it may happen that a nonstochastic Hamiltonian can be turned into a stochastic Hamiltonian by a suitably chosen basis transformation H st~ = /3Hn~176 -1. Having established the existence of such a transformation one can then use whatever knowledge one might gain from the nonstochastic quantum system for the study of the stochastic system. Examples given later include systems where after a basis transformation the integrability of the model becomes manifest. Non-stochastic matrices may also give the answer to certain specific problems, rather than describe the full process. Suppose the system has an absorbing state. This is a stable state which does not change any more once the system has reached it. It is then sufficient to study the stochastic dynamics only on the subset X' C X of the full state space which excludes the absorbing state. However, restricted on this subset the process does not conserve probability and hence is described by a nonstochastic evolution operator. This property generalizes to systems with absorbing regions in state space. An important group of quantities which can be expressed in terms of nonstochastic generators are first-passage time and persistence distributions. These are expectation values which are nonlocal in time. One observes over a whole interval of time whether the system has remained within a certain subset X' of states. This gives the persistence probability Px,(t). The first-passage probability d/(dt) Px, (t) then gives the probability that the system has left the subset at time t for the first time. To understand the relationship to nonstochastic generators some more tools have to be introduced and we postpone the discussion until Section 3.

Comments Section 2.1" In the numerical literature the discrete time evolution is often referred to as parallel updating. In each time step all lattice sites are updated simultaneously according to the stochastic rules. Continuous-time processes are usually simulated by means of a random sequential updating scheme where in each updating step a minimal set of sites is chosen randomly and changes in the configuration occur only on this set of sites. Section 2.2: For some systems, e.g. random walks in disordered media, (2.18) represents a convenient way to obtain a numerically exact calculation of expectation values by iterating on a computer the action of the transfer matrix on the starting vector I P(0) ). It is not necessary to take averages over histories and initial states as in a Monte Carlo simulation where each simulation realizes only one specific stochastic history.


29

Section 2.3: For unrestricted occupancy, i.e. allowing for infinitely many states on each lattice site, see Kadanoff and Swift (1968); Doi (1976); Grassberger and Scheunert (1980) and Cardy (1997). Here we restrict ourselves to (m + 1)-state systems, and mostly to the simplest case m = 1. For exactly solvable three-state systems see, e.g. Alcaraz and Rittenberg (1993); Simon (1995); Alcaraz (1994); Alcaraz et al. (1994); Dahmen (1995); Schulz and Trimper (1996) and Fujii and Wadati (1997).

30 3

G. M. SchQtz

Integrable stochastic processes

In the previous section we have already encountered a relationship between interacting particle systems and spin systems. This relationship has appeared in two conceptually different ways. On the one hand, one may consider systems of coupled classical spins which involve in time according to some stochastic dynamics. By identifying spin variables with particle occupation numbers, these systems are in a very straightforward manner equivalent to classical interacting particle systems. On the other hand, the description of the stochastic dynamics in terms of a quantum Hamiltonian indicated that there is also a correspondence to quantum spin systems: the stochastic time evolution is generated by a matrix involving quantum mechanical spin operators. To a d-dimensional spin or particle model with classical stochastic dynamics is associated a d-dimensional quantum spin system. Finally, there is a well-known third correspondence between classical spin systems in d + 1 dimensions and quantum spin systems in d dimensions, a special case of which is the relationship between the discrete-time and continuous-time description of stochastic dynamics. If the stochastic dynamics are defined in d space dimensions then T may be regarded as a transfer matrix describing the equilibrium distribution of a (d + l)-dimensional classical model where the additional dimension plays the role of time in the dynamical interpretation. Generally, the description of classical equilibrium spin systems in d + 1 dimensions in terms of a transfer matrix gives rise (for certain limits of the coupling constants) to a quantum spin Hamiltonian in d dimensions (Kogut, 1979). In the stochastic case the transition probabilities p parametrize the transfer matrix. In the continuoustime limit where all probabilities p = wAt vanish, the transfer matrix takes the form T = l - H At with the quantum Hamiltonian H. Hence a d-dimensional stochastic model (with its associated d-dimensional quantum spin Hamiltonian) corresponds to some (d + l)-dimensional classical equilibrium spin model.* To utilize these equivalences we recall some of the basic notions that appear in the study of exactly solvable spin models, in particular, the notion of integrability. This is a remarkable and important property as it allows for the derivation of nontrivial exact results. A many-body system is considered exactly solvable (or integrable) if there exists an infinite set of independent conserved charges in the Hamiltonian of a d-dimensional quantum system or in the transfer matrix of an associated (d + l)-dimensional statistical mechanics model. * Integrability manifests itself in the *Of course, generically the transfer matrix of a statistical mechanics model is not a stochastic transfer matrix. The correspondence between the classical model and the quantum spin system can nevertheless be made. In an integrable quantum system of finitely many degrees of freedom, such as a spin system on a finite lattice, the required number of conserved charges is, of course, finite.


31

concept of commuting transfer matrices [ T(u), T(v) ] = 0

(3.1)

where the system parameter u is a suitably chosen function of temperature, field strengths and other parameters of the model. The importance of this commutation relation for different values of u becomes explicit by expanding T(u) = ~ n (u-uo)nTn (uo) around some value u0. This expansion yields a set of matrices Tn(uo) and (3.1) implies [ Tn, Tm ] = 0 u m, n. In particular, [ Tn, T(u) ] = O. If all the Tn are independent, i.e. not polynomial functions of each other, then the commutator (3.1) proves the existence of an infinite set of (independent) conservation laws which can be constructed by expanding the transfer matrix in powers of u. These conserved charges Tn all commute among each other and any linear combination of the Tn may be viewed as a quantum Hamiltonian which is integrable. Eigenstates of the transfer matrix are also eigenstates of the Tn and their eigenvalues are conserved quantum numbers characterizing these states. One obtains an integrable stochastic process if either the transfer matrix T (Kandel et al., 1990; Schiitz, 1993a; Honecker and Peschel, 1997) can be tumed into a stochastic transfer matrix by some similarity transformation (see Appendix A) or some linear combination of the conserved charges Tn can be used to construct a stochastic Hamiltonian (see below). In the present context the most obvious question to ask is which integrable stochastic processes exist and to which extent the integrability can be used to obtain information on the stationary and dynamical behaviour of the system. There is no general answer to either of the two problems as firstly there is no complete list of integrable models and secondly it does not seem reasonable to expect that one could classify all stochastic matrices related by a similarity transformation to a given integrable system. Therefore we confine ourselves to a study of the simplest known models and point out merely a few of those generalizations which seem promising for further investigations. It is a very long way to go from the notion of commuting transfer matrices to practical applications of integrability (Baxter, 1982). This work is neither the place to explore the mathematical framework underlying integrability as such nor to review the physical properties of integrable equilibrium systems and quantum spin chains in any detail. We are concerned with the consequences of integrability for the theory of stochastic interacting particle systems. Since, fortunately, the most relevant concepts- coordinate Bethe ansatz, Hecke algebras, and quantum groups - are readily understandable by themselves in terms of standard quantum mechanical notions, we shall introduce most of these ideas in a pedestrian way where the need arises. However, it is good to have an overview at least of some very basic properties of quantum spin systems before studying how these models can be used to investigate stochastic dynamics.

32

3.1

G.M. SchQtz

The Ising and Heisenberg spin models

An interacting many-body system with an infinite set of conservation laws was first discovered by Hans Bethe in 1931 even though at the time the connection to commuting transfer matrices was not known. The history of the development of integrable systems has its roots in the study of magnetism in 1925 on the Ising model (lsing, 1925), proposed earlier by lsing's supervisor W. Lenz. A basic feature of many ferromagnetic systems is the presence of a phase transition between an ordered, magnetic state with spontaneously broken symmetry between two orientations of the magnetization and a high temperature state where the magnetization gets lost. The Curie temperature Tc where the transition takes place depends on the material, but other critical properties such as the divergence of the magnetic susceptibility X ~ ( T - Tc) -• are, curiously enough, largely universal, in the sense that the critical exponent y is the same for many microscopically very different substances. Hence there is hope that simple model systems may capture the essence of the phase transition. Magnetism has its origin in the collective behaviour of the atoms and electrons which allows for microscopic magnetic moments associated with the spins to be set up. In ferromagnets nearest-neighbour spins tend to orientate in the same direction whereas in antiferromagnets they tend to be antiparallel. In Ising's simple classical model for a ferromagnet all atoms in a solid are assumed to take spin values s = +1 (in appropriate units). To model ferromagnetic behaviour one postulates a nearest neighbour interaction between spins which favours spins that are aligned. An additional energy, proportional to the total magnetization, may arise from an external field of strength h. This behaviour is expressed in the Ising energy of a given spin configuration r / = {s1 . . . . . SL } E(rl) = - J

y~skst -h (k,t)

y~sk k

(3.2)

where the first sum runs over all nearest neighbour pairs of sites and the second sum runs over all lattice sites. With this energy function one calculates the statistical properties of the model following the usual rules of classical statistical mechanics. The equilibrium distribution

eeq (r/) (3( exp (-/~E)

(3.3)

gives the equilibrium probability of finding a state r/with energy E at temperature T = 1/(k/3). From the partition function Z = Y~contigexp (-/~ E) one obtains the free energy and all other thermodynamic properties. At zero temperature and in zero field all spins point either up or down, at finite temperatures one has domains of equal magnetization, separated by domain walls (Fig. 6). In one dimension the model can be solved exactly using the transfer matrix technique (see, e.g.


33

Baxter, 1982) and most of the relevant thermodynamical properties had already been obtained by Ising.

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l l

l

l

Fig. Ii Spin configuration in the classical Ising model with domains of predominant upspins (upper left comer) and down-spins respectively (lower fight comer). The domain walls are shown as full lines. In one dimension an equilibrium phase transition in a system with shortranged interactions can occur only at zero temperature and hence Ising's solution is of limited relevance to the investigation of phase transitions. Nevertheless both the existence of essentially one-dimensional solids - chains of real, threedimensional a t o m s - and conceptual reasons related to the general description and properties of phase transitions make it worthwhile investigating such spin models in all dimensions, including d -- 1. For this purpose exact solutions for special models which clarify the status of more widely applicable approximate theories are rather important, and particularly were so in the 1920s when there was no unified framework for the understanding of phase transitions. In fact, the ensuing history of the study of spin systems has an amusing twist. The Ising model is a classical model and as such was regarded unsatisfactory at a time when quantum mechanics was just being developed and celebrated its first great successes. Heisenberg proposed in 1928 a quantum spin model where the classical, twostate Ising spins are replaced by spin-(1/2) Pauli matrices (Heisenberg, 1928). The Hamiltonian of this model reads

(k,l)

k

34

G . M . SchQtz

where~k 96/ = ~kx o).~ + tykYcriy + t~zt~lz and the physically immaterial constant 1 has been introduced for later convenience. The one-dimensional version of this model was solved in 1931 by Bethe with what is now known as the Bethe ansatz (Bethe, 1931) (see below) and which has become the first of two starting points in the study of integrable systems. Bethe's motivation for working on the Heisenberg model was the feeling that the Ising model, being classical, was inadaequate. He used his recently gained experience in quantum mechanical scattering theory to successfully construct a trial wave function for the spin-wave eigenstates of the Heisenberg Hamiltonian which represents the first exact solution of an integrable system in the sense explained above. However, after this success Bethe did not pursue his studies of magnetism. Instead he turned to nuclear physics which he found more exciting (Bethe, personal communication). For a long time Bethe's results on the Heisenberg model had little impact. Thus the second starting point in the history of integrable systems came in 1944 with Onsager's solution o f - ironically - Ising's classical model (in two dimensions). This solution was an essential step forward in statistical mechanics, particularly in the understanding of critical phenomena, as it showed for the first time that second-order phase transitions with divergent correlation length scales and associated divergences in thermodynamic properties such as the specific heat or magnetic susceptibility could be described within the unified framework of the Gibbs distribution and the resulting partition function. In the late 1950s Onsager's solution set into motion an avalanche of work on exactly solvable models which soon incorporated Bethe's work and which later included contributions also of field theorists, mathematical physicists and pure mathematicians. One of the most basic mathematical properties of the Heisenberg quantum spin system in the absence of the magnetic field (h = 0) is the SU (2) symmetry which generates continuous rotations in spin space. It is easy to verify by direct calculation that each interaction matrix ~k" ~l commutes with S • = )--]-ks~:, SZ = ~-]~ka~ /2 = ~-~k( l /2 -- nk ). These matrices form a spin-(1/2) tensor representation of the Lie algebra SU (2) defined by the relations [ S + , S - ] = 2S z, [ S • , S z ] = + S • Hence H is symmetric under the action of S U (2), i.e. [ H , S +'z ] = 0.

(3.5)

The representation theory of SU(2) reveals that the ground state of H is (2L + 1)-fold degenerate, corresponding to parallel ordering of the quantized spins in arbitrary space direction at zero temperature. In the presence of the magnetic field the symmetry reduces to U (1), generated by S z. This symmetry corresponds to invariance under continuous rotations around the z-axis of the magnetic field. The ground state is the state where all spins point parallel to the field. An important generalization of the isotropic Heisenberg ferromagnet (3.4) is the anisotropic Heisenberg Hamiltonian where the coupling in the z-direction of


35

the spin variable is anisotropic H XXZ = - J ~ [cr~crtx + cr~crty + A(cr/zcr/z -- 1)] -- h ~ cr/z. (k,/> k

(3.6)

In one dimension this model is also exactly solvable using the Bethe ansatz (Yang and Yang, 1966; Takahashi, 1971; Takahashi and Suzuki, 1972) and has the same U(I) symmetry as the isotropic Heisenberg chain with nonvanishing field. By changing A one finds a (zero-temperature) quantum phase transition into a disordered, critical ground state with algebraically decaying correlations. The integrability of the Heisenberg spin chain follows from its connection with the two-dimensional classical statistical mechanics model known as the sixvertex model (Lieb, 1967a,b,c; Sutherland, 1967; Baxter, 1982). The quantum spin Hamiltonian belongs to the conserved charges of the transfer matrix. The existence of the infinite set of conservation laws is ultimately responsible for the success of the Bethe ansatz described in the following section. The relationship is indirect, though. The commuting components Tn of the transfer matrix of the six-vertex model do not appear explicitly. For this reason we do not comment further on vertex models. The interested reader is referred to Baxter's 1982 book on exactly solvable models in equilibrium statistical mechanics.

3.2

Bethe ansatz

The SU (2)-symmetry of the Heisenberg chain yields the ground state properties (Yang and Yang, 1966), but by itself does not provide any insight into the finitetemperature regime. In the case of the anisotropic chain the situation is even worse. For IAI < 1 only the ground state in the presence of a sufficiently strong field can be found on the basis of symmetry considerations. It is the rather dull state where all spins point in the z-direction. Hence one needs to diagonalize H and calculate the excited states of the system. Except in some limiting cases (e.g. J ~ 0 with J A fixed in which case the quantum model becomes equivalent to the classical Ising model) this is a nontrivial undertaking. The problem was solved by Bethe by starting from the fully ordered state with all spins up (in the z-direction) which is evidently an eigenstate and then making a suitable ansatz for the wave functions in the sector with N spins down.* We consider a finite system with periodic boundary conditions. Since the term proportional to the field strength h commutes with the zero-field Hamiltonian it does not change the eigenstates. Hence we set h - 0. It is convenient to express the spin-flip term or/,i xcrx +cr~cry in terms of the spin ladder operators s + introduced

*Bethe considered only the isotropic case A = 1. The same idea, however,yields the eigenstates for arbitrary A (Yangand Yang, 1966).

36

G . M . SchOtz

earlier (2.8) which yields for the one-dimensional system L

Hxxz

= -J Z

[2(s[s~-+, + .3,- +s , + , ) +

A(o/ ai+z l - 1 ) ] .

(3.7)

k=l

This representation gives another intuitive interpretation of the spin-spin interaction. The diagonal part a[a~+ l - 1 is just proportional to the number of domain walls, i.e. antiparallel neighbouring spins and therefore equivalent to a classical - 1 + s k- Sk+ + l describes spin exchange Ising energy. The off-diagonal part S~Sk+ resulting from quantum fluctuations. Consider now the set of states with one spin pointing down (Fig. 7). For notational convenience we shall denote lattice sites by x, running from 1 to L.

ItTtt

I

I

Fig. 7 Single down-spin in the Heisenberg quantum chain. There are two possibilities for spin flip (arrows) with exchange interaction 2J and two domain walls (vertical lines) with energy 2J A. N = 1. We denote the state with spin pointing down at site x of the chain by Ix ) and look for an eigenstate l e ) = Y~'~x~ ( x ) l x ) with eigenvalue e. By definition of the spin-flip terms the action of H on the state Ix) is readily calculated as HIx) = -2J(Ix+l) + Ix-l)) +4JAIx) which implies theeigenvalue equation - e ~ ( x ) = 2 J [ ~ ( x + 1) + q~(x - 1 ) ] - 4 J A ~ ( x ) (3.8) for the wave function 9 (x). This equation is solved by the plane wave ansatz qJ (X) = e ipx

(3.9)

which yields ~p -- 4 J ( A - cos p).

(3.10)

The eigenstates are plane waves, known as magnons with 'momentum' p. Imposing periodic boundary conditions qJ (x) = 9 (x + L) quantizes p since one has to satisfy e ipL = 1. (3.11)


37

This yields the translational lattice modes p = 2zrn/L. N = 2. The two-magnon sector where two spins point down is more complicated due to the nearest-neighbour interaction. For down-spins at a distance of more than one lattice unit one has to solve the two-particle eigenvalue equation for independent down-spins f q J ( X l , X2)

- 2 J [qJ(Xl - l,x2) + q/(Xl, X2 -- 1) -- 2AqJ(Xl, x2) + qJ(Xl + l , x 2 ) + q / ( X l , X 2 + 1 ) - 2AqJ(Xl,X2)] (3.12)

which can be formally obtained by calculating the action of H on the states IXl,X2).

For neighbouring down-spins x2 = x l + 1 - x one has to satisfy 1,x + 1 ) + q J ( x , x + 2 ) - 2 A ~ ( x , x + 1)] (3.13) with the s a m e ~. The reason for the change in the form of the equation is the absence of spin exchange between the two neighbouring down-spins (off-diagonal part of the eigenvalue equation) and the corresponding change of the diagonal energy term which is proportional to the number of domain walls in the state (Fig. 8). This can be captured in a unified equation of the form (3.12) by demanding (3.12) to be valid for all x l , x2 and imposing the b o u n d a r y condition on the e~(x,x

+ 1) = - 2 J [ ~ ( x -

wave f u n c t i o n

9 (x, x) + ~ ( x + l , x + 1) - 2AqJ(x, x + 1) = 0 V x.

(3.14)

This relation reduces (3.12) to (3.13) for neighbouring spins. Together with the bulk equation (3.12) it also defines the value of the wave function 9 in the unphysical region of ghost sites x l > x2. To solve (3.12) Bethe made the ansatz qlpl,p2(Xl, X2) = e ipixl+ip2x2 ~- S(p2, p l ) e ip2xl+ip~x2

(3.15)

for the two-spin wave function which was inspired by quantum mechanical scattering theory. The bulk equation (3.12) yields e = el + e2 with the single-particle 'energies' Ei = 4 J ( A -- cos Pi), (3.16) but leaves the function S(pe, Pl) undetermined. In quantum mechanical language this is a spin wave scattering amplitude (Mattis, 1965). Next, one has to satisfy the boundary condition (3.14) which fixes 1 + e ipl +ip2

_

_

2Aeip2

S ( p 2 , P l ) = - 1 + e ip~+ip2 -- 2 A e ipl "

(3.17)

38

G. M. SchQtz

T

(a)

(b)

t

I

t T T

I

I Tit I

I

2, (q + q - l ) h k

(3.42) (3.43)

with the c-number q appearing as a parameter. These are the defining relations for the generators of the so-called Hecke algebras HM(q) (Alcaraz and Rittenberg, 1993). From a matrix representation of these generators one can always construct an integrable quantum Hamiltonian H hk. This is guaranteed by a mechanism called Baxterization (Jones, 1990) which we do not review here. The important conclusion we want to stress is that the one-dimensional versions of some of the models presented in the previous section are integrable since on a submanifold of the parameter space the reaction matrices (3.33) satisfy the algebraic relations of the Hecke algebra. Any stochastic process of the form (3.35) where the hk satisfy the Hecke relations (3.41)-(3.43) is an integrable stochastic process. We point out three examples already discussed above. Consider matrices hk which satisfy the additional relation =

zM=I

hkhk+lhk = hk.

(3.44)

Together with (3.42), (3.43) these are the defining relations of the TemperleyLieb algebra (Temperley and Lieb, 1971), a quotient of the Hecke algebra. This algebra is satisfied by the hopping matrices (3.28) of the asymmetric exclusion process with q = x/DR~Dr,, but also by the generator of the m-species exclusion process with hopping rates

AiR ~ ~Ai

with rate

DR

~Ai "+ Ai~

with rate

DL.

(3.45)


51

This process is the tagged-particle version of the usual exclusion process. Another hopping process, corresponding to a different quotient of HM (q) is the priority exclusion process, t In addition to the transitions (3.45) one allows for hierarchical particle interchange with rates

AjAi ~ AiAj

with rate

DR u j > i

AiAj ~

with rate

DL u j > i

AjAi

(3.46)

For m -- 2 this is the asymmetric exclusion process with particles of species A2 and with second-class particles A1 (Ferrari et al., 1991). The second-class particles move like the ordinary first-class particles with respect to the vacancies. However, seen from the particles A2, the second-class particles behave like vacancies. Other quotients give rise to reaction--diffusion systems. These include pairannihilation-creation processes of the type AiAi ~

Ai+lAi+l

Ai+lAi+l ~

AiAi

with rate

/z

with rate

k

(3.47)

and the constraint/z = DR + DL -- )~. For m -- 1 this is the pair-annihilationcreation process (3.39) related to Glauber dynamics. All algebraic properties are independent of the choice of basis. We conclude that by checking the algebraic relations satisfied by the local transition matrices hk of a stochastic Hamiltonian one finds sufficient (but not necessary) conditions for integrability of the system. This is a first step towards an exact solution of the model. One may use the Bethe ansatz (Section 6.4), symmetry under quantum algebras (Section 7.1.1) or free fermion methods (Section 9) for the calculation of correlation functions. We shall consider almost exclusively single-species processes. The consequences of integrability for multi-species processes have so far remained essentially unexplored.

Comments Section 3: For an introduction to integrable quantum spin systems and vertex models which play such a pivotal role in our discussion we refer to Baxter's work on exactly solvable models (Baxter, 1982). For a deeper understanding of integrability beyond what is necessary here we refer the interested reader to the existing literature (Thacker, 1981; Baxter, 1982; Gaudin, 1983; Izyumov and Skryabin, 1988; Korepin et al., 1993). tThe corresponding additional relations satisfied by the matrices h k for this process are not of further interest, for details see Alcaraz and Rittenberg (1993).

52

G . M . SchQtz

Section 3.3: (i) The asymmetric exclusion process has additional applications to those mentioned above. By the same mapping as for the symmetric exclusion process (Meakin et al., 1986; Plischke et al., 1987) (Appendix B) the system describes the stochastic growth of a one-dimensional interface in the universality class of the Kardar-Parisi-Zhang equation (Kardar et al., 1986) and, in yet another mapping, directed polymers in two-dimensional random media (Krug and Spohn, 1991; Halpin-Healey and Zhang, 1995). The boundary-induced phase transitions discussed in Section 7.2 correspond to unbinding transitions of directed polymers in two-dimensional random media (Krug and Tang, 1994). A mathematical motivation for studying the process is the integrability of the quantum chain (3.28) by means of the Bethe ansatz (see Section 7) and its equivalence to the integrable asymmetric six-vertex model (Sutherland, 1967; Sutherland et al., 1967; Gwa and Spohn, 1992b; Nolden, 1992; Bukman and Shore, 1995). (ii) Some of the processes described here have also been studied in discrete time. The generator of the exclusion process with a sublattice-parallel update is the transfer matrix of the six-vertex model on a diagonal square lattice (Kandel et al., 1990; Sch/itz, 1993a,b; Honecker and Peschel, 1997; Rajewsky and Schreckenberg, 1997). Diffusion-limited pair annihilation--creation corresponds to the eight-vertex model. With this update scheme one obtains an integrable process. Different discrete-time updating schemes for the asymmetric exclusion process have also been considered (Yukawa et al., 1994; Jockusch et al., 1995; Schreckenberg et al., 1995; Hinrichsen, 1996; Rajewsky et al., 1996, 1998; Tilstra and Ernst, 1998; deGier and Nienhuis, 1999). A different discrete-time description of diffusion-limited pair annihilation has been introduced by Privman and collaborators (Privman, 1993, 1994; Privman et al., 1995). lntegrability of models with these updating schemes has not been investigated.

Section 3.4: (i) We mention another (sufficient, but not necessary) criterion for integrability which can be checked for a given stochastic process. This is the Reshetikhin relation (Kulish and Sklyanin, 1982) [ hk + h k + l , [ hk, hk+l

]] =

gk -- gk+l

(3.48)

with arbitrary matrices gk, acting on sites k, k + 1. (ii) The totally asymmetric tagged particle process (DL = 0) is integrable also if the individual particle species have different hopping rates D~ ) provided that the particle exchange rates are given by D- (Ri ) _ D~/) for D~ ) > D~ ) and zero otherwise (Karimipour, 1999c). In this process fast particles pass slow particles. (iii) Detailed analysis of the totally asymmetric priority exclusion process ( D L = O) using a matrix product technique (Derrida et al., 1993b) allows for a study of the structure of shocks. The same technique, reviewed briefly below, allows also for a study of the m = 3 process (Mallick et al., 1999).

53


4.1

A s y m p t o t i c behaviour

The infinite-time limit

4.1.1 Stationary states One of the most basic questions to ask is the behaviour of a system out of equilibrium at very late times of the stochastic evolution. One would like to know quantities like the mean density, density fluctuations, or the spatial structure of the density distribution and its correlations. For transition rates that are constant in time the asymptotic distribution I P* ) is invariant under time translations,

HIP*) = 0 ,

(4.1)

and hence called stationary. This is the distribution to which the system relaxes after a very long time. For a model with discrete-time update the analogous relation for the stationary vector reads T I P * ) = I P* }.

(4.2)

A useful representation of the stationary vector in terms of the diagonal matrix p* P* = E

e*(r/)l r/)( 171

(4.3)

7/

with stationary probabilities P* (7/) on the diagonal is given by the expression

I P* ) = P * l s ).

(4.4)

The product state (2.24) can also be written in this form with P* = Ilk (1 - p + ( 2 p - l)nk).* For a different interpretation of the stationary distribution we recall the correspondence between quantum spin chains in d dimensions and transfer matrices of statistical mechanical models in d + 1 dimensions. Any stochastic quantum spin chain with local interactions may be derived from some transfer matrix with local interactions; see Appendix A for an example. A transfer matrix describes, by its definition, the equilibrium behaviour of the associated (d + 1)-dimensional model. The time of the associated stochastic system, i.e. the tth power of the transfer matrix is nothing but the length of the statistical mechanical model in the dimension d + 1. Hence the stationary state of the d-dimensional stochastic *Using the same notation for the matrix P* and the distribution P*(r/) will not give rise to confusion since context always dictates unambigously what is meant. In general we shall make no distinction in notation between diagonal matrices and the functions which give the entries on the diagonal of this matrix.

54

G.M. SchQtz

process gives the equilibrium distribution of some (d + l)-dimensional model which is infinite in space direction d + 1. Therefore, stationary expectation values of the stochastic process correspond to equilibrium expectation values in the associated (d + l)-dimensional model. This seemingly innocent remark is important for understanding the stationary behaviour of one-dimensional stochastic processes. In one-dimensional equilibrium systems with short-range interactions, long-range order cannot exist at any finite temperature and hence there is no second-order phase transition with a corresponding divergent correlation length. Since the stationary states of the onedimensional systems that we are interested in actually correspond to the equilibrium states of two-dimensional models, there is nothing to tell us that long-range order should not occur. This example demonstrates that anything that can happen in a two-dimensional equilibrium system may, in principle, also occur in the stationary states of one-dimensional interacting particle systems. The dynamical interaction of the one-dimensional system encoded in the local transition rates has no a priori relationship to the nature of the equilibrium distribution of the corresponding two-dimensional system" local interactions may very well give rise to long-range order. One may wonder whether a stationary state exists and if so, then how many linearly independent stationary states there are. In a system with finite state space it is easy to prove existence" by construction there is at least one left eigenvector with vanishing eigenvalue (see (2.14)). This guarantees the existence of at least one right eigenvector (4.1) with vanishing eigenvalue. Also by construction, the eigenvalue of H with the lowest real part is zero and there is no eigenvalue with vanishing real part but nonzero imaginary part. This follows from a theorem by Gershgorin (Gradshteyn and Ryzhik, 1981) and ensures that a stationary distribution is indeed a limiting distribution I P * ) - l i m / ~ e-Htl P(O) ) for some initial distribution P(0). In quantum mechanical language the stationary vector corresponds to the ground state of H. However, if H is not Hermitian this vector is not the transposed vector of ( s I, but a more complicated object.

4.1.2

Ergodicity

There is no equally simple general argument which gives the number of different stationary states (i.e. linearly independent eigenvectors with vanishing eigenvalue). Evidently, uniqueness is an important property of a system, as, if the stationary distribution is not unique, the behaviour of a system after long times will keep a memory of the initial state. Also a time average over an expectation value is then not equal to the (not uniquely defined) stationary ensemble average, i.e., the system is nonergodic. It is therefore of interest to gain some general knowledge how uniqueness and ergodicity is related to the microscopic nature of


55

the process. To this end one has to study the possibilities of moving from one given state r/to some other state 17' after a finite time.* A discussion of related results and proofs of various theorems can be found in Chapter II. 1 of Liggett (1985). An important theorem for discrete-time systems asserts that if one manages to identify a subset X' of states such that one can go from each of these states to any other state within this subset with nonzero probability after some finite time, then there is exactly one stationary distribution for this subset. Furthermore, the support of the distribution is identical to X', i.e., the stationary probability P* (1/) is strictly larger than zero for all states r/ E X t. Restricted on such a subset, the system is also ergodic. To illustrate the theorem, consider first a lattice gas on a finite lattice with particle number conservation. If the dynamics are such that for fixed particle number each possible state can be reached from any initial state after finite time with finite probability then there is exactly one stationary distribution for each subset of states with fixed total particle number (Fig. 11).

(Xl) Fig. 11 Separation of the state space X into disjunct subsets X i. Transitions can only occur within each subset. There is exactly one stationary distribution for each subset. If instead of particle number conservation one allows also for production and annihilation processes of single particles with configuration-independent rates, then one can move from any initial state to any other state, irrespective of particle number. In this case there is only one stationary distribution for the whole system. For the pair-creation-annihilation process (3.39) there are two stationary distributions, corresponding to even and odd particle numbers respectively. In each case the system is ergodic within the respective connected subsets. tAt this point we would like to remind the reader that we are considering systems with finite state space. Much of what is discussed in this section does not hold for infinite systems.

56

G.M. SchQtz

If only annihilation processes occur then the particle number will decrease until no further annihilations can take place. Such a system is nonergodic on the full state space, and ergodic only on the subset of states in which no further annihilations occur. Such a subset is called absorbing (Fig. 12). Hence uniqueness of a distribution does not imply ergodicity on the full subset of states which evolve into the absorbing domain. By relabelling of the basis vectors the time evolution operator for such processes can be brought into a block structure with blocks on the diagonal corresponding to states with a given particle number and blocks only above or only below these diagonal blocks. The off-diagonal blocks correspond to the annihilation transitions connecting blocks of different particle number.

(Y) \

Fig. 12 A stochastic system with absorbing subspaces X l, X2. Transitions are possible within each of the three sets and from states in the transient set Y to either X I or X2, but not out of X I and X2. With the help of ergodicity we can investigate the limiting behaviour of a process on the level of the time evolution operator exp ( - H t ) . The matrix T* -- lim

e -Ht

(4.5)

t---~ (x)

is a projection operator, (T*) 2 = T*. By its definition T* maps any initial state to a stationary distribution. For an ergodic system all columns of T* are identical and have as entries T* the stationary probabilities of finding the state 17. In this rl , rl ' case one may write T* = I e * ) ( s I. (4.6) An analogous expression can be obtained for systems which split into disjunct subsystems. In this case T* is a sum of expressions of the form (4.6), but with


57

summation vectors and the stationary vectors restricted to the respective ergodic subsets. For systems with absorbing states there is no generic expression for T* in the presence of more than one absorbing subset.

4.1.3

Detailed balance

For many applications it is important to construct a process such that a given probability distribution is stationary. This is the case, e.g. in Monte Carlo simulations of equilibrium systems. Then the distribution function is a Gibbs measure P*(r/) cx exp (-fiE(r/)) where/3 = 1/(kT) is proportional to the inverse temperature and E(r/) is an energy function. In order to construct such dynamics one has to ensure that the proposed time evolution of the system approaches the equilibrium distribution. One possibility of solving this problem is implementing detailed balance on the transition probabilities (or rates respectively). A system is said to satisfy detailed balance if the ratio of the hopping rates w(O', rl) between two states r/, 7/' equals the exponential e x p ( - / 3 A E ) of the energy difference AE = E ( r / ' ) - E(O) resulting from the hopping event (Fig. 13). The detailed balance condition then reads

P(~?)w(~7', ~) = P(rl')w(o, 0') V )7 # rf E X.

(4.7)

It is clear from the master equation (2.2) that P is a stationary distribution, i.e. P = P*. Thus the hopping ratio is the equilibrium ratio of the probabilities of finding these states. Systems satisfying detailed balance are called equilibrium systems and their stationary distribution an equilibrium distribution or equilibrium state.* An example is Glauber dynamics introduced above. It is easy to check that the transition rates defined in Section 3.3.3 satisfy detailed balance with respect to the equilibrium distribution (3.3), i.e. this is the state the Glauber system will relax into after sufficiently long time. To avoid confusion we stress that this notion of equilibrium system refers to the nature of the d-dimensional dynamical system that one is studying. It is conceptually unrelated to the associated (d + 1)dimensional equilibrium system defined by the transfer matrix as discussed above. Equilibrium systems have a number of special properties. The restriction to an ergodic subspace Xi where P*(r/) # 0 V r/ e Xi allows one to invert the matrix P* defined in (4.3). With this matrix we can formulate the following three equivalent statements. (i) The process generated by H on Xi satisfies detailed balance with respect to P* (r/). +Fordiscrete-time dynamicsone replaces the transition rates by the transitionprobabilities p(r/t, r/) in the definitionof detailed balance. Notice that in the probabilistic literature the notion 'equilibrium' refers more generally to what we call 'stationary'.

G.M. Sch0tz

58

w(o'~

11

17'

Fig. 13 Stochastic transitions between two states of different equilibrium energies E, E'. (ii) H can be written in the form H = S ( P * ) - 1 for some symmetric stochastic matrix S. (iii) H T = ( p , ) - I

HP*, where H T is the transpose of H.

The proof of the equivalence is elementary and requires not more than using the invertibility of P* and the insertion of unit matrices )--~ I r/)( 171 at suitably chosen positions in the detailed balance condition (4.7). Notice also that the similarity transformation H -~ (P*)-1/2H(P*)I/2 = (P*)-I/2S(P*)-I/2 proves that the Hamiltonian can be symmetrized. Since this symmetric Hamiltonian has real matrix elements it is also Hermitian. Hence detailed balance implies that the eigenvalues of the generator are all real and that the related symmetrized generator can be interpreted as a Hamiltonian of some quantum system. Important quantities characterizing the equilibrium behaviour of a system include not only the stationary expectation values, but also the time-delayed equilibrium correlation functions lim ( Fl (r + t) Fx(r) ) = ( FI (t)/72(0) ) p,.

(4.8)

r---~ o o

What one calculates with this quantity are time-dependent fluctuations in a system which had sufficient time to reach equilibrium. Since the system is assumed to have a unique stationary distribution, this expression is independent of the initial state. From property (iii) we can derive a time-reversal symmetry for timedependent equilibrium correlation functions. Time-reversal symmetry (or reversibility (Liggett, 1985)) means

(Fl(t)F2(O) )p, = (F2(t)Fl(O) )t'*

(4.9)

The proof is not difficult since by definition Fl, F2, P* are all diagonal and hence commute and are invariant under transposition. Therefore

( FI(t)F2(O)}p,

=

(S IFle -ntF2l P* )


=

(SIP*F1e -HrtF2(P*)-1I P*)

=

( P * IFle - H r t F 2 l s )

=

(SJF2e-ntFll

=

(F2(t)FI(O))p,

59

P*) (4.10)

where we have used invertibility of P* and the representation (4.4) of the stationary vector. For interacting particle systems time-reversal symmetry has an interesting corollary. The time-delayed correlation function where F1 = nk, F2 = n/ describes the evolution of a local density perturbation in equilibrium. One obtains the relation (nk(t)nt(O) ) p, = (n/(t)nk(O) ) p,. Thus the local equilibrium density fluctuations are necessarily also invariant under interchange of the space coordinates. Time-reversal symmetry can be extended straightforwardly to multi-time correlators. In the absence of detailed balance there remains a nonvanishing net 'current' j (rl, O') = P ( o ) w ( o ' , 11) - P(o')w(rl, 0') between some states 1/, r/' even if stationarity is reached. Such a stationary currrent is the signature of a nonequilibrium system. Only the sum over all such currents in and out of a given state 1/vanishes in the stationary state. This can be seen from the stationary form of the master equation (2.2) which may be written Y]o' J (r/, r/') = 0.

4.2

4.2.1

Late-time behaviour Density relaxation and the dynamical structure function

The next fundamental question after understanding the stationary behaviour of a system is its late-time approach to this state. There are two different ways to probe the characteristics of a system at late times. One either prepares the system in a non-stationary state I P0 ) and then observes, e.g. the approach of the particle density or of particle correlations at late times to their stationary values. For the local density this behaviour is given by the late-time behaviour of the expectation value Apk(t) = (nk(t)) Po - P~ = ( S Inke-nt I Po ) -

lim ( S Inke -U'l

Po>. (4.11)

Alternatively, one can measure time-delayed correlation functions in the stationary state. Experimentally this is done by first waiting for the system to relax. Then one introduces a small perturbation and measures how quantities like the density decay again to their stationary values. A basic quantity of interest is the dynamical structure function. This is the Fourier transform of the time-delayed connected density--density correlation

60

G.M. SchQtz

function C* (k, l; t) in the stationary state defined by C * ( k , l " t)

-

=

lim ((nk(r + t ) n l ( r ) ) p o

t---~ o o

- (nl,(r + t ) ) ( n t ( r ) ) p 0 )

( S I(nk - p ~ ) e - H t ( n l - PT)I P* ).

(4.12)

In an ergodic system the choice of the initial state is immaterial. The physical meaning of this quantity can be understood as follows. One waits until the system has reached is stationary state and then follows the time evolution of only those states which have a particle at site l. After a time interval t one then measures the density at site k, averaged over many realizations of the process. (For systems with multiple occupancy or different species of particles the averaging after time evolution is weighted with the number of particles at site l.) In the absence of correlations in the steady state one may give the structure function another interpretation: one perturbs the stationary state by introducing a local chemical potential which corresponds to an injection of extra particles at this point I. This generates a state which is only locally nonstationary and has a density profile with a delta-peak at site I. After some time one measures again the spatial density distribution at sites k and observes how the local perturbation spreads within the system and ultimately disappears. Loosely speaking one may generally say that in the absence of long-range order the dynamical structure function measures the spreading of a localized perturbation in the stationary state. From the dynamical structure function one can read off the collective drift velocity Vc and the collective diffusion constant Dc of the particle system. To define these quantities in a translationally invariant infinite one-dimensional system we normalize the correlation function C*(k, l; t) -- C*(r; t) by its spatial average 1

R Z

--

lim C (t) -- R~oo 2R

C*(r; t)

(4.13)

r=-R+l

to obtain the normalized spatial distribution function 6"(r; t) = C*(r; t ) / - C ( t ) . The moments of the distribution function are the expectation values (Xn(t)

) -- ~

(4.14)

rn~7(r 9t). r

The collective drift velocity and diffusion coefficient respectively are then defined by (Fig. 14) Vc

=

d lim - - ( X ) dt

(4.15)

t---~ o o

Oc

=

1 li m d (

-~ t ~

d---t (

X2 )

- (X

)2)

.

The definition of these quantities for other lattices is analogous.

(4.16)


.

p(x)

tl

-L ..,-

61

/% ~. ,

9

11)19

~"

.

.

2 .

!

-ll

.

.

"

"

.

9

.

.

.

I

.

.

.

.

.

.

.

. . .

.

.

t

xo

xo -k- Vc ( t2 -- t l )

Fig. 14 Diffusive spreading of a density perturbation in the stationary state at two times t2 > t l. The collective velocity describes the motion of the centre of mass of the perturbation, the collective diffusion constant gives the width w - x/2Dct of the profile 9

4.2.2

Collective velocity and particle current

In systems with particle number conservation the particle current plays an equally important role as the local density itself 9 Writing pk(t) = (nk(t) ) they are related via the lattice continuity equation

-~pk(t) = Z

-

-

(4.17)

p

where each partial current j~P) results from elementary hopping events between sites k to k + p. The precise form of these currents follows from the system dynamics through the equations of motion (2.17). Each partial current is composed of two parts j~P'+) corresponding to hopping from k to k + p and back respectively, i.e., j2 p) = j2 p'+) - j2 p'-). From these partial currents one obtains the current across a bond (k, k + 1) as

Jk = Z

P J2 p)"

(4.18)

p

In the stationary state these currents are independent of k. Notice that, for systems with particle number conservation, C is independent of time and proportional to the variance of the panicle number in the stationary distribution since -C = ( S I N e - H t n o l P* ) - p* ( N ) = ( Nno ) - p* ( N ) = 1 / L ( ( N 2 ) - ( N )2). Hence, if we assume stationary correlations to decay faster than l / r , where r = k - 1 is the lattice distance, then (4.17), (4.18) and translational invariance allow us to write Vc = ~-~k ( S IJk,e - H t ( n o -- P*)I P* ) / C = ( S IJo e - H t (N - ( N ))1 P* )/C. Using again particle number conservation results in the expression Vc = ( J N ) - ( J )( N ) (4.19) (NZ)-(N) 2

62

G.M. Sch0tz

where J = I / L ~ k Jk is the space-averaged current. Consider now a 'grand-canonical' stationary distribution It)* ) = E

zNIN* ) / z

(4.20)

N

where IN* ) is the stationary distribution of a large, but finite system of exactly N particles and the 'partition function' Z = ~ N zN ( S I N * ) normalizes I P* ). The density dependence of the current depends on the details of 'canonical' distributions IN* ). However, the relation a vc = -z- J (P) op

(4.21)

is generally valid. It follows in a straightforward manner from the definition of I P* ) by taking the derivative of the current with respect to the density. Equation (4.21) may be seen as a nonequilibrium fluctuation-dissipation relation expressing the current response to a change in the density in terms of the drift of local fluctuations in the stationary state of the system. In an equilibrium system satisfying detailed balance the current and hence the collective velocity vanishes. Notice that in the continuum limit of vanishing lattice spacing (known as the hydrodynamical limit) the continuity equation (4.17) turns into the partial differential equation 0t p = igxJ. In terms of the scaling variable u = x / t this equation has the scaling solution p(u) given by u = Vc(p).

(4.22)

For a lattice system this yields the time evolution of density profiles on large space-time scales. For some systems, including the exclusion process, there are rigorous proofs for the validity of this description (Kipnis and Landim, 1999).

4.2.3

Relaxation times in finite systems

Often it is too difficult to find explicit expressions for correlation functions, but it may still be possible to determine the spectrum of the evolution operator. By inserting a unit operator in the form of a complete set of eigenstates y~'~ l e )( E I in the expression (2.16) of an equal-time correlator one obtains the spectral decomposition (F(t) )eo = Y~ ( S IFI ~ )( E I P0 ) e -~t (4.23) ~f

of the expectation value. In this sum the term with energy zero is the stationary value ( F)*. One realizes that the approach to stationarity at very late times is governed by the lowest energy gap Emin of H since for t >> 1/Emin one can


63

neglect all terms with e > Emin in the sum.* Thus in a finite particle system the decay to stationarity is always exponential, ( F (t)) Po ~ ( F )* + ( S IFI Emin ) ( Emin I P0 ) e - Emint

(4.24)

and Emin gives the longest relaxation time r - 1/Emin. Finally we consider the relaxation times for integrable systems with a quantum Hamiltonian in the form (3.35) where the local interaction matrices are generators of some quotient of the Hecke algebra (3.41)-(3.43). We have seen that particular representations of this quotient define a certain stochastic process. One can show that these processes have, up to the degeneracies, the same spectrum (Alcaraz and Rittenberg, 1993). Consequently, such processes have the same longest relaxation time r.

4.2.4

Infinite systems and dynamical scaling

A whole series of relaxation times may increase with system size such that in the infinite volume limit the spectrum becomes continuous. This indicates algebraic (or even slower) approach of correlation functions to their stationary values rather than the exponential decay which characterizes all systems with finite state space. Hence one can read off important information from the finite-size scaling behaviour of the energy gaps. Before considering many-particle systems it is instructive to study this phenomenon for biased single-particle diffusion on a ring with periodic boundary conditions. The master equation (2.3) for this process is readily solved by discrete Fourier transformation. Since the process is defined on a finite lattice we make the ansatz Px (t) = Y~p A p (t)e ipx where p takes discrete values p = 2zr n / L. Inserting this in the master equation gives an ordinary first-order differential equation in time for the amplitude Ap(t) which is readily solved by Ap(t) -- Ap(O)e -~pt with the 'energy' ~p = D R ( I -- e - i p ) + D L ( I -- eiP). (4.25) It is then easy to verify that the initial amplitude Ap(O) = e-ipY/L yields the solution 1 Px(t) = -~ y ~ e-'pt e ip(x-y) (4.26) p

of the master equation with initial condition P(x; O) = 6x,y where the particle is placed on site y. This solution is then the conditional probability P(x; tly; O) defined above (2.19). tln case of complex eigenvalues we mean by 'lowest energy gap' the eigenvalue with the lowest (positive) real part.

64

G.M. SchQtz

In the infinite-time limit only the zero mode with p = 0 contributes to (4.26). Hence in the stationary state the particle can be found with equal probability anywhere on the lattice, Px = I/L. For a large system the real parts of the lowlying energy gaps scale ~,, ,~ n 2 / L z with z = 2. Thus for times large compared to L2 the decay to stationarity is exponential. This raises the question of what happens in the infinite-volume limit L cxz. The corresponding master equation is solved in essentially the same way, except that the discrete sum over momenta p is replaced by an integral. With the asymmetry q = ~ / D R / D L (3.29) and the time-scale factor Do = ~/DRDL one finds the conditional probability in the infinite system P(x; tly; O)

'f0

=

-~

d p e -~'t+ip(x-y)

=

e-(q+q-t)DotqX-Ylx_y(2Dot)

(4.27) (4.28)

where In ('C) is the modified Bessel function In(r) = ~

'F

~r

dp e ipn+r cos p

(4.29)

The representation of (4.27) in terms of the Bessel function (4.28) is obtained by an elementary contour integration. It is easy to verify that both expressions satisfy the same differential-difference equation with the same initial condition P(x; 01y; 0) ~x,y. Without loss of generality we assume q > 1, i.e. the particle moves preferably to the fight. The conditional probability (4.27) describes the decay of the 'density profile' defined by the spatial distribution of the probability of finding the particle at time t t To investigate the asymptotic behaviour of the exact expression (4.27) for large times we note first that the 'momentum' p takes real values - J r < p < Jr. For large times only the contributions with small p (corresponding to the slowly decaying modes with small energy gap) contribute to the integral. Expanding Ep ~ (DR + D L ) p 2 / 2 + i(DR -- D L ) p to the lowest order in the real part and setting r = x - y, v = DR -- DL and D = (DR + D L ) / 2 one obtains asymptotically for large t the Gauss distribution for a random walker =

1 _(r_vt)2/(2Dt) . Pr(t) ~" ~ e ~/2rr Dt

(4.30)

*Talking about a 'density profile' in the context of a single particle may sound contrived. However, the density expectation value of a system of noninteracting particles satisfies the same differentialdifference equation (2.3) as the single-particle probability. Up to an overall amplitude p this leads to the expression (2.3) for the time-delayed correlation function (4.12) in the stationary state with density O. This correspondence justifies the use of many-body language for the single particle. We have chosen just a single particle because we do not wish to obscure the triviality of the discussion.


65

The particle moves with average velocity v, but fluctuates around the centre of mass r = vt with diffusion constant D (cf. Fig. 14). By going into a comoving frame with velocity v = D ( q - q - l ) _ DR - Dr., i.e. by studying the behaviour of the distribution around r' = r + vt, one finds algebraic decay ~ 1/4~ of the conditional probability, as argued above. We note that the conditional probability transforms covariantly under the scale transformation r' ~ )~r', t w+ )~zt with z = 2 since Pr'(t) )~P~r'().2t). This dynamical scale invariance means that the conditional property does not change its form if measured on different length- and time scales. The scaling exponent z which appeared already in the finite-size scaling behaviour of the system is called the dynamical exponent. Generally, this quantity relates the scaling behaviour in spatial direction to the temporal scaling on large scales. Rescaling spatial coordinates by a factor ~. and at the same time rescaling time by ,kz leaves correlation functions invariant up to an overall amplitude. Systems with this scale invariance are dynamical critical systems. Quantities like the dynamical exponent are universal for such systems, i.e. do not depend on the microscopic realization of the processes which are in the same universality class. In the case of the random walk this is apparent in the irrelevance of the lattice description. Brownian motion, i.e. random walk defined on the real line by a Fokker-Planck equation has the same dynamical exponent. =

4.2.5

S o m e caveats

Thermodynamic limit The results of the previous two subsections may suggest the following" if the infinite-volume limit of the lowest energy gap of the particle system is finite, then the decay of the infinite system to stationarity is exponential with relaxation time r~ =

lim 1/Emin(L) = L--~ o o

lim rL.

(4.31)

L---~ cx:~

Unfortunately however, sometimes nature is unkind and a certain amount of caution is necessary before relating the spectral properties of finite particle systems to the relaxation times of the corresponding infinite systems. Since we have just asserted that for any finite system rt~ = !/Emin(L), it may come as a surprise that (4.31) is not generally valid. This somewhat paradoxical statement has its mathematical explanation in the fact that the eigenvalues of an infinite-dimensional operator are determined by the boundary conditions on the wave functions. In the present context they are determined by the requirement that I P ( t ) ) is normalizable, ( S[ P ( t ) ) = 1. This is the analogue of the quantum mechanical requirement ( 9 [ qJ ) = 1 which determines the spectral properties of quantum mechanical operators.

66

G.M. SchQtz

To get insight into the physical meaning of this explanation and to show that counterexamples to (4.31) are by no means restricted to particularly exotic processes, we discuss two different ways of resolving the apparent contradiction. The conceptually simplest possibility is vanishing matrix elements in the spectral expansion (4.24), i.e. ( Emin I P0 ) = 0 or ( S IFI Emin ) = 0. This may happen in nonergodic systems which split into disjunct subsets Xi. The lowest energy gap of the complete system is not necessarily equal to the lowest energy gap of the sector Xi to which the initial state belongs and to which the dynamics are restricted. A less obvious possible scenario is a decay of expectation values with the longest relaxation time rL only after a crossover time t* which increases with system size. Before this crossover time expectation values could decay at a slower rate. A mechanism which results in such behaviour is a Galilei transformation into a moving frame of reference. For illustration we consider the biased hopping process on a finite chain of L sites (Fig. 3), but with reflecting boundaries where hopping attempts out of the system are rejected. This modification leads to the master equation (2.3) for the bulk, but with the modifications d dt PI (t)

=

Dr. P2(t) - DR el (t)

(4.32)

d dt Pt. (t)

=

De Pt.-i (t) - Dr. PL (t)

(4.33)

at the boundary sites. As a result, the stationary distribution is not constant, but exponential, P* oc q2X

(4.34)

and satisfies detailed balance. Because of the reflecting boundaries the system relaxes to an equilibrium state where no stationary current can flow. The stationary probability of finding the particle decays exponentially from the boundary towards which the particle is driven. To obtain the relaxational behaviour we note that the reflection property may be reformulated by artificially extending the range of validity of the bulk equation to all integers and at the same time imposing the boundary conditions De Po(t) = DL PI (t) and DR PL (t) = DL PL+I (t) for all times t. This strategy ensures that both the bulk equation and the boundary equations are satisfied at all times. Within the range l _< x _< L the solution of this extended master equation yields the probability Px(t). Outside this physical range the expression is still well-defined as a solution of the master equation, but does not have the physical interpretation of a probability. This technique of extending the system by 'ghost coordinates' will be used extensively below also for the solution of many-particle problems. The point is that then the dynamics can be solved with a plane wave


67

ansatz of the form

Px(t) = Z

Ap(t) (e ipx 4- Bpq2Xe-ipx).

(4.35)

p

This gives the dispersion relation (4.25). Satisfying the boundary conditions fixes Bp = - ( 1 - q-2eip)/(l - e -ip) and at the same time requires the momentum to take quantized complex values Pn -- zr n~ L - i In q. Thus the spectrum (4.25) is real and one finds a finite energy gap Emin(L) ---- Do(q + q-] _ 2) + O ( 1 / L 2) for all system sizes. In particular, in the thermodynamic limit Emin :

lim Emin(L) -- Do(q + q - l _ 2) > 0.

L---~oo

(4.36)

This indicates exponential relaxation of the conditional probability, in apparent contradiction to the algebraic decay that one obtains if the infinite-volume limit is taken from the outset. To understand the relationship between a large, but finite system and the infinite system we assume the particle to start at a site y which is far away from the boundaries. We note: (i) By going into a comoving frame with velocity v, i.e. by studying the behaviour of the distribution around y' = y 4- vt, one finds for times in the range 0 1/2 the transformation leads to a random initial state I P ) with positive probabilities only in the range 0 < a < In ( l / p ) .

5.2

Enantiodromy

A second type of equivalence between two processes /~, H is defined by the relation ISI = B H r B - l (5.7) which relates a stochastic Hamiltonian by some similarity transformation to the transposed Hamiltonian of some other system. Such processes shall be called e n a n t i o d r o m i c with respect to each other.* If H = H, then the process is selfenantiodromic. All systems satisfying detailed balance are self-enantiodromic with respect to the diagonal transformation P* (4.3). In the case of enantiodromy the expectation value ( F ) ~; for the process H is given by an expectation value for H where the transformed initial state /5 is determined by the choice of observable F and the transformed function /~ is determined by the initial distribution P. For nondiagonal transformations this can be useful since properties of the enantiodromic process with the transformed initial condition may be much simpler than that of the original process. Such a simplification, if it exists, manifests itself in the equations of motion for the expectation values of suitably chosen functions F. For equal-time correlation functions this property gives rise to what in the theory of interacting particle systems is known as duality (Liggett, 1985). We avoid using this notion which we want to reserve for the domain-wall duality between Glauber dynamics and DLPA. Moreover, enantiodromy plays an important role beyond applications to the usual equal-time expectation values, viz. for the calculation of multi-time correlation functions. Relations analogous to (5.2) can be obtained by taking the +The notion has its origin in the time-reversal of the temporal order of the stochastic evolution of the enantiodromic process. It refers to the consideration that the transposed matrix describes the motion of a process 'backwards' in time, i.e. a path (dromos) of the original forward process in state space is followed in opposite (anti) direction by the backward process. I thank C. Likos for help in finding a suitable notion for this relationship.


75

transpose of the correlator (2.25) and inserting the definition (5.7) of the enantiodromic process. Below we apply self-enantiodromy to the symmetric exclusion process and to diffusion-limited pair annihilation to show how the calculation of time-dependent k-point correlators for many-particle distributions reduces to the solution of a problem involving only k or 2k particles respectively.

5.3 First-passage-time and persistence probabilities Important expectation values besides the correlation functions (2.25) are firstpassage-time and persistence probabilities. These quantities are not obtained by measurements at some given instant in time, but by observing the system over a finite period of time. The notion of persistence refers to the tendency of a stochastic process to remain in its current state. Thus it plays an important role in the investigation of stochastic dynamics. For illustration consider a discrete-time exclusion process defined on a finite lattice. Initially, one assumes, e.g. site 1 to be empty and one asks the question, what is the probability g ( 1 ; t ) that site 1 remains empty up to time t, i.e. has never been visited by a particle up to time t? Obviously, this persistence probability is given by the expression g ( l ; t ) = (SI131T131...131T131TIP0) = ([131T131]t) which is zero for all stochastic paths which involve configurations where a particle has moved to site 1 at any time step from zero to t. Defining the nonstochastic matrix 7~ -- 131 T 131 one may consider g(1; t) to be generated by the time evolution of this nonstochastic generator. Next we consider the probability f ( l ; t) that site 1 is visited by a particle at time t for the first time. f (1 ; t) is the expectation value f(1; t) = ( S Inl (T 131)tip o> -- g ( 1 ; t - 1) - g(l; t)

(5.8)

which, since 132 = 131, is also generated by T. In the continuous-time limit with a Hamiltonian H one obtains lim ~t/~

=

131

exp (-/-)t)

(5.9)

~ ----~ o o

with /-) = 01 H 131. Hence the persistence probability is given by g(1; t) 1/2 and a second-class particle located in that region moves to the left, again in direction of the shock. A second-class particle at the shock position moves with the shock velocity v = 1 - P L -- ,OR which is positive in this figure. (From Popkov and Schlitz (1999).)


119

The Uq[SU(2)] symmetry has a remarkable consequence which allows us to actually calculate the full time-dependent distribution for a certain family of shock initial states. Since above the quantum group symmetry was defined on a finite system with reflecting boundaries and now we intend to take the thermodynamic limit, we consider here a lattice of 2L sites labelled from - L + 1 to L. We define the family of shock distributions

I lzk ) =

l+z )k q-2Nk zN I s ) / CL 1 + zq -2

(7.48)

which are product states with density Pl from - L + 1 up to site k and density ,02 from site k + 1 up to site L. Here the normalization factor CL = (1 + z)L(1 + z q - 2 ) t" and z = p 2 / ( l - P2). For reasons which will soon become clear we restrict ourselves to the special case /92(1 --,Ol)

= q2.

(7.49)

pl(l -,o2) We write the Hamiltonian as H =

L-1 Z f/i + (De - Dt,)(n-t,+l - nt,) i=-L+I

(7.50)

where the transposed matrix /.~r

L-I Z ~/r i=-L+l

(7.51)

generates the reflected process with preferred hopping to the left. In other words: H T = H1/q + (DR -- D L ) ( n - L + I -- nL).

(7.52)

It is our aim to calculate the time evolution of the shock distribution I U ~ ( t ) ) -- lim e - n t l / z k ) L---~oo

(7.53)

where we denote the thermodynamic limit of a shock distribution by I/z~ ). Let (/zk I be the transposed vector of[ #k ). Then for - L + 1 < k < L - ( lzk IHl/q = t ~ l ( / Z k - I I ~- t~2(/Zk+l I -- (DR + DL)( lzk I

(7.54)

where S1,2 -- (DR -- DL)

Pl,2(1 -- Pl,2) P2-- Pl

9

(7.55)

120

G . M . SchOtz

To prove (7.54) we transpose (7.48) and note that N and hence ZN c o m m u t e s with H. The next step is to apply the commutation relation (7.19). The quantum algebra symmetry then reduces the action of H to the left on a linear combination of one-particle states:

_,o,(

/01

DL

s+

+DR

\i=-L+I

i=-L+l

i=-L+I

(7.56)

Reversing the same sequence of steps leads to (7.54) since

81 - -

Dt~(l + z)/(l +

zq -2) and 82 - DR(I + zq-2)/(1 + z). Now we define shock distributions with a boundary perturbation, i.e. a shock distribution I ~ k ) p with density 1 at point p = - L + 1, L and k # p. The inhomogeneity at the boundary will evolve in time, thus eventually destroying the simple structure of the distribution. However, this perturbation spreads with finite speed from the boundary. Hence, by taking the thermodynamic limit for fixed t, any finite region around the shock position k will remain unaffected. Therefore we conclude lim e-Htl #j, ) - L + l = L-~oo

lim e-Htl t.zk )L -- lim e-Htl #k ) L--~ oo

(7.57)

L--~ oo

and d -tdl u k ( t ) )

:

lim e -HtHI k )

-

(7.58)

L---. ~

lim e -Ht [~llk -- 1 ) + ~21 k + 1 ) - (DR + DL)I k ) L--*~

- ( D R - DL)(pllk )-L+I -- ~ l k )L)].

(7.59)

This is because nLIk ) -- p21k )L and the analogous statement for p -- - L + 1. Taking the thermodynamic limit and using the constraint (7.49) which allows us to use the quantum algebra symmetry leads to d -tdI ~k(t) )

-

-

~1 I/~k-I (t)) + ~21/zk+l (t)) -- (~1 d- 62)1 #k(t) ).

(7.60)

Solution of this differential-difference equation with initial condition I lzk (0)) = I#k ) yields the main result

S 1) (k-l)~2 I # Y ( t ) ) - e -(a'

+62)tZ l

~22

I k - t ( 2 ~ 1 6 2 t ) l lz~ )

(7.61)


121

where I / i s the modified Bessel function encountered earlier in the investigation of the lattice random walk. Indeed, by inspection we see that a shock distribution I # ~ ) evolves into a linear combination of such distributions with weights satisfying the evolution equation of a lattice random walk with hopping rates ~1,2. Loosely speaking, the shock performs a random walk with these rates. Seen from the shock position, one has product measures with densities pl,2 to the left and right, respectively. This is true for all finite lattice distances r = 1,2 . . . . . Similar results can be obtained for an initial distribution with multiple shocks with consecutive increasing densities satisfying the constraints Pi+l(1 -- Pi) = q2. pi(1 -- Pi+I)

(7.62)

Because of the reduction to a k-particle problem the time evolution of the distribution with k shocks can be calculated explicitly using the Bethe ansatz. It is of interest to investigate the stationary distance distribution of the two shocks, i.e. the stationary probability p ( r ) of finding the shocks a distance of r sites. Work by Ferrari et al. (2000) for general densities suggests that the mean distance should be zero on the hydroynamical scale, i.e. on time scales of order t. Indeed, analysis of the equations for motion of two shocks indicate that on the lattice scale the stationary distance distribution p ( r ) is geometric with a finite mean (Belitsky and Schiitz, 1999). We conclude that on large scales (large compared to the lattice spacing) multiple 'small' shocks coalesce into one 'big' shock which may be seen as a (classical) 'bound state' of small shocks. On a physical basis this phenomenon can be understood by considering the relative velocity between subsequent shocks, which are all positive and hence try to decrease the distance between shocks. For general limiting densities of the shock the time evolution of a shock distribution on the lattice scale is more complicated and requires a careful microscopic definition of a shock. For any given initial configuration the shock position can be defined by introducing a 'second-class' particle (Ferrari et al., 1991) which we label by the symbol B. This particle moves with the same unit rate as all other 'first-class' particles A with respect to the vacancies. However, the first-class particles treat the second-class particle like a vacancy. This yields the two-species priority exclusion process (3.46) A0

~

0A

AB

~

BA

Bf3

~

OB.

As a result of these dynamics, the second-class particle is always driven into a region with a high density gradient (Fig. 21). This defines the shock position. By

122

G.M. SchQtz

tracing the motion of the second-class particle one can study the dynamics and the stationary structure of the shock (Derrida et al., 1993b, 1997). The second-class particle moves with average velocity (Lebowitz et al., 1988; Spohn, 1991 ) v = ( D R --

DL)(I -- PL -- PR)

(7.63)

and with a diffusion coefficient (Lebowitz et al., 1988; Spohn, 1991; Ferrari and Fontes, 1994a) 1 D = -(DR 2

-- D L )

p R ( I - P R ) + p L ( I -- P L )

.

(7.64)

PR -- PL

Not surprisingly, these are exactly the quantities one can read off the random walk rates for the special shock discussed above. However, the local structure of the system close to the shock is much more complicated in the general case. Seen from the second-class particle the distribution is uncorrelated only asymptotically; at finite lattice distances from the second particle the density profile becomes nontrivial in the long-time limit and nonvanishing correlations build up (Derrida et al., 1997). The shock velocity (7.63) is positive if the incoming particle current j L is less than the outgoing particle current j R , but negative for j L > j R . In this case, incoming particles pile up at the shock position and cause the shock to move to the left. This is a situation analogous to a back-moving shock front in a traffic jam. If incoming and outgoing currents balance each other, i.e. if PL -- 1 -- OR, the shock velocity vanishes. The expression for the drift velocity of the shock can be understood nonrigorously on very general grounds as resulting from conservation of mass. Suppose the shock has moved a distance Ax -- v a t in a finite interval of time. The change in area A = A x ( p R -- P L ) under the density profiles equals the change in particle number AN = ( j R -- j L ) A t due to the current. Hence one obtains the general expression for the shock velocity j R -- j L

v= ~

.

(7.65)

PR -- PL

Since asymptotically (for k ~ 4-r the state over which one averages is an uncorrelated product state, the left and right limiting values of the current are given by the current-density relation (7.25) a s j L , R - - ( D R -- D L ) R L , R ( 1 -- P L , R ) respectively and one recovers (7.63) for the ASEP. To understand the expression (7.64) for the diffusion coefficient we adopt a coarse-grained point of view from which the motion of the domain wall (defined by the position of the second-class particle) performs a biased random walk. The combinations j R , L / ( P R -- P L ) which determine domain wall velocity (7.65) can then be interpreted as being the effective jump rates D R , L to the fight (left). Thus


123

the diffusion constant is given by

1 jR + jL

D = - ~

,

(7.66)

2 PR -- PL

in agreement with the rigorous result (7.64) for the diffusion coefficient of the second-class particle and with the special case discussed above. The microscopic definition of the shock by means of the second-class particle furnishes us one with further important insight. The expression for the diffusion coefficient of the shock assumes that an average is taken not only over the histories of the time evolution, but also over initial states. This is implicit in the choice of the initial shock distribution which represents not a single initial configuration for which a shock position could not be properly defined without second-class particle. For a f i x e d random initial state with space-averaged asymptotic densities PR,L the shock velocity (7.65) is the same as for an initial shock measure since this expression follows from mass conservation alone. However, the fluctuations of the shock position and hence the diffusion coefficient a r e - for a fixed initial state - d r i v e n only by the small current fluctuations (van Beijeren, 1991; Johansson, 2000) that the system produces. There are no fluctuations originating in the averaging over random initial states. As a result, the fluctuations in the shock position become subdiffusive, growing only proportional to t 1/3 rather than t 1/2 (Gfirtner and Presutti, 1990; van Beijeren, 1991 ).

7.1.5

Small fluctuations

The behaviour of localized perturbations in a homogeneous stationary environment can be probed by examining the structure function (4.12) which measures the density relaxation of a local perturbation in the stationary state, or, equivalently, the motion of a second-class particle in a stationary environment. The first quantity of interest is the collective velocity (4.15), averaged over a grandcanonical stationary distribution of uniform background density p. For the ASEP one finds from (4.21) and (7.25) Vc = ( D R --

DL)(I -- 2p).

(7.67)

Nonrigorously one may derive this relation from the shock velocity (7.65) by taking the limit PR --~ PL of the limiting densities of the shock. Notice that Vc changes sign at the maximal current density p = 1/2. Intuitively this can be understood by imagining the following situation in traffic flow along a long road: a small perturbation (e.g. caused by a car which has just joined the traffic by coming from a side road) will move with positive velocity (in direction of the flow) if the overall traffic density is sufficiently low. However, in a highdensity regime, such a perturbation causes incoming particles to pile up behind

124

G.M. SchOtz

the perturbation (traffic jam), and thus leads to a negative collective velocity of the centre of mass of the perturbation. The diffusive spreading of the perturbation around its mean position is much harder to treat. By taking the limit ,OR ~ Pt. in the evolution of a shock initial state one expects from (7.64) a divergent collective diffusion coefficient (4.16), which means superdiffusive spreading of the perturbation. This is indeed seen numerically in a divergent effective time-dependent diffusion coefficient Dc ~" t 1/3 (van Beijeren et al., 1985; van Beijeren, 1991). Using the exactly known diffusion coefficient of the second-class particle in a finite system (Derrida and Evans, 1994) this power law divergence can be understood from a scaling argument. One assumes the collective diffusion coefficient D c ( L , t) to have the scaling form Dc ( L , t) = t ~ D* ( L z / t) because only such an expression transforms covariantly under a dynamical scaling transformation. For 0 1/2. The spatial density profile Pk approaches P b u i k exponentially with k, the distance from the origin. When the particles are supplied not too fast, c~ < 1/2, and removed faster than supplied,/3 > t~, there results a low-density phase for which particle supply is the dominant process. The bulk density is left boundary density ct; the density profile Pk approaches Pbuik exponentially within the distance r = L - k from the fight boundary. These phases are related by particle-hole symmetry. The line a = / ~ < 1/2 defines the phase boundary between the high- and low-density regimes, and then mean-field predicts a jump in the density profile from a to 1 - a as shown in Fig. 19, but located at the centre of the chain. When particles are supplied and removed sufficiently rapidly, a > 1/2, fl > 1/2, there results a maximal-current phase C for which transport is bulkdominated; the bulk density is p* = 1/2, and the current takes on its maximal value of jmax - 1/4. In this phase mean field predicts a power law approach cx l / x of the density profile to its bulk value from above (left boundary, x = k) and below (fight boundary, x = L - k) respectively. These mean field results predict two kinds of boundary-induced phase transitions. Considering the bulk density as order parameter there is a first-order transition along the line a = /~ < 1/2 where the bulk density has a discontinuity. A second-order transition takes place for both a,/~ > 1/2 along the lines ~ = 1/2 and /~ = 1/2. The mean-field calculation is instructive on a qualitative level, but does not provide a satisfactory understanding of the physical mechanism that leads to the predicted phase diagram. Thus we are left with two questions. It remains to clarify firstly to which extent these predictions are correct and secondly how the origin of the phase transitions and their location in the phase diagram can be understood. The first question has been answered affirmatively (and rigorously) by Liggett (1975) who obtained the exact bulk density and

127


All

1/2

AI

B.

0

1/2

1

Fig. 22 Phase diagram of the model in the ct -/~ plane. Region A is the low density phase, region B the high density phase and region C is the maximal current phase. The phases are separated by the curves ct = 15 < 1/2 and ct = 1/2, 15 > 1/2 and/~ = 1/2, ct > 1/2 respectively. The low (high) density phase is divided into two phases A I and AII (BI and BII) along the curve/~ = 1/2 (~ = 1/2). The mean-field phase diagram shows the exact phase transition lines between phases A, B, C, but not the nonanalytical behaviour along the dashed lines within the phases A and B respectively.

current from recursion relations for the full stationary distribution. His approach also suggests that the first-order transition originates in the dynamics of the shock in the ASEP, even though the precise nature of the exact stationary distribution along the phase transition line remains obscure. Also a physical understanding of the second-order transition cannot be achieved from knowledge of the bulk densities alone. Fortunately, it turns out that the much more recent exact solution for the density profile (Derrida et al., 1993a; Schlitz and Domany, 1993) reveals

G.M. SchQtz

128

additional structure within the phase diagram which gives the decisive clue to a rather complete answer to both questions.

7.2.2

Exact solution for the density profile

This model is integrable (Inami and Konno, 1994; de Vega and Gonzalez-Ruiz, 1994), but this fact alone does not help to find the solution. The Bethe ansatz as introduced above relies on particle number conservation which is violated for this process. Indeed, there is no known constructive method to calculate even the ground state of the corresponding quantum chain (which is the stationary state we are interested in) directly from the integrability. However, it is possible to express the exact stationary state for arbitrary a and/3 in terms of recursion relations in system size (Liggett, 1975). From these recursions for the full stationary distribution one can then extract useful recursion relations for various expectation values (Derrida et al., 1992). Recursion relations which give the stationary density and hence the phase diagram were solved by Schlitz and Domany (1993) for general values of the boundary parameters a and/3. Before discussing this solution we consider the special case of equal right and left boundary densities c~ = 1 - / 3 . Since the translationally invariant product measure (2.24) is stationary for the periodic system it is tempting to try to show that it is also stationary for the open system. This is indeed straightforward to prove by the same calculation that leads to (7.23) together with the action of the boundary matrices (3.31 ) on the product state. There is no equally trivial way of calculating the stationary state off the line a = 1 - / 3 . Hence we present the exact solution of this generic case in more detail. For the derivation of the stationary density profile of the system with L sites from recursion relations it turns out to be convenient to work with unnormalized weights f,7 (L) related to the actual stationary probabilities P,~ (L) = fo (L) / ZL through the normalization ZL = Y~'~ofo (L). We denote the unnormalized stationary density at site k by TL,k = (nk)ZL and the (unnormalized) empty interval probability

YL,k = ( (1 - n k ) . . . (1 - nL) )ZL

(7.69)

and the joint probability XPL,k = ( n p ( l

-- n k ) . . .

(1

--

From these quantities one obtains the density profile by YL,L+I

X Lp, L + I

= TL,p 9

(7.70)

nL) )ZL.

:

ZL and


129

For technical reasons we extend the range of definition of YL,k by setting Y/.-1,1.+l = /~-l YL-I,L for /~ ~: 0. One finds the following closed recursions (Derrida et al., 1992):

YL.I YL,k

---

fYL-],l YL,k-] +otfYL-],k

(7.71) for2 < k < L

(7.72)

with the initial condition Yo, l = 1

(7.73)

and also X p

for p + 2 < k < L

_

=

_

_

(7.74)

xP,L+,

=

xP,L +axP_I,L

forl_ 1/2

(7.97)

138

G.M. Sch0tz

which has a single maximum jm = 1 at p* = 1/2.

7.3.2 The phase diagram From (7.96) one realizes that the system changes its behaviour if Of = / 3 . The bulk densities on the even and odd sublattices respectively have a discontinuity in the thermodynamic limit L --+ oo at Of = / 3 # 0, 1. One finds p(even)

=

I Of I1

p(Odd)

._

[0

Of < fl Of>/3

I 1-/~

Of < ,B

(7.98)

~>/~.

For Of < /3 (more particles are absorbed than injected) the system is in a low1 density phase with average bulk density p = 89 ~_ p(Odd)) = Of/2 < ~, while for Of > fl it is in a high-density phase with p = I - r

> 89(Fig. 25).

/3 1/2

1/2 Of

1

Fig. 25 Phase diagram of the model in the a - fl plane. Region A is the low-density phase and region B the high-density phase. The phases are separated by the line c~ =/~. (From Schlitz (1993b).)


139

In the thermodynamic limit L ~ c~ the current j is given by j = min (ce,/3).

(7.99)

This result is reminiscent of the behaviour of the current in the usual exclusion process (7.95): there is no discontinuity at a = /3 in the current, but its first derivatives with respect to a and/3 are discontinuous. Now we turn to a discussion of the density profile. We first study the case c~ 13 which is related to the low-density phase by the particle-hole symmetry the profile is given by (nzx)

=

l-(l-c~)e

(nzx-l)

=

1 - / 3 - (1 -c~)e -(2x-1)/~.

_(even)

-2x/~ (7.102)

_(odd)

The bulk densities are Pbulk = 1 and Pbulk = 1 --/3 respectively. On approaching the phase transition line ce = /3 the localization length diverges. On the line the profile is linear and up to corrections of order L-1 given by

(n2x)

=

2x ce + (1 --c~)--~--

(nzx-1)

=

2x-- 1 (1 - - c ~ ) ~ . L

(7.103)

One can write ~-l = In (jL/jR) in terms of the currents to the left and to the fight of a domain wall of densities PL = ce/2 and PR = 1 - / 3 / 2 , in agreement with the stationary bulk densities in the low- and high-density phases respectively. With the exception of the sublattice structure one recognizes a qualitatively identical behaviour of this process with the usual ASEP in the phases Al and BI . The domain wall picture developed in the last section can now be verified directly by investigating the behaviour of density correlation functions.

140

7.3.3

G. M. SchQtz

Stationary correlation functions

Equal-time correlators are of very simple form if expressed in terms of shifted densities dr defined by n 2 x -- ot

d2x = ~ ,

1 -o~

dxx-I =

n2x- 1

1 -/5

(7.104)

instead of using the density operators nx. In the bulk of the high-density region both (d2x) and (d2x-l) take the value 1, while in the bulk of the low-density region both expectation values vanish. Hence these quantities indicate whether the system is in the high- or low-density regime respectively. The two-point correlation function satisfies the exact relation (dxtdx2) = (dx~) (x2 > X l ) (Schlitz, 1993b; Hinrichsen, 1996). Iterating this remarkable relation yields the m-point correlation function

(dxl ...dx,, ) = (dxi) where xi - rain {Xl . . . . .

Xm}.

(7.105)

In a domain of constant density this result implies ( nxny ) - ( nx )( ny ) -- 0, i.e., absence of correlations. We have argued above that the density profile can be understood by considering the steady state as composed of 'constituent profiles' with a region of constant low density up to some point x0 in the chain followed by a high-density region beyond this domain wall. This scenario is confirmed by (7.105) since it explains why the correlator depends only on min {Xl . . . . . Xm}: suppose without loss of generality that Xl < x2 < ... < Xm. For a 'constituent profile' for which x l is in the low-density domain the operator dx~ has vanishing expectation value and therefore the whole expression (dx~dx2) is zero if X I , independent of dx2.* If, however, x i is in a region of high density, then, according to our assumption, also x2 > Xl must be in region of high density. Thus, dxtdx2 again does not depend on x2 and takes the value 1. We conclude that the product dxl dx2 is either 0 or 1, depending on whether x i is in a region of low or high density. The same follows for products of the dx and leads to the expression (7.105) for the expectation value. In particular, for a = fl one obtains the correlation function (nxny)

-- ( n x ) ( n y

) = (1 - - o~)2--ff

1 --

for y > x.

(7.106)

In the absence of a domain wall the correlation function in a linear density profile has the same functional form, but the amplitude of the correlations is only of the order of the inverse system size (6.66). Explicit calculation of the connected *Because of vanishing correlations this is correct for all x 2. The argument can be extended to constituent distributions with short-range correlations by considering a distance Ix2 - X ll larger than the correlation length.

141


correlation function (d~dx2) - (dx~)(d~2) from the exact expressions for the density profile also shows that the localization length ~ is identical with correlation length of the connected two-point function. Its amplitude depends on the position in the bulk (Schlitz, 1993b). According to our argument the expectation value ( dx ) itself contains the information about the position x0 of the domain wall. The exponential decay of the density profile implies that the probability of finding the domain wall decreases exponentially with localization length ~ with the distance from the boundary. On the phase transition line the domain wall may be found anywhere with equal probability. This leads to the observed linearly increasing density (7.103).

7.3.4

Domain-wall fluctuations

We turn to a study of the time-delayed two-point correlation function in the stationary state G * ( x l , x z ; t ) = (Sidx~TtdxzIP *) (7.107) where T t denotes the tth power of the transfer matrix T. A standard way of computing this correlation function would be the insertion of a complete set of eigenstates of T, evaluating the matrix elements ak(xl) = (S Idx~l Ak) and t]k(X2) = ( A k [dx2[ P * ) and summing over akhkA~. Since we do not know the eigenstates and eigenvalues this is not possible. Instead one can try to solve the equations of motion for the operator nx, i.e. one can try to solve the equation nxT t = T t Qx(t) for the operator Qx(t) by using the commutation relations (A.13) given in Appendix A. Evaluating nxT t is not an easy task. The number of terms contributing to Qx(t) increases extremely fast with t. It is only the simplicity of the multi-point correlators (see (7.105)) that makes this rather unorthodox approach promising. We restrict our discussion to both Xl -- 2yl - 1 and x2 = 2 y 2 - 1 odd. By iterating (A.13) t times one finds that nzy~-i Ttnzy2-1 is of the form n2y~ - 1 T t n 2 y 2 - 1

T t

{1 - ( V 2 y l _ 2 t l ) 2 y l _ 2 t + l

-(...)

-

. . .) -

(V2y,-2t+2kV2y,-2t+2k+l

(1)2yl-2t+2V2yi-2t+3...) ...)

-- ( . . . )

-- V2yl--4+2tV2yl--3+2t } n 2 y l - 2 + 2 t n 2 y l - l + 2 t n 2 y 2 - 1

(7.108)

where the dots denote some complicated sums of products of operators n y and Vy = l - n y acting on sites y between 2yl - 2t and 2yl - 1 + 2t. In order to avoid additional complications through boundary effects we choose t < yl - 1. The r.h.s, of (7.108) defines Q2y~-I (t)n2y2-1 which is the quantity we need. The interesting region is the interior of the 'light cone' defined by 2y2 - 3 2t _< 2yl - 1 _< 2y2 - 1 + 2t. The perturbation of the stationary distribution represented by the projection on particles on site 2y2 - 1 does not propagate

142

G.M. SchQtz

outside this domain and hence the correlation function outside this light cone is equal to the steady-state correlator. The calculation inside the light cone is nontrivial. With Xl = 2yl - 1 as above and x2 increasing beyond 2yl + 2 2t more and more contributions from the r.h.s, of (7.108) are nonzero. There is hardly any hope to find a closed expression for Q2yl-l(t)n2y2-1. Instead one can evaluate G*(xl, x2; t) for small t inside the light cone using computer algebra. One calculates the exact form of (7.108) and then implements the fusion rules (7.105) on the multi-point correlators on the r.h.s, of (7.108). This leads to an explicit expression of the correlator (7.107) for small t as a function of a and /5. For a = 1 - / 5 the exact general form of the correlator can be guessed by generalizing the result from t = 1, 2, 3 to arbitrary t (Schiitz, 1993b): G*(x,x+2y't)

ll-#(t-y,t+y)

=

+ (L-~) L-/-2y+I

ll-#(t + y , t - y)

(7.109)

with the incomplete/~-function l#(m, n) (Abramowitz and Stegun, 1970). The choice c~ = 1 - / 5 is not restrictive as far as the physics is concerned: since this curve runs across the phase diagram it covers both the high-density phase and the low-density phase and crosses the phase transition line at a = / 5 = 1/2. The most interesting behaviour of G*(xl, x2; t) is seen in the low-density phase along the curve/5 = 1 - c ~ > 1/2. For large times t (such that lyl/t 1/2 the incomplete/3-function has the asymptotic form (Abramowitz and Stegun, 1970)

ll_#(t + y , t - y) =

(l)- ~~r3

el/r

r

4rrt

e_(y/~+t/r)e_

y2/t

(7.110)

with the relaxation time r -1 = - I n (4/~(1 - / 5 ) )

(7.111)

and the spatial correlation length (7.100). In terms of the relaxation time r the inequality/5 > 1/2 has to be understood as 1 1/2) that connects the highdensity stationary state to the maximal-current phase (example 2). This second, distinct type of domain wall is a notion necessary for a full understanding of the system (Kolomeisky et al., 1998). The bulk densities far to the left and fight of the (011) domain wall are reached exponentially fast with length scale ~. As we increase the entering rate ~ (holding the exit rate t3 < 1/2 fixed), the localization length ~ characterizing the low-density behaviour of the domain wall increases, going to infinity (Schiitz and Domany, 1993; Derrida et al., 1993a) at c~ = 1/2: The (0ll) wall undergoes a continuous phase transition into the maximal-current/high-density domain wall (ml 1) described in our second example. Because the maximal current phase is algebraic, the stationary density profile for a > 1/2 approaches its bulk value not p u r e l y exponentially, but with an algebraic correction. Thus the domain wall transition explains the nonanalytical change of the stationary density profile within the high-density phase at a - 1/2 for any value of/~ < 1/2. We turn now to a quantitative analysis of the physical origin of these observations in terms of the drift velocity Vs (7.63) of the domain wall and of the collective velocity vc (4.21) of the lattice gas. To this end we will show how the late-stage dynamics of the system and the approach to the true high-density stationary state are governed by the motion of the two types of domain walls. In the introductory examples the domain wall was easy to visualize because the wall is sharp if the entering and exit rates are small. To compute the drift velocity in the general case we recall the derivation of the expression (7.65) for the drift velocity of the domain wall. In a finite macroscopic system the assumption p ( x , t) = p ( x - vt) breaks down near the boundaries, and therefore the parameters jR,L and PR,L should be understood as stationary bulk values of the current and density in the far left (L) and far fight (R) parts of the domain wall. For the low-density/highdensity domain wall (011) one has jR = / ~ ( 1 - / 3 ) , PR = 1-/3, and jL = o t ( l - a ) , pL = a. Substituting these in (7.65) we obtain the drift velocity Vs (7.63) for the TASEP Vs = / 3 - a.

(7.116)


One realizes that the shock velocity changes sign at ~ = same scenario for the first-order transition as in example For the maximal-current/high-density domain jR = /~(1 -- 13), PR = 1 -- 13, and jr. = 1/4, PL initially empty lattice Vs --/~

1

2"

147

fl < 1/2, leading to the 1 for small a, ft. wall (m Il) we have = 1/2. Hence for an (7.117)

The expression for v changes its functional form at c~ = 1/2 because the wall has a different form beyond the phase transition (011) --+ (roll). To understand why the transition takes place at a = 1/2 we consider the collective velocity (4.21 ) of a general lattice gas. The collective velocity measures the drift of the center of mass of a momentary local perturbation of the stationary distribution. For the TASEP Vc = 1 - 2,o changes sign at p = 1/2 where the current takes its maximal value j = 1/4. To appreciate the significance of the collective velocity for the phase diagram of the TASEP consider first the low-density phase along a line with fixed 13 > 1/2. For a left boundary density ct < 1/2 a small perturbation of the stationary state (corresponding to a fluctuation in the injection of particles) travels with positive speed into the bulk where it will eventually dissipate. However, if the perturbation is maintained, i.e. the constant left boundary density is increased by a small amount, the perturbation will continuously penetrate into the bulk and lead to an increase of the bulk density. This happens until ~ = 1/2. Further increase of the left boundary density results in a negative collective velocity and the perturbation does not spread into the bulk. The system has entered the maximal current phase where it remains even if the left boundary density is further increased. This phenomenon is the underlying mechanism that leads to the approach to the maximal transport capacity, i.e. to the onset of an overfeeding with particles. The overfeeding originates in the change of sign in the collective velocity. For/~ < 1/2 the system does not enter the maximal current phase (because of the negative shock velocity), but the overfeeding still occurs for c~ > 1/2 and leads to the domain wall transition (011) ~ (m[1). The overfeeding implies that further increase of the left boundary density beyond 1/2 does not result in any change of the characteristic length scales in the high-density phase. This is seen in the behaviour of the domain wall velocity v (7.116), (7.117) and also in the divergence of the localization length ~,~. Particle-hole symmetry can be used to extend our results to the low-density phase. Thus we can explain both the location of the second-order phase transition lines in the TASEP and the nonanalytic changes in the density profile within the low- and high-density phases. Numerical simulations which support these ideas are given in Kolomeisky et al. (1998). The domain wall approach to the determination of the phase diagram makes little reference to the microscopic details of the dynamics. The crucial ingredients,

148

G . M . Schfitz

the domain wall velocity (7.65) and the collective velocity (4.21) are generally valid. We conclude that the phase diagram of the ASEP is universal in the sense that systems with a single maximum jmax in the current-density relation j (p) have a similar phase diagram. Hence, given j (p), one finds the domain wall velocity (7.65), the collective velocity (4.21) and thus the location of the phase transition lines. Coupling to boundary reservoirs of left density PL and fight density PR gives a low- and a high-density phase separated by a coexistence line which is determined by j (OL) = j (PR) < jmax. In the low density phase (j(PL) < j(PR)) the bulk density equals the left boundary density Pt., in the high-density phase (j (PL) > j (PR)) the bulk density equals the fight boundary density pR. Expressed in terms of the current we recover the extremal principles (7.114), (7.115). Within these phases there are domain-wall transitions at PR,L - P* which is the density that maximizes the current and at which the collective velocity changes sign. For both PR, PL > P* the system is in the maximal current phase where the bulk density takes the value p*. The domain wall velocity v vanishes at all phase boundaries. For a continuous-time random walk motion of the domain wall also (7.119) is generally valid close to the phase transition lines. Thus one can predict the shape of the density profile from the fluctuations of the domain wall motion. The diffusion coefficient D is singular along second-order lines. This picture is well supported not only by the exact solution of the TASEP, but also by exact results for other exclusion processes (Schlitz, 1993b; Sandow, 1994; Rajewsky et al., 1998; Tilstra and Ernst, 1998; Evans et al., 1999; deGier and Nienhuis, 1999) for which the current-density relation and location of phase transition lines is known. The first-order transition is consistent with the experimental data obtained for protein synthesis, see Section 10.2, and has been observed directly in traffic flow (Popkov et al., 1999), see below.

7. 4.2

Density profiles

We may go further and check this picture by considering the consequences of the fluctuations in the domain wall position. The domain wall is a compromise between the particle injection and extraction processes that attempt to enforce their own distinct stationary states. From our random walk discussion of the domain wall dynamics we expect that a superposition of the domain wall localized at the left boundary and uniform bulk density Pbulk = 1 --/~ capture the physics of the high-density stationary state, in agreement with the intuitive arguments developed above. Thus it is tempting to derive the localization length (7.86) which determines the postulated distribution p(x) of the domain wall position and hence the decay of the density profile directly from the stationary distribution of a biased lattice random walker in a large, but finite system. For a continuous-


149

time random walk with right and left hopping rates ~R,L (4.34) yields the exact equality ~ = 1/In (~R/SL). For the domain wall motion (7.55) gives ~ - l = In ( j + / j - ) .

(7.118)

Somewhat surprisingly this simple-minded ansatz is in agreement with the exact expression (7.86) in all phases and explains the origin of the two independent length scales (7.85) ~a = 1/ In 4 j - and ~ = 1/ ln(4j +) in terms of the domain wall diffusion. For a genetic lattice gas we take again a coarse-grained approach and introduce a localization time r that characterizes the length of time the wall spends away from the boundary. We argue that the drift distance los Ir and the diffusional wandering ( D r ) 1/2 each are of the same order of magnitude as the localization length ~ itself; then r ~ D / v 2 and (7.119)

~ O/lvsl.

Thus we obtain not only a foundation in terms of the domain wall motion to the phenomenological derivation of this result by Krug (1991) for the transition from the low-density phase to the maximal current phase, but also an extension of the validity of (7.119) to the other phase transition lines. This domain wall approach is legitimate whenever the localization length ~ is much bigger than the (unit) lattice spacing and than other, internal bulk correlation lengths which may result from particle interactions in the lattice gas. The validity of (7.119) can be checked for the TASEP. In the TASEP with boundary densities PR,C we obtain from (7.66) the domain wall diffusion constant D

=

1/3(1 - / 3 ) + c~(l - o~) ~ l-~-ct 4/~(1-/4) + 1 4(1 - 2/~)

for c~, fl < 1/2

for o~ >_ 1/2,/~ < 1/2

(7.120) (7.121)

We note that (7.120) reproduces the exact diffusion coefficient in an infinite system (Ferrari and Fontes, 1994a) and the diffusion coefficient of the open system on the coexistence line ,~ = /~ < 1/2, proposed independently on the basis of current-fluctuation arguments (Derrida et al., 1995). We get from (7.119) expressions for ~ which are much larger than unity (and thus trustworthy) in the vicinity of the coexistence line ot = /~ < 1/2 (case 1) and close to the phase boundary with the maximal current phase/~ = 1/2 respectively (case 2). In these limits ~ coincides with the exact expression (7.86).

7. 4.3

Current with two and more maxima

To understand the origin of these extremal principles in a more general context consider a driven lattice gas with hard-core repulsion. At/9 -- 1 no hopping can

G.M. Sch~z

150

take place and hence the current vanishes. Two maxima can arise as the result of sufficiently strong repulsion between nearest neighbour particles as opposed to the pure on-site repulsion of the usual TASEP which leads to a single maximum (Fig. 26). 0.2 0.4 0.6 0.8 ,

,

,

,

l

|

u

,

t

1

,

,

,

i

.

0.2

0.1 /

r

jmi .

/,

/ ,|

Pmin

i

9

P2

,

P2 1

P Fig. 26 Exact current--density relation of the TASEP with nearest-neighbour interaction for E = 0.995, ~ = 0.2 (equations (7.123)-(7.126)). (From Popkov and Schiitz (1999).) At first sight one might not expect such a little change in the interaction radius of the particles to affect the phase diagram. However, the theory developed above - e v e n though valid only for systems with a single maximum in the current- indicates that the local minimum in the current-density relation leads to a qualitative change in the nature of the shocks and their interplay with density fluctuations. Indeed, the full phase diagram (Fig. 27) generically consists of seven distinct phases, including two maximal current phases with bulk densities corresponding to the respective maxima of the current and the minimal current phase in a regime defined by j(PR), j(PL) > j(Pmin); PL < ,Omin < PR. (7.122) Here the system organizes itself into a state with bulk density bulk corresponding to the local minimum of the current. As in the maximal current phases no finetuning of the boundary densities is required, t In order to understand the more complicated structure of this phase diagram we use the concepts of coalescence and branching of shocks (Popkov and Schlitz, 1999; Ferrari et al., 2000). A single large shock (with a large density difference tWe consider the situation where the first maximum in the current-density relation at density p~ is higher than the second maximum at p:~. The opposite case can be treated analogously. In the case of degenerate maxima the two maximal current phases have additional transitions to a maximal-current * PR < Pl* section of the phase diagram. The spatial structure of coexistence phase in the PL > P2' the steady state can be described as a superposition of two regimes with bulk densities p~' and p~ respectively, separated by a slowly fluctuating domain wall which performs an unbiased random walk.


1

& 0.8

Pl

Pmin

~1~

'

.

,

.

.

I

.

0.6

0.4

0.2

,

Cc~ P b u l k

MINIMAL CURRENT

~p

I

~k high density c~ phase:

= P+

~_'~ o ~,~,~,o o ~,~, Pbulk = p-

Pbulk = Pmin

Pmin

P2 ,

PHASE p+

151

.ooooooooooooooo,~

t

P2

maximal

~~-~

~ current

cI phaselI:

9

Pbulk=P2

b~tk = P+

low density

~, 0

phase:

_

"-=-o0800388888

maximal current phase I: Pbulk = Pl

Pbulk = P |

0.2

Pl

i

i

i

0.4

0.6

0.8

p-

Fig. 27 Exact phase diagram as a function of the boundary densities PL, (PR). Full (bold) lines indicate phase transitions of second (first) order. Circles show the results of Monte Carlo simulations of a system with 150 sites where ~ = 0.995, & = 0.2. (From Popkov and Schiitz (1999).)

P2 - Pl) may be understood as being composed of subsequent smaller shocks i with narrow plateaux at each level of density (Fig. 28). In the case of the ASEP all these shocks move in the same direction (say, to the right), b u t - as investigation of the respective shock velocities vs(i) (7.63) shows - all with negative relative speed os(i + 1) - vs(i) < 0. Hence eventually they coalesce into one 'big' shock as discussed above. However, the same analysis shows that in the presence of a minimum in the current-density relation a single shock may branch into two distinct shocks, moving away from each other. With these observations the dynamical origin of the phase transition lines can be understood by considering the time evolution of judiciously chosen initial states. Because of ergodicity, the steady state does not depend on the initial conditions and a specific choice involves no loss of generality. We turn our attention to a line PR = c with Pmin < c < p~ in the phase diagram which crosses the minimal current phase. Along this line it is convenient to consider an initial configuration with a shock with densities PL and PR on the left and on

152

G . M . Sch0tz

p+

p

--~ .

p

.

.

.

~.f-.

~

F

-

_

_

Pmin

-

x

Fig. 28 Schematic drawing of the decomposition of a large shock into small shocks and their velocities, leading to branching and coalescence. Here ,o~' < PL < ,~ < PR < P~. (From Popkov and Schiitz (1999).)

the right respectively, which is composed of many narrow subsequent shocks at various levels of intermediate densities (Fig. 28). (i) We start with equal boundary densities in which case the system evolves into a steady state with the same bulk density Pbulk = PL = PR(ii) Lower PL a bit below PR with just a single shock separating both regions. According to (7.63) the shock travels with speed Vs = ( j + - j - ) / ( P R -- PL) > 0 tO the fight, making the bulk density equal to PL. At the same time, small disturbances will, according to (4.21), also drift to the fight, as Vc = j ' ( P L ) > 0 in this region, thus stabilizing the single shock. (iii) Now, lower PL slightly below Pmin. While the shock velocity Vs is still positive, so that one expects the shock to move to the fight, the collective velocity Vc = jt(pL) < 0 indicates that disturbances will spread to the left. This discrepancy marks the failure of a single shock scenario. In order to resolve it, we return to the picture with many subsequent shocks at each density level between PL and PR (Fig. 28). Equation (7.63) shows that all small shocks below Pmin will move to the left, while all those above Pmin will move to the fight. The left-most of the left-moving shocks will merge in a single one, and so will the right-most of the fight-moving shocks. The result is two single shocks (PL, or) and (o', PR) respectively moving in different directions. The density levels or, or' are determined by the stability criterion and satisfy PL < r < Pmin and Pmin < ~r' < PR). In the intermediate density intervals (o, Pmin) and (Pmin, o") shocks are not stable. Here the profile approaches Pmin, thus expanding the region with the density Pbulk = Pmin- The system enters the m i n i m a l current phase. Qualitatively the same scenario will persist for any left boundary density in the range PL ~ [151,Pmin]. This picture is well supported by the Monte Carlo simulations shown in Fig. 29, demonstrating the branching of a single shock into two distinct shocks moving in opposite directions. Notice that the change of bulk density is continuous across the point PL = Pmin tO the minimal current phase, so the transition is of the second order. (iv) As we lower PL below/51, the shock velocity Vs = ( j m i n - j ( P L ) ) / ( P m i n - PL) > 0 becomes positive. The shock is moving to the right, leading to a low-


153

t=O

1 0.75 P 0.5

0.25 _...~~::"." ............... j 0

300

Fig. 29 Snapshots of a particle density distribution at the initial moment of time and after 300 Monte Carlo steps, showing expansion of the minimal current phase. Simulated is the system of 150 sites, with particles initially distributed with average density PL = O. 1 (PR = 0.85) on the left (on the right); 3000 different histories are averaged over. (From Popkov and Schlitz, 1999.)

density phase with bulk density Pbulk = ,OL which drops discontinuously from ,Obulk - - Pmin at PC = / 9 1 - + - 0 to ,Obulk = Pc at PL = f 3 1 - 0. The system undergoes a first-order phase transition. On the transition line the shock performs an unbiased random walk, separating coexisting regions of densities Pmin and PL respectively. (v) Let us start again from PL = PR and now increase Pc. Until one reaches PL = P29 the collective velocity Oc = j l (PL) > 0 is positive, leading to hulk = PL.

(vi) As soon as PL crosses the point PL -- P~, the sign of the collective velocity Vc changes and the overfeeding effect occurs: a perturbation from the left does not spread into the bulk and therefore further increase of the left boundary density does not increase the bulk density. The system enters the maximal-current phase II through a second-order transition. Using analogous arguments one constructs the complete phase diagram (Fig. 27) and obtains the extremal principles (7.114) and (7.115). The velocities (4.21), (7.63) which determine the phase transition lines follow from the current-density relation. This behavior can be checked with Monte Carlo simulations. A model with two maxima of the current is a TASEP with nearest-neighbour interaction defined by the bulk hopping rates (Katz et al., 1984) 0100

~

0010

with r a t e l + ~

(7.123)

1 100

~

1010

with r a t e l + E

(7.124)

01 0 1

~

001

1

with r a t e l - e

(7.125)

1 1 01

~

1 01 1

with r a t e l - ~

(7.126)

with lel < 1; I~1 < 1. The injection at the left boundary site 1 and extraction of particles at the right boundary site L is chosen to correspond to coupling to

154

G.M. SchOtz

boundary reservoirs with densities ,OR,L respectively. Along the line P R = PL the stationary distribution is then exactly given by the equilibrium distribution of a one-dimensional Ising model with boundary fields and the bulk field such that the density profile is constant with density p = P n = P L (Antal and Schlitz, 2000). The current j = (1 + 8 ) ( 0 1 0 0 ) + ( 1 +E)( 1 1 0 0 ) + ( I - E ) ( 0 1 0 1 ) + ( 1 - 3 ) ( 1 1 0 1 ) as a function of the density can be calculated exactly using standard transfer matrix techniques. The exact graph is shown in Fig. 26 for specific values of the hopping rates. Monte Carlo simulations confirm the validity of the theoretical prediction for the phase diagram (Popkov and Sch/itz, 1999). For systems with more than two maxima in the current the interplay of more than two shocks has to be considered in the same manner.

7.5

Traffic f l o w models

Traffic flow may be viewed as a system of interacting particles with exclusion, moving according to certain dynamical rules in a quasi one-dimensional geometry. Even though the motion is continuous in space (Helbing, 1997), contact can be made to lattice gases by dividing the road into cells of the average length of a car (plus some minimal distance between consecutive cars) and considering such as cell as occupied if at a given instant in time a car (or the larger part of a car) is found in that cell. This automatically gives rise to a description of traffic flow in terms of some exclusion process. For a two-lane road the description may be extended to exclusion processes where each lattice site can take more than two different states. This description in terms of an exclusion process also suggests using discrete-time updating, corresponding to taking snapshots of the traffic configuration at constant intervals of time. If all cars moved with the same average speed, one could choose a time window such that one time step would correspond to a hopping by one lattice unit as in the TASEP. To capture the possibility of speed changes, however, a more elaborate modelling is required. Even though the TASEP discussed above is certainly not a realistic model for traffic flow, it partially incorporates what appears to be the most basic mechanism, viz. the competition between the desire to travel with an (individual) optimal velocity, while, at the same time, attempting to keep a (velocity-dependent) safety distance to the next driver. At low densities there is no conflict between these requirements and one has essentially free flow of noninteracting particles. However, at sufficiently high densities, the safety distance at the optimal velocity becomes incompatible with actual traffic density and free flow breaks down. A further important feature is a certain amount of randomness due to individual driver behaviour, particularly when braking or accelerating. In the TASEP these mechanisms are incorporated in a simple manner: the desired velocity (the velocity of a single particle in an empty system) is the same for all particles, the


155

safety distance is one lattice unit, and randomness is described by the exponential waiting time distribution of the particles. Despite these simplifications some qualitative features of real traffic (Hall et al., 1986; Kemer and Rehborn, 1996) can already be seen: shocks exist and the stationary current j (p) =/9(1 - p) as a function of the particle density p has a single maximum. An apparently unrealistic feature is the absence of correlations in the steady state (Schadschneider and Schreckenberg, 1993; Schreckenberg et al., 1995). An unrealistic feature of the current-density relation is the reflection symmetry with respect to the maximalcurrent density p* -- 1/2 and its rounded shape close to the maximum. Various more elaborate one-dimensional lattice gas models have been introduced for the study of realistic traffic flow (Nagel, 1996; Chowdhury et al., 1999, 2000). The following is part of the picture that emerges: (i) The existence of a shock in the ASEP and the maximum in the currentdensity relation is genetic and appears to be the consequence of the hard-core repulsion (site-exclusion) in conjunction with biased hopping. (ii) The round shape of the current-density relation at p* is specific for the ASEP. Deterministic discrete-time exclusion processes (Krug and Spohn, 1988; Schlitz, 1993a; Yukawa et al., 1994; Rajewsky et al., 1998; Tilstra and Ernst, 1998) also show a symmetric current--density relation with one maximum, but the derivative of the current is discontinuous at the maximal-current density p*, see (7.97). Increasing the hopping probability in a probabilistic discrete-time process towards deterministic hopping, leads to an increasingly sharp jump in the current derivative at p* (Schadschneider and Schreckenberg, 1993). In this respect the current in these models resemble the shape of the current in real traffic (Hall et al., 1986) and of more realistic traffic flow models like the NagelSchreckenberg model (Nagel and Schreckenberg, 1992). These observations suggest that the strength of the velocity fluctuations is responsible for the roundness in the shape of the current-density relation at p*. An exponential waiting-time distribution leads for a single car with average velocity v to a diffusion coefficient D = v, i.e. velocity fluctuations are of the order of V/-~. For a random walk in discrete time with hopping probability p to the fight one finds o = p D = p(l - p). Close to the deterministic limit p ~ 1 the velocity randomness is much smaller. Also in the Nagel-Schreckenberg model the relative speed fluctuations of a single car around its mean are smaller. It is interesting to note that for a single driver moving with average speed v without any obstruction, velocity fluctuations of the order ~ seem too large. This is consistent with the unrealistically round shape of the current in the TASEP. (iii) The symmetric shape of the current-density relation results from particlehole symmetry and appears in models in which cars travel with constant average speed, i.e. move with constant probability or rate, independently of the environment beyond the nearest neighbour site to which they move. This is an unrealistic assumption since clearly cars slow down when they see a slowly

156

G . M . SchOtz

moving car already some distance ahead. Numerical and mean-field results for discrete-time cellular automata (Nagel and Schreckenberg, 1992; Schadschneider and Schreckenberg, 1993; Schreckenberg et al., 1995) which allow for reduction of speed that depends on the occupation of sites further ahead show an asymmetric current-density relation resembling the shape of the current-density relation of real traffic. (iv) For parallel update, but not for sublattice parallel update, the same mechanism of increasing the hopping probability also increases antiferromagnetic particle correlations (Schreckenberg et al., 1995; Rajewsky et al., 1998), i.e., cars are less likely to be found on nearest-neighbour sites than some distance apart. Neither the 'antiferromagnetic' correlations nor the asymmetry in the current can be attributed to a discrete-time update alone. This can be shown with a toy model with exponential waiting-time distribution like the TASEP, but with a nextnearest-neighbour interaction which describes slowing down of a car if the nextnearest-neighbour site is occupied as well. A particle hops to the right with rate r if the next-nearest-neighbour site is empty and with rate q if it is occupied: A00

~

0A0

with rate r

(7.127)

AOA

~

OAA

with rate q.

(7.128)

This model corresponds to the case e = S of the exclusion process (7.123)(7.126) considered above. Hence on a ring with periodic boundary conditions the stationary distribution is given by the equilibrium distribution of the onedimensional Ising model. The stationary probability of finding a state n is given by 1 (q)~=l(nini+l+hni' . (7.129) P* (-n-) = ~ L -r Here Z L is the partition function and the 'chemical potential' h parametrizes the fixed bulk density p. This stationary state is identical to that of the discrete-time ASEP with parallel update for suitably chosen hopping probability p (Yaguchi, 1986). It appears that the correlations have their physical origin in speed reduction rather than in the nature of the updating scheme. This is in agreement with similar conclusions drawn from the study of steady states in a different class of cellular automata models for traffic flow (Schadschneider and Schreckenberg, 1998). According to the dynamics described above the local current is given by

jk = (nk(l - nk+l)[qnk+2 + r(l -- nk+2)]).

(7.130)

The stationary particle current j is readily calculated using standard transfer matrix techniques for the one-dimensional Ising model (Baxter, 1982). In the thermodynamic limit L ~ cx~ one finds the exact current density relation

j=rp

1+

~/1 - 4p(1 p)(1 - q / r ) - 1 ] 2(1-p)(l-q/r) "

(7.131)


157

In the repulsive case the current-density relation becomes asymmetric (Fig. 30a) in a way which is closer to real traffic data as the symmetric relation j = p (1 -/9) for the ASEP with r = q = 1. There is no discontinuity in the derivative at the maximal-current density p*, in agreement with the arguments given above, since in this model a single particle moves in the same way as in the TASEP. The same phenomena occur in a continuous-time model where slowing down is modelled by particles which may hop over a distance of either one or two sites (Klauck and Schadschneider, 1999).

0.15 0.075[~ 0.05 0.025

.. 0.2 0.4

0.6

0.8

Fig. 30 Stationary current j as a function of the density p for r = 1, q = 0.1. (From Antal and Schiitz (2000).) Of course the ring geometry of a periodic system is not relevant for modelling any specific road geometry. More important are open systems (Nagatani, 1995) as investigated above. An interesting question concerns the traffic density on a piece of road between two junctions where cars enter and leave with certain rates. The theory of boundary-induced phase transitions makes a prediction of the stationary phase diagram in terms of effective 'boundary densities' which can be determined empirically from traffic flow data. For various model s y s t e m s - including the toy model (7.127) (Antal and Schiitz, 2000), the Nagel-Schreckenberg model (Santen, 1999) and a random exclusion process where each particle moves with its own intrinsic rate (Bengrine et al., 1 9 9 9 ) - Monte Carlo simulations confirm the theoretical prediction. The first-order transition has been observed directly in traffic data taken close to an on-ramp on a motorway near Cologne, Germany (Popkov et al., 1999). The data (Neubert et al., 1999) for the current follow closely what one expects when crossing the phase diagram through the first-order transition line along a curve PR = c o n s t , corresponding to constant on-ramp activity.

Comments Section 7: Diffusive lattice gases have been treated by Spohn (1991), with particular emphasis on a macroscopic approach to the large-scale structure of interacting particle

158

G.M. Sch0tz

systems. A broad overview specifically on driven lattice gases is given by Schmittmann and Zia (1995). Some important applications of one-dimensional diffusive systems to interface growth and to directed polymers in random media are extensively reviewed in Krug and Spohn (1991) and Halpin-Healey and Zhang (1995). S e c t i o n 7.1: (i) Historically, the asymmetric exclusion process represents the first known stochastic many-body process where a q-deformed classical Lie algebra plays a role (Alcaraz et al., 1994; Sandow and Schlitz, 1994). More recently, the generators of quantum algebras have appeared in the dynamics of a ballistic annihilation process (Richardson, 1997) and in a single-particle exchange process (Schulz et al., 1997). There is also an unexpected connection of the quantum algebra symmetry of the ASEP to the continuum KIwZ equation for interface growth in 1+1 dimensions (Schiitz, 1997a). The transformation 2S[ ~ q-2S~ (or 2Nk ~ Qk) is the lattice analogue of the Hopf--Cole transformation which turns the nonlinear KPZ equation with additive noise into a linear diffusion equation with multiplicative noise (Hopf, 1950; Cole, 1951). This curious observation may hint at a link between q-deformed symmetries and properties of certain stochastic partial differential equations. The interested reader is referred to Fuchs (1992) for a discussion of quantum algebras. (ii) Since each local hopping matrix commutes with the generators of the algebra the results of Section 7.1.1 hold also for disordered systems with space-dependent hopping rates s from site k to site k - 1 and with rates rk from site k to k 4- 1 such that the hopping asymmetry q = x/rk/ek+ ! = e #~E across a bond (k, k + 1) at inverse temperature = l / ( k T ) is constant. On a ring the presence of disorder limits the current and gives rise to a nontrivial current-density relation and density profiles (Tripathy and Barma, 1997, 1998). (iii) The calculation of the density from an initial... 11110000... step-function profile in the hydrodynamical limit was extended by Jockusch et al. (1995) to the ASEP with discrete-time parallel updating. The resulting density profile is not linear, but has a circular shape, bulging downwards. (iv) More detailed investigation of the time evolution of the local current with this initial state has led in very remarkable paper by Johansson (2000) to a surprising connection between the probability distribution of the current at site k at time t to the distribution of the largest eigenvalue of certain random matrices. This approach yields not only the mean current at time t, but also an explicit expression for its fluctuations around the mean on time scales of the order t 2/3. This result implies the first direct derivation of the dynamical exponent z = 3/2 without reliance on scaling arguments. It also demonstrates a link between the determinant representation of the solution of the master equation and the eigenvalue statistics of random matrices, since the current fluctuations can be represented as sums of determinants which in turn can be written as a determinant with a similar structure. (v) Also the current fluctuations in the stationary state of a finite system have been calculated by using the Bethe ansatz (Derrida and Lebowitz, 1998), using the same strategy as Gwa and Spohn (1992a). The generating function of the moments is given by the spectrum of a nonstochastic version of the Hamiltonian for the TASEP where the ratio of the coefficient of the hopping term to the diagonal term is not equal to one. The Bethe


159

ansatz solves also exclusion processes with long-range hopping (Alimohammadi et al., 1998). Current fluctuations in the stationary state of the ASEP on an infinite lattice have been calculated by Ferrari and Fontes (1994b). (vi) There are various interesting generalizations of the ASEP which can be solved by the Bethe ansatz, including models without exclusion (Sasamoto and Wadati, 1998a,b,c) and models of particles with different sizes (Alcaraz and Bariev, 1999). It is shown that they all belong to the same universality class. A common feature of the totally asymmetric versions of these models appears to be the possibility of a determinant representation (7.38) first found in (Sch/itz, 1997b) for the usual TASEP. It seems likely that the results of Johansson (2000) (see remark (iv)) could be extended to these generalized totally asymmetric models. (vii) Other two-species driven lattice gases, which are presumably nonintegrable, include models for ionic conductors (Sandow et al., 1995) and models for gel electrophoresis (see Section 10). For these models the quantum Hamiltonian approach can be utilized by employing variational methods (PrS.hofer and Spohn, 1996) and for the derivation of fluctuation--dissipation relations (Katz et al., 1984; Pr~ihofer and Spohn, 1996). Section 7.2: (i) The exact solution of the ASEP with open boundaries, i.e., the stationary density profile as a function of ct and 13 was obtained independently by Derrida et al. (1993a), who derived the matrix ansatz for the stationary state discussed above in the context of the time-dependent algebraic formulation of the exclusion process. By constructing an explicit representation of the matrix algebra the density profile and other quantities could be calculated (for a detailed review see Derrida and Evans (1997)). The recursion relations of Liggett (1975) and Derrida et al. (1992) can be obtained from the matrix algebra without constructing a representation. With hindsight, the solution of Schiitz and Domany (1993) which we review here may be viewed as a representation-free treatment of the matrix algebra. (ii) The differences of binomials which show up in the exact results are the dimensions of the irreducible representations of the generators of the Temperley-Lieb algebra (3.42)-(3.44) which constitute the bulk part of the quantum Hamiltonian for the process. It is not known whether this correspondence is coincidence or whether there is a deeper algebraic reason. (iii) Liggett (1977) and Andjel et al. (1988) have studied the late-time behaviour of an infinite system which has an initial shock profile with fight and left limiting densities PL,R. One finds a phase diagram for the late-time bulk densities which coincides with the mean-field phase diagram of the ASEP with open boundaries. The superposition of domain wall profiles that we postulate for the stationary distribution of the finite system with open boundaries can be seen in numerical simulations of the infinite system with a corresponding initial domain wall (Boldrighini et al., 1989). (iv) The motion of multiple shocks in the ASEP with open boundaries can be calculated exactly on certain manifolds of parameter space provided the shock levels satisfy the condition (7.62) (Krebs, 1999). This is quite surprising since the boundary fields break the quantum algebra symmetry. Krebs obtained this result by considering shock initial states for which the equations of motion form a closed set. In this way one identifies a submanifold of parameters which interestingly enough coincides with the manifolds (Mallick and

160

G . M . SchQtz

Sandow, 1997) where one obtains finite-dimensional representations of the matrix algebra of Derrida et al. (1993a). n consecutive shocks correspond to an (n + 1)-dimensional representation. S e c t i o n 7.3: (i) In the mapping of Kandel et al. (1990) this model is equivalent to a two-dimensional four-vertex model in thermal equilibrium with a defect line where other vertices, not belonging to the group defining the six-vertex model or eight-vertex model, have nonvanishing Boltzmann weights. The two steps describing the motion of particles define the diagonal-to-diagonal transfer matrix T(o~,/3) in the vertex model (see Appendix A). The partially asymmetric exclusion process with sublattice parallel update corresponds to a bulk six-vertex model with a defect line. One obtains qualitatively the same phase diagram as for the usual ASEP (Rajewsky et al., 1996; Honecker and Peschel, 1997). (ii) We do not discuss the nature of the recursion relation for the stationary state derived in Sch/itz (1993b). But there is one technical ingredient which may be of interest for the solution of other systems with deterministic bulk dynamics. In Schiitz (1993a) it was shown that for the deterministic bulk dynamics defined above one has n 2 x - I V2yl A ) = 0

(7.132)

for 1 m vanishes. Hence a m-point correlator is determined by the dynamics in the sectors with less than or equal to m particles in the (nonstochastic) time evolution generated b y / 4 . This is the reason behind the decoupling of the equations of motion from higher-order correlators. The matrix elements appearing in the expressions for the density and density correlations respectively can be calculated using the Schwinger-Dyson formula (4.37) by treating H - as perturbation. In each matrix element the series terminates at an order p < m, making the derivation of exact results feasible. In particular, the two-particle transition amplitudes satisfy the relation (k, j le-Aqtl m, n ) = (k, j le-HXXZtlm, n ). They can be calculated using the Bethe ansatz in essentially the same way as in the exclusion process" one derives a differential-difference equation by taking the derivative with respect to t first for particle separation j - k > 1, and solves it with a two-particle Bethe wave function. This fixes the energy term, but leaves the scattering amplitude S as free parameter. Then one derives the differential-difference equation for j = k + 1 and fixes S by demanding that this nearest-neighbour equation should have the same form as the general equation for arbitrary separation. The resulting phase shift S is given by

bz + b3 eipl+ip2 - (bl -1- b4 - b5)e ip2 S ( p l , P2) --- - b 2

d- b3 eipj+ip2 - (bl q- b4 - b5)e ipl

(8.9)

with b5 = lo14 q-- 1/)24 + 1/)34 d- 1/342 d- 1/)43 - w41. In a further step one investigates the long-time behaviour, i.e. one studies S for small pl, p2. Depending on the reaction rates this leads to three distinct classes of systems: (1) S ~ 1 corresponding to an asymptotically noninteracting system as in the case of the symmetric exclusion process; (2) S ~ P2/Pl like in the ASEP, corresponding to interacting particles; (3) S ~ - 1 . This defines a new class of systems, studied in detail in Section 9. The second decoupling mechanism which allows for an exact treatment of the hierarchy of equations of motion (8.2) is a decoupling of string expectation


167

values. In one dimension string expectation values of the form S k , r ( t ) = ( (a -- b n k ) ( a

-- b n k + l ) . . - (a - b n k + r - l )

)

(8.10)

play a special role. The best known examples are the empty-interval probabilities (a = b - 1) which led to the exact solution of the random sequential adsorption process (3.40) (Cohen and Reiss, 1963) and of the diffusion-limited fusionbranching process (Doering and ben-Avraham, 1988; ben-Avraham, 1997). In these (and other) processes the equations of motion for m of such strings form a closed set, involving strings of different length, but equal or less in number. A general discussion of this decoupling mechanism for processes of the form (3.34) was given by Peschel et al. (1994). The relationship to quantum spin systems can be utilized in the cases where the equations of motion for a single string take the form Sk,r(t) = A S k - l , r ( t ) - l - B S k + l , r ( t ) + C S k , r - l ( t ) - + - D S k , r + l ( t ) - - E S k , r ( t )

(8.11)

with constants A, B, C, D, E depending neither on k nor r and with boundary condition Sk,0(t) = 1Yt. (8.12) A subset of these models are the free-fermion systems discussed in Section 9. In another interesting class of models one has a diagonal constant E of the form E = a -t- b ( r - k ) . This includes the diffusionless random sequential adsorption process A A ---, 1313with the unusual density relaxation from an initially empty lattice p ( t ) = 1 - exp ( 2 e - t - 2 ) into a large set of absorbing states, defined by all states without two neighbouring vacancies (Evans, 1997). Also models with diffusion and a variety of reactions fall into this subclass of models. The equations of motion (8.11) are related to the quantum mechanical problem of the motion of an electron in a finite one-dimensional crystal in a uniform electric field (Peschel et al., 1994). Since it is not clear how to solve the equations for several disconnected strings we do not study this interesting class of models further.

8.3

Field-induced density oscillations

For a study of the density profile from (8.7) one needs the explicit expressions of the constants ai, bi (8.5) of H in terms of the original rates. The similarity transformation yields al

-"

1/321 + 1/341

a2

=

11331 + 11)41

168

G. M. SchQtz

bl

=

w l 2 + w32 + w21 + w41

b2

--

1/323 + 1/343 - / / ) 2 1

b3

--

//)32 -+-//342 -

b4

=

w13 + w23 -+- w31 + w41.

-

l/)41

1/)31 -

1/341

(8.13)

We assume the system to be defined on a hypercubic lattice in d dimensions with nearest-neighbour interactions. The equations of motion (8.2) for the density profile (nk(t)) reduce to an ordinary lattice diffusion equation with a constant inhomogeneous term c - a l + a2. Therefore it is sufficient to study the onedimensional case. The density expectation value satisfies the equations of motion d dt(nk(t)) =al+a2+b2(nk_l(t))+b3(nk+l(t))-(bl+ba)(nk(t)).

(8.14)

This immediately yields the stationary density o* =

2w41 -+- w21 + w31

.

2w41 + w21 + w31 + 2w14 +//)24 -if- 1/334

(8.15)

Analysis of the stationary equations for the two-point correlator shows that density correlations decay exponentially over a correlation length ~ defined by cosh (1/~)

= ( b l q-

b4)/([b2 -b b 3 [ ) .

Despite its simplicity, (8.14)contains interesting physics. Consider the random initial state [p* ) (2.24) with stationary density p* (8.15). For bl +b4 # [b2+ b3l this uncorrelated state is not stationary, correlations build up with time. We study how a local perturbation caused, e.g. by injection of a particle at site I - 0, spreads and decays in the system. The normalized initial state representing this setup is the state [ P0 ) = no~p*[ p* ) and we calculate Apk(t) = (nk(t) -- p* ) Co, i.e., the time-dependence of the approach of the local density to stationarity. Equation (8.14) is readily solved by Fourier transformation and yields Apk (t) = 1 2zr - 0"

dpe -(bl +b4-b2eip-b3e-iP)t-ipk .

(8.16)

The combination of rates Vc = b 2 - b3 is the drift velocity of the local perturbation in the uncorrelated background and Dc -- (b2 + b3)/2 is the collective diffusion coefficient. The space-averaged total density Ap(t) -- Y~k A p k ( t ) / L relaxes exponentially A,o(/) = (1 - p*)e -(bl+b4-b2-b3)t. (8.17) This follows immediately from (8.16) by summing over k. We study now a diffusion-limited fusion model AA ~ AO, OA with continued production of particles in pairs 00 ~ AA. First we assume all events to take


169

place with the same rate as the hopping rate. With 1/)23 = 1/)32 = 1/)24 - " //334 = w41 = D we find from the table (8.13) the surprising result Vc - Dc = 0. Despite the diffusion of all particles the collective diffusion coefficient vanishes and the perturbation remains localized: Apk(t)

=

3k , 0Ap(t)

=

(1

--

p*'~ )t)k,OCA - 4 D t

9

(8.18)

In a further step we introduce a driving field which causes particles to hop to the right (left) with rate w23 = D(I + r/) (w32 = D(1 - 17)) and which we assume to change the fusion rates to w24 = D(I + 2r/), (for A A ~ OA) and 1/334 = D(1 - 20) (for A A --+ AO). The rate of pair production remains unchanged. One finds now again for the collective diffusion coefficient Dc = 0, but Vc = 20D. The full density relaxation is given by (8.19)

Apk(t) = J k ( 2 r l D t ) A p ( t ) .

The Bessel function Jk is an oscillating function with the asymptotic behaviour Jk(2r) ~ l/x/-~-r-cos (2r - r r k / 2 - zr/4) (Gradshteyn and Ryzhik, 1981). Such spontaneous microscopic density oscillations in the presence of a spatially and temporally constant driving field are an unexpected feature of stochastic singlespecies reaction-diffusion systems.

8.4

Field-driven phase transitions

The presence of a driving force may also lead to other interesting nonequilibrium phenomena. We consider the fusion-branching process where particles hop with rate D -- 1/2 and coalesce instantaneously when they meet on the same site. This leads to fusion rates w24 = 1/)34 = D = 1/2 for A A ~ OA, AO. In order to maintain a nonempty equilibrium distribution we allow particles to create offspring OA, AO --+ A A on nearest-neighbour sites with branching rate w42 = o343 - - b . T h i s process belongs to the class of processes where the equations of motion for vacancy strings (i.e. the empty-interval probabilities) decouple (Krebs et al., 1995). For a translationally invariant system, (8.11) reduces to a linear equation in the difference coordinate r, Sr(t)

=

Sr-l(t)

+

(1 +

2b)Sr+l(t)

-

2(1

+

b)Sr(t)

(8.20)

and boundary condition So(t) = l'v't.

(8.21)

In the infinite system this equation is not difficult to solve. By setting the time-derivative equal to zero, one finds two stationary solutions Sr* - 1 and Sr* =

170

G . M . SchQtz

(1 + 2b) -r. The first solution is trivial, it corresponds to the empty lattice. The second solution corresponds to a product measure with density p* -- 2b/(1 + 2b).

(8.22)

To determine the dynamics one subtracts the stationary part from Sr (t). This does not change the bulk equation (8.20), but the boundary condition for the shifted probability 7Sr(t) = S r ( t ) - S* becomes S0(t) = 0. One recognizes in (8.20) with the transformed boundary condition a lattice diffusion equation with absorbing boundary. A simple exponential ansatz

St(t) = l/(2n')

f

dpe(-Ept-ipr~

ipr -- Bpe -ipr]

yields Ep = 2(1 + b) - e ip - (1 + 2b)e -ip, Bp = identity (4.28) for modified Bessel functions

(8.23)

- 1 and hence with the

oo

Sr(t) = y~. Ss(O)yr-Se-Ut [ I r - s ( 2 y t ) - Ir+s(2yt)]

(8.24)

s=l

with y = x/l + 2b and/z = 2(1 + b). Setting r - 1 one determines the approach of the density to its equilibrium value Ap(t) = p(t) -- p* -- - S I (t). From the integral representation (4.27) of the modified Bessel function one obtains another useful identity oo

1

I

oo

y ~ yk Ik-n(r) = yne~(Y+Y- )r _ Z k=l

y-k Ik+n(r)

for y > 1.

(8.25)

k=0

Asymptotically, y~C~=oY-klk+n(r ) ~. e-n2/2r/[

2V/2-~(l -- y - l ) ] . For an uncorrelated initial state with density O0 one has Ss(O) = (1 - p0) s - (1 - p,)s. Inserting this into (8.24) gives three distinct relaxational regimes (ben-Avraham

et al., 1990) Ap(t) = A(y, po)e -b(p~

(8.28)

y(l-

Po) < 1"

with the inverse relaxation time b(p0) = p0[(l p , ) - I -k- (1 -- p0)-]]. For completeness sake we give the initial-value dependent but otherwise not very illuminating amplitudes A < ( y , 00) = {[y-l ( 1 - / 9 0 ) - I ] - 2 - [ Y - 1]-2}/v/4yrF 3, A ( y ) = l / v / ~ y 3 and A>(1/, Po) = 1 - Po - (1 - p*)/(1 - Po). A remarkable feature of this process is the dependence of the relaxation time on the initial -


171

density. The longest relaxation time cannot simply be read off the lowest energy gap of the time evolution operator of the finite system. This energy gap determines the relaxation of the finite system with small initial densities /90 < 1 - l / y only after a crossover time which diverges in system size (Doering and Burschka, 1990). In the presence of an external driving field the rates become biased. Defining the fight and left hopping rates W 2 3 - - DR, w 3 2 = Dr. with asymmetry q = x/DR~Dr., the assumption of local detailed balance yields the modified rates //)24 = DR, //)34 = DL for fusion and //)42 - - 2bDL, //)43 - - 2bDR for branching respectively. In the infinite system the equations of motion for the empty interval probabilities remain unchanged. Hence the relaxation of the density for a translationally invariant initial state does not depend on the bias. However, in a finite system with reflecting boundaries the situation changes dramatically. The driving force pushes the particles to the boundary where they can coalesce. If the bias is sufficiently strong a phase transition from a finite-density phase to a zero-density phase sets in and one obtains a nontrivial phase diagram which is characterized by three different length scales (Hinrichsen et al., 1996a). One length scale ~1 - - 4 / / I n q is set by the driving field and two further branchingdependent length scales ~2 = 2/In (qy), ~3 = 2 / I n (q/y) are related by space reflection, sending q - - ~ q - l . To discuss this phenomenon we assume (without loss of generality) q > 1. Following Hinrichsen et al. the fusion process dominates for sufficiently strong bias q > y. The system is in a low-density phase with afinite number of particles, i.e. vanishing density in the thermodynamic limit. The particle density decays exponentially to zero as one moves from the right boundary into the bulk. At the phase transition point q - y the correlation length ~3 diverges. One finds a linear decay of the density profile with a space-averaged density ~ = p*/2 where p* is the stationary density (8.22) of the infinite system. In the high density phase the density decays on both boundaries exponentially to its bulk value p*, but with different length scales ~2 and ~'3 respectively at each boundary. These properties suggest to try to understand the phase transition in terms of domain wall diffusion, similar to the first-order transition in driven lattice gases with open boundaries. The principal difference, however, is the absence of a local conservation law which allows for stationary regimes of arbitrary density. Here only two densities of the homogeneous system are stationary, viz. Pl - 0 and/92 defined by (8.22). With regard to our attempt to understand the phase transition in terms of a driven domain wall this is encouraging as these are precisely the bulk densities between which the first-order transition takes place. If the picture of a driven, diffusing domain wall is indeed correct then the transition is expected to take place when the velocity of an upward shock (0, p*) (for q > l) changes sign. As long as the velocity is positive, the shock moves towards the right boundary, leaving the bulk of the system empty (low-density phase). For negative shock

172

G.M. Schfitz

velocity the system would be in the high-density phase. On the phase transition the shock would perform an unbiased random walk, thus leading to the linear density profile. Notice that this reasoning applies for q > 1 and under the assumption that a downward shock (if it exists) would not play a role in the latetime dynamics of the system. Therefore the next question to ask concerns the velocity of a shock. We cannot use mass conservation to derive the shock velocity. Instead, we take into account mass production through the branching process and consider first the case of vanishing driving field. Suppose there is an upward shock (0, p*) located at time t -- 0 at x - 0. There is no net mass production in either of the two stationary regimes, but at the shock position there is a net production which drives the shock to the left. Therefore, after time t the shock has moved a distance r = - v s t where vs is the velocity of the upward shock. To calculate os notice that the number N = p*r of new particles created is determined by the area under the new density profile. This yields v7 = - N / ( p * t ) . On the other hand, the branching process implies (in the simplest approximation) N = 2bDp*t which with hopping rate D = 1/2 finally yields vs = - b . Because of symmetry, a downward shock (p*, 0) will move with velocity v+ = b. In the presence of a bias the situation is slightly more complicated. The branching process for the upward shock yields N = 2 b DL p * t . In addition to this intrinsic velocity the whole frame of reference moves with velocity DR - Dr., thus leading to vs = DR - Dt.(I + 2b). In a similar manner one obtains v+ = DR(I + 2b) - Dr.. To discuss the implication of this result for the possibility of a first-order phase transition we first note that v+ - v7 = 2b(DR + DL) > 0, i.e. in an infinite system a finite region of density p* will extend at a constant rate. Secondly, for DR > Dr., i.e. q > 1 as assumed throughout this discussion, v + > 0 and hence a phase transition in a finite system can be driven only by the behaviour of vs . Any intermediate downward shock moves to the fight boundary and therefore does not play a role in the late-time bulk behaviour. Finally, we find that the condition v7 -- 0 reproduces the exact relation y = q for the phase transition, thus confirming the shock picture of this first-order field-induced phase transition. One may treat this reaction-diffusion system in the rate equation approach obtained by the continuum limit of the mean-field approximation to the exact equations of motion for the local density. The resulting nonlinear partial differential equation has shock solutions with fixed densities known as Fisher waves (Fife, 1979). So far we have established the exact shock velocities, but the previous discussion gives also insight in the fluctuations of the motion of the Fisher wave which are not accessible in the rate equation approach. The precise shock dynamics do not follow from the arguments put forward above, but it is natural to assume that the shock performs a continuous-time random walk. From the shock velocities one reads off shock hopping rates DR + c, Dr. (1 + 2b) + c for the upward shock and D R ( l + 2 b ) + c ' , DL + c ' for the downward shock respectively,


173

with undetermined constants c, c'. On the other hand, the correlation lengths ~2,3 are nothing but the localization lengths of the driven up- and down-shocks respectively, i.e. the respective logarithms of the ratios of the shock hopping rates. We arrive at the conclusion that a shock performs a random walk with c -- c' = 0 and therefore diffusion coefficents D + = (DR(I + 2b) + D L ) / 2 and D s ----(DR + Dt~(1 + 2 b ) ) / 2 respectively. Remarkably this argument can be shown to be exact by going to the continuum limit and solving for the corresponding empty-interval probabilities with shock initial profile (ben-Avraham, 1998a). It is even possible to consider a shock initial distribution on the lattice and solve for the full time-dependent distribution (ben-Avraham, 1998a; Krebs, 1999). A phase transition of a similar nature takes place in the semi-infinite system with a particle trap at the origin (ben-Avraham, 1998b,c).

Comments Section 8:

Perhaps the most prominent class of reaction-diffusion systems which are

not discussed here are diffusion-limited two-species reactions, in particular the annihilation

process A + B -~ (3 (Bramson and Lebowitz, 199 l). The exclusion version of this process behaves essentially like the process without hard-core exclusion (Belitsky, 1995). The corrections in the decay of the density are of the same order l/x/7 as the corrections of the nonreacting symmetric exclusion process of Section 6 to noninteracting particles. The quantum Hamiltonian formulation of such processes has been exploited in Alcaraz (1994) and Dahmen (1995), but concrete consequences of integrability (where applicable) are largely unexplored.

Section 8.1: Further enantiodromy relations have been derived in Schlitz (1995a); Sudbury and Lloyd (1995) and Fujii and Wadati (1997). Section 8.2:

When implementing the initial value in the Bethe ansatz expression HXXZ t

for the matrix element ( k, j leIra, n ) some care needs to be taken in the treatment of the pole for bl + b4 - b5 # 0. For certain processes the pole has been shown to give rise to a longer relaxation time as expected from the continuous spectrum (Santos, 1997b; Henkel et al., 1997). An alternative treatment of the 10-parameter process where the equations of motion decouple from higher-order correlators is the time-dependent matrix algebra of Section 6.5.

Section 8.4: (i) The solution (8.24) of the lattice equation as well as the exact result for finite lattices was obtained in Krebs et al. (1995) in a slightly different, but essentially equivalent way. This equation was also solved by ben-Avraham et al. (1990) in the continuum limit where the lattice spacing vanishes. (ii) The stationary distribution of the system has the interesting property that it can be presented as a matrix product state with 4 • 4 matrices satisfying the stationary version

174

G . M . SchQtz

/) = 0 of the algebra (6.36) (Hinrichsen et al., 1996b). No attempt has yet been made to calculate time-dependent correlation functions with the full algebra. Some eigenvalues and eigenfunctions of the generator were obtained (Hinrichsen et al., 1996a) using the free~ fermion technique discussed in the next section. Not surprisingly the localization lengths ~2,3 arising from the shock hopping rates appear in the wave functions. The physical interpretation of the localization length ~1 (corresponding to single-particle diffusion) both in the stationary state and in the eigenfunctions of excited states is not yet understood.


175

Free-fermion systems

In special cases the scattering phase S(pi, pj) (3.17), (8.9) in the Bethe ansatz can be seen to become independent of the momenta Pi, Pj for a suitable choice of the interaction parameters. In the anisotropic Heisenberg chain (3.6), (8.6) this happens when A = 0. In this case S = - 1 and the Bethe wave function is a totally antisymmetric free fermion wave function. This observation is readily understood after a Jordan-Wigner transformation (Jordan and Wigner, 1928), an old technique developed in the early days of quantum mechanics which transforms spin-(1/2) raising and lowering operators (which commute when acting on different lattice sites) into free-fermion annihilation and creation operators satisfying anticommutation relations. Bilinear expressions like s~sk~ 1 or nk -- s k s~- in the Heisenberg chain become bilinear expressions involving local fermionic operators c~, ck. In particular, the term Aoi3 ty~+l 3 transforms into a piece containing the quartic fermion interaction ckckck+lck+l. + + Thus the Heisenberg chain can be interpreted as a system of interacting spinless fermions. However, if A = 0 this quartic term disappears and all that remains are quadratic expressions. Therefore the resulting system is a free-fermion model. The description of a stochastic process by a free-fermion model neither implies that the (classical) particles described by such a stochastic Hamiltonian would be noninteracting, nor does it mean that one is actually dealing with a stochastic process involving fermions. The only property classical hard-core particles have in common with fermions is that each lattice site can be occupied by at most one particle. A physical understanding of the appearance of free fermions in one-dimensional models (and only in one dimension) is based on the interpretation of the particles as annihilating random walkers. This will become clear after some preliminary thoughts given in Section 9.5 which are based on the infinite reaction limit discussed in Section 4.3. We therefore postpone this discussion and treat the free-fermion description at this point just as a technical device. The main goals of this section are (1) a classification as complete as possible of all the free-fermion systems and (2) to provide the machinery necessary to treat these systems beyond what can conveniently be achieved with string probabilities (8.9). Before tackling this program it is necessary to obtain some general information on the structure of general free-fermion systems (Section 9. l). The strategy is then to turn the corresponding nonstochastic Hamiltonians into stochastic Hamiltonians by suitably chosen similarity transformations. Sections 9.2 and 9.3 clarify how the various free-fermion processes studied in the literature are related to each other by similarity transformations. The systems investigated below are the only known translationally invariant free-fermion models with two-site nearest-neighbour interaction (3.33) and what is developed

176

G . M . Sch0tz

in this section is likely to be a complete theory of this class of models. Section 9.2 deals with constraints on the parameters of a free-fermion Hamiltonian which one has to impose in order to obtain such a stochastic process by a factorized similarity transformation (5.4). There are two types (I, II) of models which are distinct in the sense that all models within each class are themselves related by a factorized similarity transformation, but models of type I are not equivalent in this way to models of type II. Having established constraints on the nonstochastic H f f , one may ask the question which constraints on the reaction rates of the stochastic process this implies. Answering this question (Section 9.3) then allows one, just by looking at the rates of a given process, to decide whether or not it can be analysed using freefermion techniques. It turns out that Glauber dynamics is a representative of type I models, whereas diffusion-limited pair-annihilation-creation represents type II models. This is interesting since these models are related to each other by the domain-wall duality transformation (R~icz, 1985) which does not factorize into single-site transformations (Santos, 1997a). We review the tools for investigating domain wall duality and state, partly without derivation, the main results relevant for free-fermion systems. It emerges from these considerations that studying any representative of type I or type II models is sufficient to obtain any desired information about the other free-fermion models. How free-fermion techniques work for the calculation of dynamical properties is then discussed in Section 9.4 and in more detail in Section 9.5 for the process involving biased diffusion and pair annihilation. Using the transformation technique, the methods developed in the context of this model generalize with little modification to the other free-fermion systems of Section 9.3, in particular to the fusion process A A ~ AO, OA which describes exciton dynamics on N(CH3)nMnCI3 polymers (Kroon et al., 1993; Kroon and Sprik, 1997; Kopelman and Lin, 1997) (Section 10). It is worth pointing out at this stage that remarkably enough, the experimental exciton reaction and diffusion rates are indeed consistent with the free-fermion condition on this process. This is made plausible by the discussion at the beginning of Section 9.5 which gives a physical interpretation of the free-fermion condition. Besides the intrinsic theoretical interest for the understanding of the effects of diffusive mixing in low-dimensional media, this application lends also direct experimental justification to the study of free-fermion systems.


9.1

177

The Jordan-Wigner transformation

For the definition of the Jordan-Wigner transformation (Jordan and Wigner, 1928) and also for other purposes it is useful to introduce the particle-parity operators l

o,,,

=

l-I j

(9.1)

j=k

Qt

=

Ql,t

(9.2)

O

=

QI,L.

(9.3)

The eigenvalues of these operators are +1, depending on whether there is an even number of particles in the interval [k, l] (Qk,t = 1) or an odd number of particles respectively (Qk,l = - 1 ) . With these definitions the Jordan-Wigner transformation is given by Ctk

=

Sk a k _ 1

(9.4)

ck

=

Qk-lS-~.

(9.5)

Using the commutation relations for the Pauli matrices and the relations (or i)2 = 1 for i = x, y, z one easily verifies the anticommutation relations

{c,, c,I - {c[, c7} = 0

(9.6)

{c~, Cl} -- 8k,t.

(9.7)

t nk -- ckck

(9.8)

and The fermionic number operator

takes values nk = O, 1 and is identical to the usual particle projector nk = (1 a{)/2- sis~. By applying the definitions (9.4) and (9.5) to the transition terms of reactiondiffusion systems one notices that expressions of the form s~s~+ 1 are given in terms of the fermionic operators -

S~Sk+ 1 --

"t"

Ck+lCk,

s-~s~+ , = ck+ , ck,

-

Sk S~+l

i"

--

CkCk+l,

s k s k+ , -- ctk ct,+ , .

(9.9)

There are, however, no other products of two Pauli matrices which transform into bilinear fermion operators besides these four and the number operator nk. Hence non-nearest-neighbour hopping processes s - ~ s f or reactions cannot lead to expressions which are bilinear in the fermionic creation and annihilation operators.

178

G.M.

Schs

For the same reason this transformation cannot be used for, e.g. two-dimensional systems. Notice that on a ring of L sites with periodic boundary conditions for the Pauli matrices O ' Li + l - - O"li one has c Lt +l = c ~ Q and CL_t_ l = Qc 1 . Q may be written Q - ( - 1 ) N where N = ~ n k is the number operator. Therefore periodic boundary conditions translate into antiperiodic boundary conditions in fermionic terms in the sectors with an even number of particles. This is relevant for the spectral properties of finite systems. It is now easy to see that the most general Hamiltonian with two-site interaction for free fermions of a single species is of the form

,,-

=

-E,,Lr k=l L

=

-

[c + o,s

G, +

-

+

s,+,

k=l

+/ZlS~'S~l + / z 2 s k sk+ 1 -t- hlnk -t- h2nk+l]. with periodic boundary conditions s~:+l -

(9.10)

s~: or antiperiodic boundary condi-

tions SLa:+l = --Sli respectively. The free-fermion Hamiltonian has a Z2 symmetry generated by [ H f f , Q] = 0. It splits the Hilbert space into a sector with an even number of down-spins (particles) and an odd number of down-spins. This is obvious from the form of the local interaction matrices H f f which change the total spin only in units of 0, 4-2. The parameters DI,2,/Zl,2, h 1,2 and c may take arbitrary space-dependent, in principle even complex, values. Any stochastic model described by this Hamiltonian for a suitable choice of parameters or by an equivalent Hamiltonian obtained through a similarity transformation is then a free-fermion system. Here we study those models, but restrict ourselves to space-independent coupling constants and to stochastic systems with two-site interaction (3.35).

9.2

Stochasticity conditions

It is not a priori clear that a similarity transformation which relates the nonstochastic Hamiltonian (9.10) to some stochastic process really exists. One has first to consider possible restrictions on the parameters of H f f coming from the requirement that H -- H f f or some transformed H = B H ff13 - l has real positive transition rates and satisfies conservation of probability. This simplifies the classification of stochastic free-fermion system obtained from H f f . We use conservation of probability as a means to derive conditions on the parameters of H f f . One could go further and use also positivity of the real part of its


179

eigenvalues. This approach would be necessary for a complete description of

all possible free-fermion processes. However, by restricting ourselves to translationally invariant two-site processes conservation of probability alone appears to yield sufficiently strong conditions. For the derivation of the stochasticity conditions on H f f one then has to show that ( S 113Hff = 0 which follows from

0 = ( S In = ( S IBH ff13 -1 . The simplest transformations are the homogeneous factorized transformations 13 = B | (5.4) where B -= B |

=

bll b21

b12 b22

(9.11) "

Transformations of this form preserve the desired locality of the two-site interaction. A complete classification of transformations which have this property does not exist. The only other known class of transformations which preserve locality is the domain-wall duality transformation discussed in detail below and there is no indication that other classes of transformations might exist. Therefore we focus on these two classes of transformations and consider first factorized transformations (9.11) with arbitrary constants bij. In this case one has to choose periodic boundary conditions for H f f since antiperiodic boundary conditions would generate negative rates on sites 1, L. A crucial step in most of what follows is the factorization of both B and ( S I together with the decomposition of H f f in a sum of matrices H / f which act nontrivially only on two sites. This reduces the problem of finding constraints on the 2 Ldimensional matrix H f f to a discussion of constraints only on H / f since one only needs to show (SIBH ff =0. (9.12) This is a vector equation on the four-dimensional space on which H / f acts. The solutions to these four equations give necessary conditions for H to be stochastic. The four equations (9.12) have two distinct types of solution, depending on the matrix B. We discuss the origin of these two solutions which we shall use for classification purposes. Defining F = (bll + b21)/(bl2 + b22) one has to consider IFI -7: 0, ~ (type I), IFI = 0, ~ (type II). The cases IFI = 0 and IFI - oo are related by the factorized spin-flip transformation exchanging the role of particles and vacancies. It is therefore sufficient to consider only F = 0 for solutions of type II. For further analysis it is convenient to use h = (h] +h2)/2, a -- (h] - h2)/2, D = (D] -t- D2)/2, 17 ~ (Dl -- D2)/2. For type I solutions one has both (bl] + bzl), (b12 -k- b22) r 0. The equation (9.12) then yields F2

=

D-h /z]

(9.13)

180

G. M. SchOtz

F2

=

lz2

(9.14)

c

=

D+h -D-h

(9.15)

a

~

-T~.

(9.16)

Relations (9.13), (9.14) imply/21//,2

D 2 - h 2 and allow for the parametrization which then implies 1-`2 = r 2. The factor r may be absorbed in a diagonal factorized transformation/~ w i t h / ~ 1 1 = r and /~22 = l / r leading to a transformation B' -- /~B with 1-" - 1. Hence the most general free-fermion matrix of type I which leads to a stochastic process is of the form =

121 -- ( O - h ) / r 2, 122 -- ( O -k- h ) r 2

f f ,I

+

+

(O + q)s~-s~-+l + (D - rl)s~s-~+ 1 + (D - h)s k sk+ 1

+ ( D + h)s k sk+ l + (h - 0)nk + (h + rl)nk+l -- D - h. (9.17) The transformation B satisfies F 2 = 1. By choosing the normalization of B such that bll + b21 -- 1 the transformation matrix may be parametrized:

BI--(

1-blbl

1-b2)b2 "

(9.18)

For processes of type II, the equation (9.12) yields the conditions

122

=

-2h

(9.19)

=

0.

(9.20)

There is no relation involving a and we choose without loss of generality a = 0.* This leads to a free-fermion Hamiltonian

H f f ' l ' = D,s-~sk+ l + D2s;s~+ 1 + 12,s-~s-~+ l + h(nk + nk+, - 2).

(9.21)

Normalizing B such that b12 -+- b22 = 1 allows for the parametrization

BI i = (

bl -bi

b2b2 ) 1 -

(9.22) "

It has to be stressed that this classification into type I and II processes has no physical significance for the properties of these processes. It serves only as an ordering scheme for their systematic study. These two types of processes are in fact related by the nonfactorized similarity transformation of Section 9.3.3. +The choice of a has only technical significance. The term proportional to a is a lattice divergence and cancels for periodic boundary conditions. It is included only to ensure that (9.12) is not only a sufficient but also a necessary condition for conservation of probability. If (9.12) yields no relation for a, its choice is irrelevant.


181

9.3 Equivalences For real and positive off-diagonal elements the Hamiltonian defined by (9.17) is itself already stochastic and describes diffusion-limited pair annihilation with pair creation A A ~ 00 (Table 1) (Grynberg et al., 1994). Indeed, it turns out that the processes of type I are all equivalent to this diffusion-limited annihilationcreation process H ~ (3.39) which is studied in detail in Section 9.6. By similarity, diffusion-limited fusion HOLF is also among the type I processes (Section 5.1). For this process the transformation/3 not only gives expectation values of this process in terms of the annihilation process, but also shows why the emptyinterval approach (8.9) to the fusion process is equivalent to the free-fermion approach described in detail below. As outlined in the introduction to this section, the main purpose here is to establish relations between the rates tOij of the transformed, i.e. stochastic Hamiltonian H which guarantee that H is in fact equivalent to a free-fermion system. The relations derived below are exhaustive in the sense that no other processes than those listed here can be transformed to free-fermion form I by a factorized similarity transformation/3. Just by counting parameters one might expect a fiveparameter family of processes, resulting from the free parameters D, r/, h of H f f and bl, b2 of B. However, the requirement that llJij >__ 0 turns out to lead to several distinct classes of three-parameter processes. Since there is no complete classification of free-fermion systems in the existing literature, we present the derivation both here and for type II processes in some detail.

9.3. i

Classification of free-fermion systems (I)

It is convenient to write the stochastic Hamiltonian to be obtained from Hf.L/ in the form H 1 - Y~k H] with the nearest-neighbour stochastic matrices H / (3.35). So far we have made use only of conservation of probability which is built into the structure of the stochastic matrix (3.34). This has given rise to constraints on H f f , leading to H f f , / . In what follows we employ the opposite strategy and use the knowledge we have about H f f ' t to obtain information about H]. It is important to recall the technical issue that the matrix elements of the transformed matrix free-fermion matrix HI gives the rates l13ij of the stochastic matrix H~ only up to an arbitrary divergence term Ak - Ak+l (3.36). A set of very strong constraints on the matrix elements 113ij of n / arises from the Z2 symmetry of H ff't combined with the positivity condition on the rates OJij. The Z2 symmetry, which has been of little importance so far, implies that cr[ O'k+ z 1 commutes with H f f ' l gets

Since type I processes satisfy ( S I H f f ' ! - 0 one ( S Icr~z a~+ z l H f f , I -- 0

(9.23)

G.M. SchOtz

182

On the other hand, Hkt =

13HIf'I]3 - 1

-[-

Ak

-

ak+l

where A is a matrix chosen

such that Hkt is itself stochastic, not only the sum H t of local transition matrices.* Inserting this relation into (9.23) and defining the transformation parameter 3 = bl - b2 yields the following three conditions: 0

=

(1 +

231/)41

(9.24)

0

=

(1 --3)(1/)24 + 1/)34)- 231/)14

(9.25)

0

=

(1 + 3)(to12 + 1/)13) + (1 --3)(1/)42 + 1/)43).

(9.26)

3)(1/)21 + 1/)31) +

Two more useful relations arise from the other two nonstochastic equations of motion for the Pauli matrices ~r[

(S Icr{H/f'l t

=

(S I{ (h + 17) + 2Da~ + (h - r])o'/~_lO'/~ }

(9.27)

(S la~H/f't

=

( S l{(h - r/) + 2Dcr[ + (h + r/)r162

(9.28)

}.

z 1 and on the r.h.s, of On the r.h.s, of (9.27) there is no term proportional to O'k_ (9.28) there is no term proportional to crkz+ l " Since the factorized form of the similarity transformation preserves the local structure of the equations of motion, (9.24)-(9.26) lead to the additional two constraints 0

0

----

--

(1 + 3)(to34 -- 1/)24 + 1/313 -- w12) + (1 -- 3)(w42 -- 1/343 + 1/)21 -- 1/331) +23(w23 -- w32)

(9.29)

tol4 +//324 + 1/)34 + 1/)21 + 1/331 + l/)41 -- 1/)23 -- 1/)32

(9.30)

for the transformed Hamiitonian. The five free-fermion conditions (9.24)-(9.26) and (9.29), (9.30) suggest further analysis in terms of the transformation parameter 3. Reality of the rates implies vanishing imaginary part of 3. This is a condition on the parameters of the transformation. Positivity of the rates then imposes various constraints of the t o i j . These constraints depend on 3. For 3 = 0 positivity of the rates and (9.24)--(9.26) imply w21 = 1/313 = 1/)24 = 1/)34 = to42 = 1/)43 = 0. Then relations (9.29), (9.30) are automatically satisfied. Hence the untransformed Hamiltonian with Dl,2 > 0, 0 < h 2 < D x and both DI,2 and h real yields the most general process that can be obtained from a transformation with 6 = 0. Indeed, except in the trivial limiting case h - 0 the transformation matrix B reduces to the identity A: 3 = 0. 1/)31 =

1/)12 =

*In the case of periodic boundary conditions which is considered here, the term A k - A k + 1 cancels in H I . For open boundary conditions this lattice divergence term can be absorbed in a redefinition of the boundary fields.


183

matrix bl - b2 = 0. In terms of the right and left hopping rates DR,t, = DI,2 and v - ( D + h ) / ( 2 D ) the stochastic transition matrix H l ' a may be parametrized:

0 0

9

0 . DL

DR .

(1 - v ) ( D R + DL) 0 0

V(DR 4- DL)

0

0

.

H~,A =

0

.

(9.31)

k

The dots on the diagonal indicate that these matrix elements are given by conservation of probability (3.34). From (9.23) and (9.27), (9.28) one reads off the equations of motion for the string expectation value (Qk,l) (9.1) d dt (Qk,t)

=

(h - 17)(( Q k - l , l )

+ ( Q k , l - I )) + (h + 17)((Qk+l,/)

(9.32)

4-( Q k , l - I )) - - 4 D ( Q k , t )

with Qk,k-l -- 1 for all t. This is a closed set of equations of the form (8.11). It can be solved by Fourier transformation in the centre of mass coordinate R = k 4- l and a plane wave ansatz with inhomogeneous boundary conditions for the remaining dependence on the relative coordinate r = l - k. The solution by a different approach which makes direct use of the free-fermion nature of the problem is given in Section 9.5. The link between the expectation value (Qk,/) and the free-fermion nature of the problem is the relation 4( CkCl ) = ( Q k , l ) + ( Q k + l , l - l ) -- ( Q k + l , l ) -- ( Q k , l - I ) for l > k. The particle density is given by (nk) = (1 - ( Qk,k ))/2. B: 0 < 8 < 1. For 0 < 8 < 1 positivity of the rates and (9.24), (9.26) imply toil = 1/312 = 1/313 = 1/242 -- 1/243 = 0 and thus leaves only nonvanishing hopping, pair annihilation and fusion rates which are related through w23 4- w32 = Wl4 + 1/224 4- to34 (9.30). Hopping and fusion asymmetry are related through w34w23 = w24w32. This relation is obtained from (9.25), (9.29) by eliminating 8 and gives

ul,B "k

0 0 0 0

0 . DL 0

0 DR . 0

(1 - v ) ( D R + DL) vDR vDL "

(9.33) k

as the most general fusion-pair annihilation process satisfying the free-fermion condition 9 The parameter v parametrizes the branching ratio v I(1 - v) between fusion pair annihilation, t *The parameters DR,L and v appearing here and below are for each class of models different functions of the original parameters D 1,2, h and the transformation parametersb 1,2. These functions are given by the transformation/3.

184

G.M. SchQtz

Analysing DR.L and v in terms of DI,2, h, bl,2 shows that this transformation is possible only if in the original Hamiltonian H f f , l all parameters are real and one has D1,2 > 0 and h 2 = D 2, i.e., if one has a pure pair annihilation or pure pair creation process respectively. The transformation matrix B has parameters bl - - 0 a n d - 1 < b2 < 0 i f D + h =0andbl - - 0 , 0 < b2 < 1 respectively if D - h = 0. For D + h = 0 the equations of motion (9.32) become equations of motion for the expectation value of Q~'fl = H~'=x[1 - 2 n z / ( l - b2)]. (A similar expression follows for D - h = 0.) In B one recognizes the transformation matrix found by Krebs et al. (1995) and Simon (1995). C: ~ = 1. As in cases (A) and (B) positivity of the rates and (9.24)-(9.26) impose severe constraints on the reaction parameters, here Wil -- Wli -- O. Equations (9.29) and (9.30) relate the hopping rates to the fusion rates, w32 = w34 and we3 = w24. The branching rates w4e, 1/)43 are left undetermined by these relations. By explicitly calculating these rates from the transformation B one finds that they satisfy//)43 u~32 = 1/3231/)42. There are no further constraints and thus the general free-fermion form of the branching-fusion process may be parametrized by H~,c _

0

0

0

0

0 0

. DL

DR .

DR DL

0

vDL

vDR

.

(9.34) " k

Also this is a three-parameter process. One has DR -- h - 17, Dr. -- h + 17 and the relation for the transformation parameter D -- h(b2 + 1/b2). Therefore the domain of parameters of H f f ' l is given by D I , 2 > 0, 0 < h 2 < D 2 and Dl,2, h real. Unlike in case (B), both transformation parameters are fixed up to a sign and given by bl = (D + h -tx / D 2 - h 2 ) / h and b2 = ( O + ~ / D 2 - h 2 ) / h. The branching-fusion ratio v = b2 2 - 1 vanishes for D 2 -- h 2. In this limit of pure fusion the quantity Q~,t,c gives the probability of finding vacancies on all adjacent sites k, k + 1. . . . . y. The particle density is given by (nk) = 1 - ( Q ~ , ,xc ) . Likewise, the expectation value 4(ckct) is transformed into the y-I

interparticle distribution function (IDPF) (nk I-Iz=x+l (1 - n z ) n t ) (Doering and ben-Avraham, 1988) which gives the probability that the next particle to the fight of a particle on site x is on site y. This establishes the equivalence of the freefermion approach with the IDPF formalism for this model, or, equivalently, with the empty-interval approach (8.11). From the transformation parameters one reads off the transformed initial densities for random initial states. Inverting the equivalence we can make use of our knowledge of the branching-fusion process (Section 8.3) to predict the behaviour


185

of the pair annihilation-creation process. The initial densities P0 < ,Ocrit for which the decay of the density in the branching-fusion process depends on P0 are mapped into negative initial densities of the pair annihilation-creation process H ~ (3.39), (9.31). In this range of initial densities the transformation/3 is not a stochastic similarity transformation (Henkel et al., 1995). As a result there is no initial-density dependence of the density relaxation for uncorrelated initial states in the pair annihilation-creation process, except that at initial density P0 -- 0 the approach to equilibrium is slower, cx t - l / 2 e -t/r (8.27) (Grynberg and Stinchcombe, 1995), than for initial densities ,o0 > 0. In this generic case the approach has a different power-law prefactor cx t-3/2e -t/r (8.26) (Santos, 1997b). D" 3 > 1. For 6 > 1, positivity of the rates and (9.24), (9.25) imply / / 3 i l - Wi4 = 0. Given these constraints, (9.30) then also implies w23 - w32 -- 0. The remaining four parameters are related by Wl2W42 = 1 / 3 1 3 1 / . ) 4 3 , which follows by eliminating ~ from (9.29), (9.26). This allows for the parametrization wl2 - av, tO13 = a v - 1 , 1 / ) 4 3 - - by, 1/)42 - - by -1. In terms of these parameters the range of definition l < 3 < o~ implies the relation b > a. There are no further constraints on the reaction rates and hence the biased voter model

H~,D

=

0

av

av -1

0

0 0

. 0

0 .

0 0

0

by -1

by

0

(9.35) k

defines a free-fermion process for b > a (Simon, 1995; Henkel et al., 1997). By rearranging terms, H 1,~ = Y~k HI'O may be written in the form H t ' ~ = Y~.k/~,o with ISI~, D

__

1 -~(1-cr] )(a + b - ( a - b ) c r [ ) ( v (

Z Z (9.36) l -~rk_lcr ~Z ) + v - - 1(1-crkZ crfi+l)).

This is a kinetic lsing model at T = 0 in a weak magnetic field and with domain wall driving.* It is interesting to observe that for this model the equations of motion for the spin variables cr~ do not decouple as they do in ordinary Glauber dynamics without magnetic field (a = b) (see Section 3.3.3). Yet this is an exactly solvable free-fermion system. The domain of parameters of H ff'1 is given by Di.2 > 0, 0 < h 2 < D 2 and Di,2, h real. One has bl = (1 + ~/(D + h ) / ( D - h ) ) / 2 and b2 = (1 ~/(D - h ) / ( D + h ) ) / 2 . *More precisely, this corresponds to a Glauber model in the limit where both the temperature and the magnetic field approach 0, but with their ratio proportional to (a - b)/(a + b) kept fixed.

186

G.M. Sch~tz

E" 3 < 0. The systems with 3 < 1 are related to those with r > 1 by a global spin-flip operation. This yields processes of the form 9

0

0

vDR

.

DR

0

vDL

DL

9

0

(1 -- v)(DR + DL)

0

HI,B '

=

0

0

0

'

(9.37)

k

and vDL

vDR

0

DR DL

. DL

DR .

0 0

0

0

0

0

9

H~,c'

=

"

(9.38)

k

The form of the kinetic Ising model I, D remains unchanged, but has b < a. This completes the list of free-fermion systems of type I.

9.3.2

Classification o f free-fermion systems (H)

For type II processes the Z2 symmetry of H f f ' t l cannot be exploited to obtain constraints on the reaction rates, yet the analysis of type II processes is more straightforward than that of the type I processes. Explicitly performing the transformation B 1t (9.22) gives the reaction rates in terms of the parameters Dl,2,/z l, h, bl,2. For a complete analysis it turns out to be sufficient to study the rates (9.39)

--

D2)b 2 - l z l b 2 -O2b22 - O l ( l - b l )2 -t-/zlb 2

w23

=

-Bib 2-

O2(1 - b 2 ) 2 4 - / z l b 2

(9.41)

win

=

(D1 + D2)(I - b e ) 2 - # l b 2

1/341

=

w32

(Ol-'t-

Taking the sum of all four equations gives w32 + wa3 + together with the positivity of the rates implies t032 =

//323 =

1/)14 =

1/341 =

O.

1/314 -t- 1/)41 =

(9.40)

(9.42) 0 which

(9.43)

There is no particle hopping and pair creation-annihilation in type II processes. This is in contrast to the type I processes which are all equivalent to the process I, A with only hopping and pair creation-annihilation. These relations admit the two types of solutions 1 - 2b2 = 0 (A) and Dl = D2 = 0 (B).


187

A" 1 - 2b2 - 0. The requirement (9.43) fixes also the second transformation parameter as 4bZtzl = DI + D2. As only the combination b2/zl enters the rates, the sign of bl is irrelevant. All the rates obtained from the transformation now take very simple expressions in terms of the parameters h, D, r / o f the original free-fermion Hamiltonian H I1. Defining a = h + D - )7, b = h + D + r/, c -- h - D + )7, d = h - D - r/one obtains the process (Henkel et al., 1995) 9

I_11/,A_

a

b

0

c d

. 0

0 .

d c

0

b

a

.

"

(9.44)

k

Since the rates are defined only up to a divergence Ak Ak+l one can redefine the rates such that, e.g. c -- d or a = b. For D _< 0 this process is Glauber dynamics at finite temperature in zero magnetic field, but with domain wall driving. Using enantiodromy, the transformation 13 shows that the simple form of the equations of motion for the magnetization ( a [ ) is related to the diffusion of a single (noninteracting) particle under the action of the (nonstochastic) Hamiltonian H f f ' I I. In general, the equations of motion of a k-point spin correlator in Glauber dynamics are determined by the sectors with up to k particles of H f f ' l l . -

D > 0 corresponds to Glauber dynamics with negative (antiferromagnetic) coupling 9 Positivity of the rates implies for both signs of D the inequalities h > 0 and 0 < D 2 < h 2. w

B: DI = D2 = 0. Also for models of type II,B the constraint (9.43) determines completely the form of the process. From the transformation one gets 4b 2#1 -- 0 and the resulting process takes the form 9

I_III, B "'k

b b

0

a

a

0

. 0 b

0 .

a a

b

.

(9.45) k

with a = h(l - b2) and b = hb2. This is an entirely trivial process where on each lattice site particles are created independently with rate b and are annihilated independently with rate a. There is no interaction between the processes on neighouring lattice sites 9 As a physical model, the system may be interpreted as a kinetic Ising model of Glauber type in the infinite-temperature limit with an infinite magnetic field, but with their ratio proportional to (a - b ) / ( a + b) kept fixed 9

G.M. Sch0rtz

188

9.3.3

D o m a i n - w a l l duality

The stochastic rules for Glauber dynamics can be reformulated as a reactiondiffusion system by identifying a pair of parallel spins on neighbouring sites with a vacancy and a pair of antiparallel spins, i.e., a domain wall with a particle (R~icz, 1985). Domain wall hopping thus tums into particle hopping, and a spinflip event between neighbouring parallel spins, i.e., creation or annihilation of two neighbouring domain walls, becomes a pair creation or annihilation event. Hence, one expects the existence of some mapping from H I t,A to Ht'A. Clearly, this is not a one-to-one mapping as both a pair of neighbouring up-spins and a pair of neighbouring down-spins are mapped into a vacancy. Nevertheless, using some of the algebraic structure of Glauber dynamics and diffusion-limited pair annihilation-creation, these two systems can be related by an invertible similarity transformation (Santos, 1997a). We shall call this the domain-wall duality transformation, or simply duality transformation.* The duality transformation is based on some properties of the affine Hecke algebras (Section 3.4). Consider the following representation of the TemperleyLieb algebra (3.42)-(3.44) with q + q - l _ 1

h2j-I hzj

1

=

~(1 + a)r) 1

~(1

+

1< j < L

ajo')+l) Z

"

1 < j < L-

(9.46) 1

(9.47)

and define gj = ( l + i ) h j - - 1 where 1 < j < 2L - l and i is the imaginary unit. Furthermore, we need the operators X+ = l and X_ - a~. These definitions prepare the ground for the definition of the duality operators (Levy, 1991 ) D+ = g i g 2 . . , g2L-I X+

(9.48)

The operators gj and X+ form an affine Hecke algebra and their algebraic relations can be used to show that

hj+l = D+hj D_7,_l

(9.49)

for 1 < j < 2L - 2. This can be verified explicitly by using the representation (9.46), (9.47). Defining h~:L by the transformation h:~L -- D + h 2 L - l D-7,-l yields a set of operators satisfying the relations of a periodic Temperley-Lieb algebra. Moreover, relation (9.49) becomes an algebra automorphism which holds modulo 2L. In the representation (9.46), (9.47)one finds 1

h?L -- ~ ( 1 + Co'~a~+ 1)

(9.50)

;This notion of duality is not to be confused with duality as used in the mathematical literature on interacting particle systems (which is a special case of what we refer to as enantiodrorny).


189

where L

C = H a) ~

(9.51)

j=l

measures the spin-flip parity of a state. It is useful to introduce V+ = ~ D +

(9.52)

where the factorized similarity transformation 7-?, = R | is defined by the local transformation matrices R = (1 + icr Y)/~/-2. Both R and V are unitary, i.e. the transposed of the complex conjugate transformation is equal to the inverse. With the definitions (9.3), (9.51) one then obtains the following transformation laws V+cr I V y : 1

=

-cs:'

V-t-t'7~g:~_ 1 =

--0"~

0"2... (7;

v;

' crjz v •

_

-t-QCrL cr 1

V+(7~ g~r_ 1

--

(7; O'j% 1

vg

Oj_ 1

x

=

v2 l ,Lv+

-

x

Via{, VZ 1

I Oj

-

... O'L_ 1 . (9.53) That this transformation acts indeed as a domain duality transformation can be seen by inspection of the various transformation laws (9.53). As an example, consider the spin-flip events (1" ,1, 1") --+ (1" 1" 1') and ($1" ,1,) --+ (,1, ,1, ,I,) on sites k - 1, k, k + 1. Both create a pair of domain walls and the sum of both events z z l )/4. The duality is represented by the matrix fik = skx (1 - crk_lcr iz )(1 - or;zcr;+ transformation maps fik into uk sk sk+l which The action together with following two

V+

-

V+gtkV~_ 1 = Sk'-XsXk+l(1 _ crZ)(lk _ ai+lz )/4 --

is indeed a pair creation event in terms of particles. of V+ on arbitrary states is given by the transformation laws (9.53) the action on the vectors 10) and (SI respectively. Below the relations will suffice" (S [evenV+

=

a(S[

(9.54)

-1

(V+)_II0)

=

a

2

(10)+IL))

(9.55)

where ( S ]even is the restriction of ( S I to the subset of states with even particle number and I L ) represents the completely full lattice. The factor of proportionality a which cancels in any transformed expectation value has the form a ~ / ~ ( - 1) L+l exp (iqS). The crucial observation is that also the nonstochastic matrix (9.10) can be written entirely in terms of the transformed Temperley-Lieb generators ~ - l h j ~ . As a result, in the transformed free-fermion Hamitonian / ~ H = V:~ l H f f V+ appears none of the nonlocal terms which the nonlocal transformation V+ might

190

G . M . SchQtz

generate. T h u s / q f f may be used as a representative for stochastic processes of the form H -- 13171ff13 - l = - Y ~ k Ilk with local interaction matrices Hk. In principle, one might expect to find a completely new class of free-fermion systems by transforming the nonstochastic H f f using first the duality transformation and then performing a factorized similarity transformation/3. Therefore, as in Section 9.2 we first discuss the stochasticity conditions o n / - ) f f . Carrying out the transformation on the local interaction matrix (9.10) leads to an off-diagonal three-site interaction term (ttl + / z 2 - Dl - D2)cr[_lcr~cr[+ I. There is, however, no corresponding diagonal part proportional to cr[_l~r[+ l which would be necessary to conserve probability. Hence one gets the condition /zl + #2 - Dl - D2 = 0. This allows for the parametrization/zl = 2 D r , /z2 = 2 D ( I - v) with D = (Dl + D2)/2 as before. For further analysis it is convenient to set h l = h2 = h. After rearranging terms and up to boundary conditions which need separate discussion one obtains the duality-transformed free-fermion Hamiltonian with /4/f

=

2D(o'[ + o'[+,)[1 + (1 - 2v)o'~:cr/,x+l] + [2r/(o'[ - cr[+,)

x --[-C Jr- h + Ak -- Ak+l. +h]o/: O'~:+1

(9.56)

Conservation of probability leads to additional equations which we write out explicitly (c + 2h)(bll + b21) 2 + 8D(I - v)(bll + b21)(bl2 + b22)

=

0 (9.57)

(c + 2h)(b21 + b22) 2 + 8 D ( I - v)(bll + b21)(bl2 + b22)

=

0 (9.58)

a D o \((bll + b21) 2 + (bl2 + b22)2~), + 2c(bll + bzl)(bl2 + b22)

=

0. (9.59)

There are the same two types of solution as in Section 9.3.1: IF[ ~ 0, cx~ (type I), [FI = 0, c~ (type II) where F = (bll + b2])/(bl2 + b22). For type I, (9.57) and (9.58) imply either F 2 = 1, (c + 2h) + 8D(I - v) = 0, c+4Dv=Oorthesetofrelations2D(l + F 2 ) - h F = 0 , c + 2 h = 0 , v = 1. In the first case one may take without loss of generality the solution F = 1, since taking 1-' = - 1 leads after a factorized transformation to the Hamiltonian

/.~/f' = a/.)/f, 1-'=-1 a _ l ~_ _/~r/f, 1-'=1. The second possible set of constraints is a special case of the first set since for c + 2h - 0, v = 1 the value of F can be transformed to F = 1 by a diagonal transformation. H e n c e / q / f (F = 1) may be taken as representative for type I processes. With the two additional constraints on the parameters/41'a ~ iSlff,! is already a stochastic Hamiltonian, without further transformation by some matrix/3. It turns out to be the Hamiltonian H I l,a for Glauber dynamics (9.44). Since H / l , a is a representative of the type II processes H / l , a , H 11,B, one realizes t h a t / 4 l , a = H / l , a and/-)/,B = H //,B are the only type I processes related to the dual H a m i l t o n i a n / 4 f f ' l .


191

The type II solution yields the two constraints c + 2h - 0 and v = 0. Here the transformation 7r t t R-1 yields a representative which turns out to be the stochastic Hamiltonian H t'a. Hence all stochastic processes related to 17-1f f ' t t are the processes H t discussed in Section 9.3.2. Therefore, up to boundary conditions, the processes of type I and type II are dual to each other, i.e. related by the duality transformation D combined with some similarity transformation/3. Thus the duality transformation does not yield new free-fermion processes. To summarize, the various equivalences between free-fermion systems found in Krebs et al. (1995); Henkel et al. (1995, 1997); Simon (1995) and Santos (1997a) are all generated by the duality transformation and a factorized similarity transformation from the free-fermion process H Dw with diffusion-limited pair annihilation and creation. This family of processes appears to constitute all onespecies free-fermion processes with nearest neighbour interaction.

9.4 Stationary and spectral properties 9.4.1

Stationary states of free-fermion systems

We have already derived the stationary distributions for the fusion-branching process which are the empty lattice and the product measure (8.22). For the mixed pair creation-annihilation process D r h a stationary state is defined by the product measure with density p = x/~/(V/-v + ~/1 - v). By projection on the sectors with even and odd particle numbers one obtains ip)even(odd) _ 1 4- Q -- 1 :k: (1 - 2p) t, IP)

(9.60)

where Q is defined in (9.3). These two vectors define the stationary distributions for the even and odd particle sector respectively. Only Glauber dynamics at T r 0, the dual process to H l,a for h :/: D, has a nonfactorized state, viz. by construction the equilibrium distribution of the one-dimensional Ising model I P*) - e -r ~:ka~~ I s ) / Z L (see Section 3.3.3 for the definition of the rates in terms of fl J). Of course, this distribution can be obtained from the product measure using the domain-wall duality transformation.

9.4.2

Relaxation times

While the analysis of the stationary distributions does not reveal much physically interesting structure the dynamical properties of free-fermion models are rather remarkable. Since the spectrum is independent of the choice of basis one may choose H f f ' t t for the diagonalization and calculate the energy gaps for each

G.M. Sch6tz

192

process in terms of the reaction rates using the similarity transformation. This last step involves expressing the parameters of H f f ' I t in terms of the respective reaction rates and is not carried out explicitly here. Instead we only calculate the spectrum in terms of DI.2 and h. This provides a simple criterion for the existence of an energy gap which we apply to the various free-fermion processes. The Fourier transforms of the fermionic annihilation and creation operators respectively are defined by bp

e-i ~ ~ e 2rrikp ~/~ ~. c~

:

(9.61)

k=l bp*

--

e- ~c~

x/r_~

(9.62)

k=l 5satisfying {bp, bq} = {bp, bq} = 0 and {bp, bq} = 6p,q. Inverting (9.61), (9.62)

yields e in~4

ck

=

~

-2nikp

Z

e

t.

bp

(9.63)

p , ck

---

e -in~4 ~ + Ep e L b p. ~

(9.64)

Thus the representation of the number operator in Fourier space is N -- ~--~bp) b p . p

(9.65)

Here the sum runs over all integers p -- 0 . . . . . L - 1 in the sector with an odd number of particles and over the half odd integers p = 1/2, 3 / 2 . . . L - 1/2 in the even sector. The empty lattice 10) is the vacuum state annihilated by all operators b p, i.e., bpl O) - 0 Vp. The operators btp create excitations of momentum p. The , states I Pl . . . . . PN ) -- btp~ . . . bpNlO) form a complete set of basis vectors for the process. As discussed in the previous section the spectrum of H ff, t t does not depend o n / z l . Since at this stage we are only interested in the spectrum, not in the eigenvectors, one may set tZl -- 0. With this choice H f L I t is diagonal in momentum space since inserting (9.63), (9.64) yields H f f ' t ! = 2 h L - Y~p ~:pbtpbp with the single-particle 'energies' r

= ( D i e ip 4- D2e - i p + 2h).

(9.66)


193

Here p is used as a shorthand for 27rp/L. The N-particle states I Pl . . . . . PN ) have eigenvalue 2h L - Y~'~piEpi. In the sector with an even number of particles the ground state with the lowest eigenvalue E0 = 0 is the completely filled state

ILl A single vacancy excitation of momentum p has eigenvalue ~p with real part 2D cos p + 2h. Since in a stochastic matrix no eigenvalue must have negative real part one must have h > 0 and 0 < D 2 < h 2. For D < 0 the lowest lying excitation is the state b pl L ) with a vacancy excitation of momentum p = -t-7r/L. Such a system has an energy gap Emin = 2h - 21DI cos zr/L which in the infinite volume limit becomes Emin = h - IDI for h - IDI g= 0. If h - IDI = 0 the real part of the energy gap vanishes for L --+ c~ resulting in a divergent relaxation time "t'max ~ 4L2/(lDIrr2). For D >_ 0 one obtains the same energy gap which in this case results from the modes with momentum p = Jr + n'/L. In the even sector the excitation with the lowest real part of the energy is the state bpb-pl L ) with total momentum zero. The energy gap is twice the energy gap of the odd sector. The main conclusion is that the spectrum becomes gapless in the infinite volume limit for D 2 = h 2, but has a finite energy gap h-IDI for D 2 < h 2. We cannot predict from this result the relaxation times without further specifying the system and the initial state. However, for the free-fermion processes as classified above this result implies that of the type I processes only the mixed annihilation-fusion process has a gapless spectrum. The only processes of type II with vanishing energy gap are zero-temperature Glauber dynamics with either ferromagnetic or antiferromagnetic coupling. Only in these models can we expect algebraic decay of correlations and dynamical scaling.

9.5

Diffusion-limited pair annihilation

A more intricate question about the dynamical properties of a system is the quantitave behaviour of expectation values. For most interacting particle systems the calculation of such quantities is a very hard task. Free-fermion models are an exception since here an explicit solution of the equations of motion for the correlation functions is possible. Nevertheless the physical contents of these models are far from trivial. The most interesting models are those with critical dynamics where the energy gap vanishes in the continuum limit. A representative of these models is diffusion-limited pair annihilation. But to put free-fermion systems into a broader context and to understand the physical meaning of the free-fermion condition we first discuss a more general pair annihilation model.

G. M. SchQtz

194

9.5.1

Definition and general properties

Consider a model defined on a ring of L sites with periodic boundary conditions where each lattice site may be occupied by at most one particle. The interaction has two components: site exclusion and pair annihilation. In addition to that we consider an interaction with an external field driving the particles in one preferred direction. These particles (denoted A) then hop with rates DR,t. to the fight or left nearest neighbouring site respectively if this site is vacant (denoted 13) and annihilate with rate ~. if it is occupied: Process

Rate

AI3 ~ ~A

DR

t3A ~

D/..

AI3

AA ~ 00

~.

Following the rules of Section 2 this process gives rise to the Hamiltonian L

H

-- y~[DR(s'~s-k+, -- nk(1 -- nk+,)) + D L ( S ; S ~ I --(1 -- nk)nk+,) k=l +

+

+~.(S k Sk+ 1 -- n k n k + l ) ] .

(9.67)

which is of free-fermion form if ~. = DR + Dr,.

(9.68)

For this choice of parameters the quartic ferrnion-interaction terms nknk+l cancel. Physically the origin of the interactions between the particles has two different interpretations" the obvious (and standard) interpretation is that one considers the exclusion principle which forbids double occupancy as a 'physical' hardcore on-site repulsion. In addition to that there is a short-range, i.e. nearestneighbour 'chemical' interaction which leads to the annihilation reaction. In this interpretation, the three rates DR,L and ~. are independent parameters. Alternatively, one may think of the particles as having no exclusion, but a short-range on-site 'physical' interaction which may be attractive or repulsive. As a result of this interaction the diffusive hopping rates DR,L depend in some nontrivial way on the particle occupation numbers. On top of that there is the 'chemical' annihilation reaction with rate ~.. The exclusion model discussed here is obtained from the interacting system without exclusion in the limit ~. --~ oo. In this limit, any multiple occupancy is reduced to occupation by at most one particle (if the site was occupied by an odd number of particles) and thus the model reduces to a two-state system. Taking this limit results in an effective


195

nearest-neighbour annihilation rate ~. which is not a free parameter. It depends like the resulting hopping rates DR,L on the interaction and the hopping rates of the original model. The three rates are related through ~. = x - l (DR + Dr.)

(9.69)

where x > 0 characterizes the interaction, x > 1 corresponds to repulsive interaction and x < 1 arises from attractive interaction. This is intuitively clear, but can be derived rigorously with the infinite-rate formalism. In the absence of interaction one has x = 1 which corresponds to the free-fermion condition (9.68). Thus the free-fermion system may be regarded as a system of physically noninteracting particles, but with an instantaneous on-site pair annihilation reaction. This interpretation helps understanding some of physical properties of diffusionlimited annihilation, particularly in the presence of drift.

9.5.2

Mean-field analysis and Smoluchowski theory

It is interesting to study first the differential equation satisfied by the local density (nk(t)). Differentiating with respect to time one finds d __dt(nk(t))

=

D

((nk+l(t)) + ( n k - l ( t ) ) -- 2(nk(/)))

--7/((nk+~ (t)) -- (nk-1 (t))) --(Jk + rl)(nk-I (t)nk(t)) -- ( ~ -

rl)(nk(t)nk+l(t)) (9.70)

with D = (DR + D L ) / 2 and 17 = (DR -- D L ) / 2 . In the linear terms one recognizes a lattice Laplacian and lattice derivative respectively. Any ~. > 0 will result in a strong dampening of the amplitude and the question arises to which extent the nonlinear effects associated with the driving continue to play a role. An intimately related question is the role of the correlations built by the pair annihilation. These correlations are of importance not only for the evolution of shocks, b u t - as already seen in the introduction - also for understanding the temporal behaviour of the density decay. In the standard rate equation approach the decay of the particle density is proportional to its square because two particles are necessary for an annihilation event. The resulting differential equation b(t) = -2~.p2(t)

(9.71)

can be derived from (9.70) in mean-field approximation by considering the average density p ( t ) = l / L Y~k (nk(t) ) for which one obtains the (exact) differential equation tS(t) = - ~ . / L Y~k ( n k ( t ) n k + ] ( t ) ) . In the mean-field approximation one

196

G.M. SchQtz

completely neglects the effect of correlations, i.e., one assumes particles to be distributed randomly and independently at all times. This yields (nk(t)nk+l (t)) = (nk(t))(nk,+l(t) ) and hence (9.71) with the solution p(t) --

P0 1 + 2~pot

(9.72)

for initial density P0. The algebraic late-time decay p(t) cx 1/t gives indeed a correct description of the process in the fast-diffusion limit DR,L --~ oo (Privman and Grynberg, 1992) where all correlations built up in the annihilation process are immediately washed out. Interestingly, this result is also confirmed by experiments on threedimensional processes, but is at variance with the measurements of the exciton luminosity (and hence density) in the effectively one-dimensional TMMC experiment described in the introduction and again in Section 10. In this and other one-dimensional systems, experiments consistently give a power-law decay with an exponent of approximately 1/2 (Privman, 1997), in agreement with renormalization group predictions (Lee, 1994). This process provides an example where slow diffusive mixing in low dimensions gives rise to anomalous relaxation, not captured by the simple mean-field analysis of the rate equation approach. Adapting Smoluchowski's improved line of reasoning to the present problem starts with the insight that the annihilation leads to large depleted areas bounded by single diffusing particles. The size of these areas increases in time and particle collisions which lead to annihilation become increasingly less likely not only because the number of particles decreases, but also because the time they need to meet increases. The decisive step is then to modify the rate equation (9.71) by an effective, time-dependent reaction 'constant' Jk(t). This effective reaction rate is, according to Smoluchowski, determined by the current flowing from a background of density ,o to a single particle which acts as perfect sink for particles. Such a current j decreases in one dimension in time according to a power law, j cx p/v/[. This can be calculated in a straightforward manner for noninteracting particles or with the tools developed in Section 6 for the SSEP. Since the probability of finding a particle which acts as a sink is also p, one obtains the effective rate equation P(t) cx t - l / 2 p 2 ( t )

(9.73)

which gives the experimentally correct asymptotic behaviour p cx t -1/2. The effective reaction rate decreases cx t-1/2 since the time it takes for particles to meet is determined by diffusion and hence increases cx t 1/2. Implicit in this argument is the presence of particle anticorrelations which extend over a range of the diffusive length scale ~D ~- x//: because of the annihilation process it is much less likely to find two particles within a distance ~D than at larger


197

distances. At larger distances these two particles had essentially no chance to meet and annihilate within the time interval t. This reasoning also suggests that at late times the bias in the hopping rate will have no significant effect on the dynamics. Shocks can occur only in the presence of repulsive interaction if many particles come close to each other and the leading particles block the motion of the incoming particles. But the particle anticorrelations imply that this is not likely to happen. After some finite crossover time (which we do not discuss here), the short-range repulsive interaction becomes irrelevant even in the presence of a bias (Privman, 1994). Verification of these arguments and more detailed results on the spatiotemporal structure of the particle distribution require an exact treatment which can be obtained for diffusion-limited annihilation in the free-fermion case (9.68). Exact results for this system have been obtained by a variety of equivalent means. For the calculation of the density from random initial states the empty-interval approach is the most straightforward procedure. However, for more complicated initial conditions or more complicated expectation values one needs more powerful tools. The basic assumptions and implications of Smoluchowski's theory turn out to be correct, hence we restrict ourselves from the outset to the free-fermion case. Since it involves no additional technical complication we include the possibility of a bias.

9.5.3

Free-fermion solution (1): operators

In what follows, it is convenient to work with the single-particle diffusion constant D = (DR + DL)/2 and hopping asymmetry 0 = (DR -- DL)/2. Since by the action of H the particle number changes only in units of two, Q (9.3) commutes with H and splits it into a sector with an even number of particles (Q = +1) and into a sector with an odd number of particles (Q = - 1 ) . The Hamiltonian may be written H = D Hs + 17Hd where the driving part lid is given by Hd = Y~=l (s~-s++l -- s~-sk+ l) and Hs is the Hamiltonian for the system without driving. In terms of the Fourier components (9.61), (9.62) of the fermion operators one has

(,

p Hd

=

--2i~-~sin

- - - ~ ) ) b t p b p + s i n ( ----~-)b_pbp} 2rrp --~

bp

(9.74)

.

p

Hd commutes with Hs which will become important below. There are now two different strategies for the calculation of correlation functions. One may either diagonalize / / and expand expectation values in a basis

G.M. Sch0tz

198

of eigenstates (4.23) or one determines time-dependent operators F(t) in the Heisenberg picture (2.15) directly from H. Since these are the quantities ultimately necessary for the calculation of correlation functions we choose the second approach advocated in Schlitz (1995b). For definiteness we discuss only the sector with an even number of particles. The calculation in the odd sector is completely analogous. The equations of motion d - - F = [H, F] dt

(9.76)

for the time-dependent operator (2.15) lead to a set of two coupled ordinary differential equations' 2rrp)

d btp(t)

Epbtp(t) + 2sin - - ~ b_p(t)

(9.77)

d dtbp(t)

--Epbp(t)

(9.78)

solved by

btp(t) = e~pt(btp+COt(~-~)(l-e-(~p+~-p)t)b_p)

(9.79)

bp(t)

(9.80)

-~

with btp(0) = btp, bp(O)

e -~pt b p

= bp and

Ep = 2 D [ I - c o s ( From nk

=

-2rrp ---~--)]-2i0sin(-~-)

(9.81)

ctkck and (9.63), (9.64) one obtains nk (t) ---- -~1 Z e2Jrix(p- p,)/Lbtp(t)bp,(t).

(9.82)

p,p'

This together with (9.79) and (9.80) solves the initial value problem for the local density and indeed for all density correlation functions. From the dispersion relation (9.81) one anticipates an algebraic decay of the density and density correlations in the thermodynamic limit. tBecause of the boundary conditions these equations describe the time evolution of the creation and annihilation operators only when applied to products with an even number of operators. This is not a restriction as all expectation values (n kl "'" nkN ) are of this form.

1 Exactly solvable models for many,body systems

199

We note that (9.79)-(9.81) demonstrate also the impact of the driving on the system in the scaling regime t ~ L 2. For large L, one may approximate ~p by

Ep

~

2rri 27r2 p2 -20---E- p + 2D--~-- .

(9.83)

Therefore, the effect of the driving may be absorbed in a Galilei transformation ri ~ ri + Ot where ri -- x i / L are the scaled space coordinates appearing in the correlation function. Thus for arbitrary translationally invariant initial conditions the 0-dependence of all correlation functions vanishes completely in the scaling limit. The only nonlinear effects which are associated with the exact form of the dispersion relation (9.81) are pure lattice effects. This observation would be somewhat puzzling if indeed one considered the particles as hard-core objects with nearest neighbour interaction. In this interpretation the free-fermion condition on the annihilation rate has no particular physical significance and one would expect some crossover behaviour from an early-time, nonlinear 'shock' regime to the Galilei-invariant regime. The absence of such a crossover has a natural interpretation in terms of particles with chemical annihilation reaction with infinite rate, but no physical repulsive interaction. Hence a crossover time after which nonlinear effects caused by shocks disappear is to be expected only fork < 2 D . For k > 2D, i.e. attractive interaction, all results should qualitatively remain the same as in the free-fermion case: In the extreme limit k --+ c~ one would simply have a system where particles annihilate with rate 2D if they are two lattice units apart rather than at a distance of only one lattice unit as for k - 2D. This corresponds to particles coveting two lattice sites which annihilate when they meet. It is intuitively clear that such a modified system cannot show qualitatively different behaviour. This conjecture may be substantiated by observing that the results for expectation values for k = 2D yield strict upper bounds for expectations of the process with k > 2D. Also the pure exclusion process with particles of size two (Sasamoto and Wadati, 1998b) or an arbitrary mixture of sizes (Alcaraz and Bariev, 1999) is in the same universality class as the usual exclusion process. At this point we can tackle the question of the appearance of free-fermions in this problem of stochastic dynamics of classical interacting particles. Consider just two particles located on sites k, l of an infinite lattice. The origin of the free-fermion character of the process becomes transparent in the calculation of the two-particle transition probability P ( m , n; tlk, l; O) = ( S [CmCne-Ht CtkC]l O ). This calculation can be done by either using the free-fermion description, or, in a less technical way, by reminding oneself of the meaning of an annihilating random walk and the description of random walks in terms of a sum over the histories of the stochastic time evolution. In discrete space and time the transition

G.M. Sch(itz

200

probability (or conditional probability) for a single particle P(m; t lk, 0) is the sum over all paths leading from k to m, each weighted with its proper statistical weight given by the hopping rates and the particular form of the trajectory. If two noninteracting particles, one starting at site k and the other at site l, move, then the transition probability that the particle which started at site k < l reaches site m < n and the particles which started at site l reaches site n at time t is still the sum over all possible trajectories which connect k with m and l with n, where each single trajectory has the same weight as in the single particle case. Hence, for noninteracting particles, P ( m , n; tlk, l; O) = P(m; tlk; 0)P(n; tll; 0). This sum includes the contribution of paths which cross each other. In an annihilating random walk of otherwise noninteracting particles the contribution of all crossing paths have to be subtracted. Since we are on an infinite, one-dimensional lattice and both particles are identical this contribution is just the one given by all paths which start at site k and end at site n (instead of m) and which start at site l and end at site m (instead of n). Therefore

P ( m , n ; tlk, l; O) = P(m; tlk; O)P(n; tll; O) - P(n; tlk; O)P(m; tll; O) (9.84) which is indeed what one obtains using the anticommutation relations in the free-fermion approach. The same subtraction scheme generalizes to higher-order conditional probabilities and is again conveniently captured in the free fermion anticommutation relations. This point of view makes also clear why free fermions correspond to the infinite reaction limit. If the reaction rate is finite, paths which cross each other get a nonzero weight. Hence the subtraction scheme would yield a wrong result. By considering the topology of paths in higher dimensions one realizes also that a free-fermion description can hold only in one dimension. The expression (9.84) and its generalization to higher-order conditional probabilities is the lattice analogue of the expression found by Torney and McConnell (1983) in a continuum description of the process.

9.5.4

Free-fermion solution (2): states

For more specific results, one needs either to reconvert the fermion operators in (9.82) into Pauli matrices or represent the initial state in terms of fermionic creation operators. Which approach is more appropriate, depends on the nature of the initial state or initial distribution. For the second strategy which we consider now one also needs a representation of (sl in terms of fermionic operators. The vector 10) is the vacuum state with respect to the annihilation operators, ckl0) -- 0. In spin language this is the ferromagnetic state with all spins up corresponding to the completely empty lattice. Acting with fermionic creation operators yields 9

"I"

ckt~ "'Cksl0) = Ikl . . . . . kN)

(kl < k2 < . . . < k s )

(9.85)


201

which are the states with particles placed on sites k l . . . . . k N. A general transla* 9. .btpN tionally invariant N-particle state is obtained by acting with products bp~ o n 10) w h e r e Y~i pi --O. Following (SchLitz, 1995b) we introduce the bilinear expressions Bpt = b t_ p b pt .

(9.86)

Bp = b p b _ p

where p = 1/2, 3/2 . . . . . (L - 1)/2. This operator creates pair excitations with vanishing total momentum. A special class of initial states are those built by polynomials in Bp*. Among these particular translationally invariant states are uncorrelated random initial conditions with an even number of particles (9.60). In order to derive a representation of these states in terms of fermionic operators we study first the representation of {sl. From the representation (9.74), (9.75) of H and using (Bp) 2 -- (Be) 2 = 0 one finds for the left zero energy eigenvector (sl of H <sl

-(Oll--I(l+cot(~--~-~)bpb_p)+(OlbO~p,p _

1. However,the figures provided by MacDonald and Gibbs (1969) suggest that the description remains qualitatively correct if one averages over this sublattice structure.

222

G . M . SchQtz

sets, just as the curve derived from the step function profile. Thus the exact solution not only reconfirms the neat physical understanding of the kinetics of biopolymerization in terms of domain walls, but also gives insight into fluctuation effects that are lost in the mean-field approach.

10.3 Exciton dynamics on polymer chains We describe the excitons discussed in the introduction as particles hopping on a lattice which for the time being is arbitrary, i.e. it may be multidimensional and possibly also disordered. We consider a somewhat more general process where particles hop with rate D and various annihilation processes take place: pair annihilation with rate Z, fusion with rate 1 - a and death with rate 1 + ct. This describes a system of particles where free particles (with no nearest neighbours) decay with rate 4 (in appropriately chosen units of time), but which form more stable composite states if they are nearest neighbours. In this case the spontaneous decay rate is reduced by an amount ct (per neighbouring particle). However, if particles come too close, they annihilate spontaneously in pairs, which is described by the effective pair-annihilation rate ~.. We set ~. = 2a, the general case has been investigated in Henkel et al. (1995). The quantum Hamiltonian H of the process is related by a similarity transformation H = / 3 - 1 H x xz 13 with 13 = e s§ to the (nonstochastic, but Hermitian) anisotropic Heisenberg Hamiltonian (3.6) in a magnetic field

HXXZ =

1 2 Z [O(a[cr; + a~a?) + (O + ot)a[a[ - (2 + O + a ) ] - Z a[. k

(10.5) This implies decoupling of the correlators (Section 8.2). For the equal-time m-point correlation function (nk~ (t)...nkm (t)) one obtains an interesting result using the quantum spin chain representation. Going through essentially the same steps as in the derivation of the relations (6.4) one finds

(nkl(t)'"nkm(t) ) = Z (nkl(O)"'nk,,(O) )(kl lk6S

. . . . .

km le-ntlll

. . . . .

Im ).

(10.6) Independent of the lattice this relation asserts that the calculation of the equaltime m-point correlator of the stochastic process is equivalent to the solution of the m-particle problem in the Heisenberg Hamiltonian, corresponding to the sector with m down-spins. This problem can be tackled using spin-wave theory (Mattis, 1965). We also note that even on quite large and complicated lattices the decoupling allows for a highly accurate numerical calculation of correlation

1 Exactlysolvable models for many-body systems

223

functions. In one dimension the problem can be solved exactly with the Bethe ansatz. For long-lived excitations with a strong separation of time scales one is left with pure pair annihilation (or fusion) process. These can be observed in laserinduced exciton annihilation on TMMC chains. A single exciton has a decay time of about 0.7 ms. The on-chain hopping rate is 1011-1012 s -1. If two excitons arrive on the same Mn 2+ ion, they undergo a fusion reaction A + A ~ A with a reaction time ~ 100 fs (Kroon et al., 1993), i.e. the experimental data suggest that the fusion process is approximately instantaneous. D sets the time scale for the diffusion. The finite lifetime r of the excitons is much larger than D - l , thus a decay term r -l y~ (s~- - n i ) may be neglected. This allows for an approximation of the process by the free-fermion condition and as shown in Section 9 one expects the average density of excitons to decay algebraically in time with an exponent x = 1/2. This is in good agreement with the experimental result x = 0.48(2). Similar exponents were found in other pair decay processes (Kopelman and Lin, 1997; Kroon and Sprik, 1997), thus experimentally confirming universality, a n d implicitly-the validity of the underlying picture of the role of fluctuations in low-dimensional systems.

Acknowledgements

This work has its roots in two rather different fields. First and foremost I would like to thank E. Domany for getting me started in interacting particle systems in general, and in the asymmetric exclusion process in particular. Some of the understanding of boundary-induced phase transitions presented here has its origin in collaboration and in most enjoyable discussions with him. Secondly, I would like to thank V. Rittenberg for an excellent introduction to quantum spin systems earlier during my Diploma and Ph.D. studies. I would like to thank also all the other collaborators whose insights have contributed to forming my views on the subject: T. Antal, D. B. Abraham, V. Belitsky, M. Henkel, A. B. Kolomeisky, E. B. Kolomeisky, T. J. Newman, V. Popkov, S. Sandow, J. E. Santos, J. P. Straley and particularly G. T. Barkema for his contribution to polymer reptation and R. B. Stinchcombe for his idea of investigating dynamical matrix product states. I am very much indebted to B. Derrida, M. Evans, C. Godr~che, J. Krug, J. Lebowitz, D. Mukamel, I. Peschel, Z. R~icz and H. Spohn with whom I had numerous helpful and inspiring discussions. I also benefited from many hints on interesting literature. Thanks are also due to U. Tfiuber for comments on a draft version of the first part of this work and for his readiness to explain renormalization group results, to K. Krebs for explaining some unpublished exact

224

G.M. SchQtz

results on the motion of shocks and particularly to N. Rajewsky for many valuable comments and suggestions for changes in the original draft. Last, but not least, I wish to express my gratefulness to H. Bethe for sharing some of his personal recollections of a time which belongs to the most exciting in the history of physics.

Exactly solvable models for many-body systems

225

Appendix A: The two-dimensional vertex model Following the idea of Kandel et al. (1990) we show how the discrete-time exclusion process defined in Section 7.3 is related to a two-dimensional vertex model (Baxter, 1982). Consider a four-vertex model on a diagonal square lattice defined as follows: place an up- or down-pointing arrow on each link of the lattice and assign a nonzero Boltzmann weight to each of the vertices shown in Fig. 34. (All other configurations of arrows around an intersection of two lines, i.e., all other vertices, are forbidden in the bulk.) The partition function is the sum of the products of Boltzmann weights of a lattice configuration taken over all allowed configurations.

al

a2

b2

c2

Fig. 34 Allowed bulk vertex configurations in the four-vertex model. Up-pointing arrows correspond to particles, down-pointing arrows represent vacant sites. In the dynamical interpretation of the model the Boltzmann weights give the transition probability of the state represented by the pair of arrows below the vertex to that above the vertex. (From Schiitz, 1993b.) In the transfer matrix formalism up- and down-pointing arrows in each row of a diagonal square lattice built by M of these vertices represent the state of the system at some given time t. Corresponding to the M vertices there are L -- 2M sites in each row represented by the links of the diagonal lattice. The configuration of arrows in the next row above (represented by the upper arrows of the same vertices) then corresponds to the state of the system at an intermediate time t' = t 4- 1/2, and the configuration after a full time step t" -- t 4- 1 corresponds to the arrangement of arrows two rows above. Therefore each vertex represents a local transition from the state given by the lower two arrows of a vertex representing the configuration on sites j and j 4- 1 at time t to the state defined by the upper two arrows representing the configuration at sites j and j 4-1 at time t 4- 1/2. The correspondence of the vertex language to the particle picture used in the introduction can be understood by considering up-pointing arrows as particles occupying the respective sites of the chain while down-pointing arrows represent vacant sites, i.e., holes. The diagonal-to-diagonal transfer matrix T acting on a chain of L sites (L

226

G . M . SchOtz

even) of the vertex model is then defined by Destri and de Vega (1987) L/2 L/2 T = H T2j-I" l-I T2j = T~ j=l

(A.7)

even.

j=l

The matrices 7) act nontrivially on sites j and j + 1 in the chain, on all other sites they act as unit operator. All matrices Tj and Tj, with Ij - j'l -~ 1 commute. (The difference j - j ' is understood to be mod L.) The bulk dynamics of our model is encoded in the transfer matrix by choosing the vertex weights as al = a 2 - - b 2

=Cl

=

(A.8)

1.

In the bulk this leads to 1 0 0 _

0

1

1

Tj -- 1 + s f sj+ l - njvj+l =

0

0

0

000

0 0 0 1

(A.9)

j,j+l

In the particle language the matrices 7) describe the local transition probabilities of particles moving from site j to site j + 1 represented by the corresponding vertices. If sites j and j + 1 are both empty or occupied, they remain as they are under the action of Tj. The same holds for a hole on site j and a particle on site j + 1, corresponding to the diagonal elements of Tj, representing vertices al, a2 and Cl. If there is a particle on site j and a hole on site j + 1, the particle will move with probability one to site j + 1. This accounts for vertex b2. Open boundary conditions with injection of particles on site 1 and absorption of particles on site L correspond to the additional vertices shown in Fig. 35 together with vertex weights corresponding to the respective probabilities of creating and annihilating particles. In a two-dimensional lattice (Fig. 36) we consider the half-vertices at the left boundary as the fight arms of the vertices shown in (Fig. 35) and the half-vertices at the fight boundary as their left arms. Thus the left arrows define the particle configuration on site L and the fight arrows are considered as site 1. Vertices a l, a2 and b2 have a different weight at the boundary: a l' = 1 - /~, a 2' -- 1 - a, b 2' - etl~. Note that vertex b2 at the boundary describes simultaneous absorption of a particle at site L and creation of a particle at site 1. With this convention TL (a, fl) acting on sites L and 1 corresponding to the

Exactly solvable models for many-body systems

/5

u

/5(1 -- u)

a(l -/5)

227

(1 - a ) ( l - / 5 )

Fig. 35 Additional vertex configurations allowed at the boundary and their Boltzmann weights. The left arrows of these vertices describe the particle configuration at the boundary site L of the system while the right arrows define the particle configurations at the origin (site 1). (From Schiitz, 1993b.)

vertex weights shown in Fig. 35 is given by

TL(oe,~)

=

-

l + o e ( S l -- Vl) + ~(s+ - - n L ) +Oe~(s+ - - n L ) ( S l l--c~ 0 /5(1 -- a ) 0 o

o

0

0

(1 - a ) ( l

oe(l --r

-13)

o

1 --~3

-- Vi)

9

L,I

(A.10) The transfer matrix T = T(c~,/5) acts parallel first on all even-odd pairs of sites (2j, 2 j + l) including the boundary pair (L, 1), then on all odd-even pairs. Thus in the first half time step T even shifts particles from the even sublattice to the odd sublattice (so far it was not occupied) and then, in the second half step, T ~ moves particles from the odd sublattice to the even sublattice again. As a result, we expect an asymmetry in the average occupation of the even and odd sublattice which is related to the particle current. In a model with transfer matrix 7" = T~ even the asymmetry will be reversed, but there will be no essential difference in the physical properties of these two systems. A possible configuration of particles in a 12 x 12 lattice is shown in Fig. 36. Note that the presence of particles at site x -- 11 and times t = 2, 3 imply the existence of particles on the left edge of their 'light cones' as long as they move in a region where the even sublattice is fully occupied, i.e. they move ballistically two lattice units per time step to the left. A particle on an even lattice site at some (integer) time t always implies the existence of a particle on the fight edge of its 'light cone' up to the boundary. The model has a particle-hole symmetry. We denote by Ix l, x2 . . . . . X L) -Sxl Sx2 . . . Sxt I O ) the N-particle state with particles on sites Xl . . . . . XL (10) is the state with all spins up corresponding to no particle). The parity operator P reflects particles with respect to the centre of the chain located between sites x = L / 2 and x = L / 2 + 1 and the charge conjugation operator C = I-IjL__l vjx interchanges

228

G . M . Schz3tz

5 4 3 2

1 0

1 2

3 4

5

6

7

8 9

10 11 12

Fig. 36 Configuration of particles (up-pointing arrows) on a lattice of length L = 12 in space (horizontal) direction M = 2t = 12 between times t = 0 and t = 5 + 1/2 (vertical direction). Down-pointing arrows denoting vacant sites have been omitted from the drawing. At time t = 0 the even sublattice is filled and the odd sublattice empty. Particles are injected at site 1 after times t = 0 and t = 4. At the boundary (site 12) panicles get stuck at times t = 1 and t = 2 and are absorbed at times t = 0, 3, 4, 5. (From Sch/itz, 1993b.)

particles and holes and therefore turns a N-particle state into a state with L - N particles. One finds ( C P ) T ( a , [3)(CP) = T(/~, ct). (A.11) In the bulk the particle current is c o n s e r v e d and can be obtained from the c o m m u t a t o r s of n2x and n2x-l with T. T h e s e relations play a crucial role in the construction of the stationary state and the c o m p u t a t i o n of the t i m e - d e p e n d e n t correlation function. Defining the current operators J2x:eVenand J2x-l'odd by j even

2x

=

n2xV2x+l

jodd 2x-I

=

(1-

V Z x - 2 V Z x - 1 ) ( l - nzxn2x+l)

(1 < x < L / 2 -

1)

(2 N 2 one finds an exponential approach to the divergent stationary width w 2 ~ N(I - c~2)/4. The second limit shows a power-law divergence of the width w 2 "~ ~ at intermediate times 1 0 and u 4- Do > 0. The symmetry is spontaneously broken, and the order parameter toe has a nonzero expectation value, of the form toe = ~"e~, where ~ is a set of orthonormal base vectors. At zero temperature, the membrane

K. J. Wiese

266

/

i '..

/ :'

"L

RG

:.

""--,,.,., '-.,

. .--""

'"'.,,,., :.

,.~ L vno,-y

""",.. ....... .,,"f/

RG

Fig. 6 Free energy for t < 0 (left) and t > 0 (fight) in the limit of large membranes. is in a flat (ordered) phase, with 1/ ~'--2

Itl u+Du"

(2.11)

This resembles the XY-model in two dimensions. There, long-range order is destroyed by spin waves. We shall see in the next section, that fluctuations renorrealize the rigidity of the membrane and render it stiffer. This renormalization is sufficient to make the membrane flat. For further discussion of the thermodynamic behaviour see Guitter et al. (1989). To incorporate self-avoidance, let us use the Flory approximation. This consists in replacing ?(x) by the radius of gyration Ro and derivatives with respect to x by I/L, as well as the integration over x by LD, where L is the size of the flat membrane. This leads (up to numerical factors) to (Fig. 6) "]-[, ,~ K L D - 4 R 2

-4r-t t O - 2

g 2 -'t- (It d- D o ) L D - 4 R ~ + DL2DRG d.

(2.12)

First of all, the bending rigidity tc can always be neglected with respect to t and It.

For t < 0 and in the physical region (D < d), the terms proportional to t and u + Do dominate and minimizing the free energy leads to

RG "~ L.

(2.13)

Self-avoidance can be neglected at large scale. For t > 0, self-avoidance prevents the membrane from collapsing, and balancing the terms of order t and b gives

RG "~ L V~lo~y

(2.14)

2+D VFlory = ~2 +. d

(2.15)

with the Flory exponent

267

Polymerized membranes, a review

2

We will show in Section 7.5 that (2.15) is a reasonable approximation in the crumpled phase. In general we will find R6 ~ L v*

(2.16)

with some nontrivial exponent v*. Let us still mention the results for v* in the crumpled phase, obtained by a Gaussian variational approximation. We shall show in Section 7.4 that this approximation becomes exact in the limit of d ~ c~ with probably exponentially small corrections. The work by Goulian (1991); Le Doussal (1992) and Guitter and Palmeri (1992) predicts: 2D Vvar = 9 (2.17) d For two-dimensional membranes (D = 2), this differs from the Flory approximation by terms of order 1/d 2.

2.4 Stability of the flat phase In the last section, we saw that a simple scaling analysis suggests the existence of a flat phase. This phase could of course be destroyed by fluctuations. We shall show here that this is indeed the case for fluid membranes, but that a nonzero shear modulus, i.e. a fixed connectivity, stabilizes the membrane in the flat phase (Nelson and Peliti, 1987). Our presentation is largely inspired by the lecture of Nelson (1989), but we will use an e-expansion here instead of a self-consistent approximation. To describe fluctuations of a membrane with inner coordinates x -- (Xl, x2) around a flat configuration, it is advantageous to use the representation

~(Xl, X2) -- (

Xl ~-ttI(Xl,X2) ) X2 ~- U2(Xl, X2) 9

(2.18)

h(xl,x2) The line element d; is

d~ - (

+

OlU2dxl + (1 + 02U2) dx2

)9

(2.19)

Olh dxl -k- 02h dx2 The deformation of this line element is described by the deformation matrix u,~t~ (Landau and Lifshitz, 1983) d r 2 = ~.2 (d2x + 2 u ~ dx~dxt~) "

(2.20)

268

K.J. Wiese

With the help of (2.19) we find: 1

1

u~# -- -~1(ig~u~ + O~uo~)+ -~(O~h)(O#h) + -~(O~u•215

(2.21)

The last term is of higher order in u and can be neglected in the following. (It has to be included at order ~2.) We shall thus use u~r ~ ~1 (iO~u/~ +

1

O~u~) + -~(Ooth)(O~h).

(2.22)

The energy of a nearly flat membrane is the sum of bending rigidity and deformation energy 2 2] . [ u , h ] - f d 2 x ~'~( A h ) 2 + ~ 1 [ 2/2u,rt~+~.u•215

(2.23)

/2 and Jk are the Lam6 coefficients (Landau and Lifshitz, 1983). (We use /2 instead of the usual notation of # (Landau and Lifshitz, 1983) to reserve # for the renormalization scale.) ~, t2 and ~. are related to x, u and u by ~ = x( 2, /2 = 4tt( 4 and ~. = 80( 4. In this expression, the displacement vector u,~ appears only quadratic and can thus be eliminated by calculating its path-integral

7"~eff[h]--kaTln[f We separate in decomposition

D[u]e-7-tlu'hl/kar].

(2.24)

u,~(x) the (q -- 0)-mode and use for the other modes the Fourier

u~(x) -- uOt~+ A~ + Z

(i

-2 [q~ut~(q) + q/~t~c~(q)] + ,4~r

)

e iqx, (2.25)

qr where

riot(q) = J dZx e-iqxua(x) and '4~t~(q) is the Fourier transform of

(2.26)

A~#(x) = 89

1 / d2x e -iqx Ooth(x)O~h(x).

/i~(q) = ~

(2.27)

For q ~ 0 , / i ~ (q) is now decomposed into its longitudinal and transversal parts. (That this is indeed possible is shown in Appendix C.) ,~c~r

i = ~ [q,~t~c~(q) + q~,~(q)] +

PdS(q)~(q),

(2.28)

2

269


where

q~ql3 q2

(2.29)

pdT ( q ) ~ , ~ ( q ) .

(2.30)

PdT~ ( q ) = 8 ~ t3

is the transversal projector and ~p(q)_

We can now absorb the longitudinal part q3t3(q) of ,~t3 (q) by shifting the variable t~c~(q): t~(q) > floe(q) - ~c~(q). (2.31) It remains to integrate over fi,~(q). To this aim expand 7" := 2/~ fi ,#~ ( q ) fi o,~ ( - q ) + ~. fi ~ o, ( q ) fi ~ ~ ( - q )

(2.32)

in the basis of rotational invariants q2, iqfi(q)l 2 and tT(q)fi(-q): 7" -- f_zq2lfi(q)2l+(fx+X)lqfi(q)12+(2fx+~.)l~p(q)12+x

(iqfi(q)dp(-q)

+ c.c.).

(2.33) By a second variable transformation fi~(q)

_ )~ iq~ ~ ( q ) , u ~ ( q ) + 2fz + )~ ~5

(2.34)

terms proportional to r and ~ are decoupled and we obtain 7" -- 4/2(/~ + ~.)

2g+x

~,(q)+(_q) + quadratic terms in ft.

(2.35)

Up to a constant, the effective Hamiltonian (2.24) thus becomes

"]'~eff[h] : ~

d2x (Ah) 2 -[- "~-

d2x

(eTfl [Ooth(x)Oflh(x)]) 2 "

(2.36)

The 'prime' indicates that the 0-mode is excluded from the integral. The coupling constant K is k -- t2(t2 + ~'). (2.37)

2~+x We see that the shear modulus /2 is responsible for the interaction. For fluid membranes,/2 = 0 and no correction appears, even if ~. r 0. We shall now study (2.36) in perturbation theory, by using an ~ = 4 D expansion. A similar technique was employed by Aronovitz and Lubensky (1988), where they study the RG-fiow for all fields. A self-consistent method was utilized in Nelson and Peliti (1987) and Nelson (1989).

270

K.J. Wiese

To carry out an E-expansion, we rewrite the effective Hamiltonian (2.36) as 7-/elf[h] = ~-

dOx (z~h)2qK -~ZK~ E

f,

dOx

[3ah(x)O#h(x)] )2 , (2.38)

where i has been absorbed into the field normalizations (h --+ h / f f ~ ) and R ZK/z , K0 = ~-~- = K ' ~ -

ho(x) = ~ ' h ( x ) .

(2.39)

The renormalization factors Z and Zk absorb the divergences and are fixed by the minimal subtraction scheme. /z is the renormalization scale, E = 4 - D the dimension of the bare coupling. Bare quantities are indexed as 'o'. The vertex is

Pl "N ~" ql ,,/

K

SD(pl + P2 +- ql q- q2)

2

(2rr)D

--X P2/-'4

A

H

i=1,2

(piqi)2 _ (pi)2(qi)2 (pi + qi)2

~,, q2

(2.40) We shall now calculate perturbative corrections. As the 0-mode is excluded from the integration, the contribution to x coming from the 'tadpole' is 0:

=0.

(2.41)

i i

The second contribution to the renormalization of x is: k

.....

(P -+-k) 2 / k4"

(2.42)

P P A divergence for k ~ cx~ is manifest as a pole in 1/E with positive residue C (which needs not be specified)" k

_(~

= (p2)2C-P-~'E

(2.43)

P P The divergence of this diagram is subtracted at scale # by choosing 2C

Z=I-mK. E

(2.44)

2


271

The sign is such that the interaction reinforces the bending rigidity. To analyse the renormalization of the vertex, we remark that due to the transversal projector, all three possible diagrams are convergent:

~---~---~

.

(2.45)

This is not evident from power counting. Hence at one-loop order ZK = 1,

(2.46)

and renormalization becomes particularly simple. The function/3 (K) and the full scaling dimension ~"(K) of the field h, the roughness exponent, are obtained from (2.39) as /5(K) = #

~'(K) =

1K = -~ K o I+K~KlnZK--2K~K

4-D 2

131 4-D 2tz~-~) lnZ = o 2

(2.47)

lnz , 2

/5(K)

0 lnZ. OK

(2.48)

Since C is positive, the/~-function possesses a positive, IR-stable fixed point at one-loop order, which we denote K*. Then ~'*

=

~'(K*)

=

4-

D 2

E 4- D 2). F O(~? 2) ~ ff O(E 4 4

(2.49)

(This result could have been obtained faster by using the method of exact exponent identities explained in Section 3.9.) In D = 2 1 ~'* -- ~ + O(E2).

(2.50)

This can be interpreted as an effective k-dependent bending rigidity //,

Xeff(k) ~ K '---. k

(2.51)

We can now analyse the stability of the fiat phase. Following De Gennes and Taupin (1982), we estimate the fluctuations of the normal to the surface projected on x3 (the component parallel to h(x)): n3(x) =

. V/I + (Vh(x)) 2

(2.52)

K.J. Wiese

272

The first term of the expansion is the mean of (Vh(x)) 2. Without interaction (K = 0) it is: {(Vh(x)) 2) -- k B T f 0

d 2q q 2 ~ ~ksT I n ( L / a ) , (2yr)2 ~q4 2rr~

(2.53)

where L and a are IR and UV cut-offs. As for many two-dimensional systems, the logarithmic divergence at large distances indicates that order is destroyed by fluctuations. For membranes with nonzero shear modulus, the estimate (2.53) is incorrect. One has to take care of the renormalization of x, hence replace x in (2.53) by Xeff(k), given by (2.51). This yields:

((Vh (x)) 2)with Kerr

=ksTf

d2q q2 (2rr)2 Xeff(q)q4 -- IR-convergent.

(2.54)

The normals keep their preferred direction parallel to X3, even for systems with infinite size. The symmetry is broken and the membrane flat. This seems to be a violation of the Mermin-Wagner theorem: in fact, the fluctuations in the membrane give rise to long-range interactions, for which the Mermin-Wagner theorem is not valid. To conclude: as soon as the membrane is in the phase of high bending rigidity, i.e. the flat phase, the in-membrane fluctuations reinforce the bending rigidity and stabilize the membrane. Stated differently: the fixed point of the flat phase is attractive. Nevertheless, the fluctuations in the height h are large and described by a nontrivial roughness exponent ( h ( x ) - h(y)) 2) ~ Ix - yl 2~ 9

(2.55)

This exponent was estimated above to be 89 It can also be calculated by an expansion in 1/d (David and Guitter, 1988), e = 4 - D (Aronovitz et al., 1989) or within a self-consistent screening approximation (Le Doussal and Radzihovsky, 1992) and can be compared with experiments (Schmidt et al., 1993), and numerics (Abraham and Nelson, 1990a,b; Guitter et al., 1990; Zhang et al., 1993, 1996; Bowick et al., 1997b). This should rule out the value of ~ = 1, proposed in Lipowsky and Giradet (1990, 1991) and Abraham (1991). This is summarized in Fig. 7. We have also mentioned above that the crumpling transition occurs at a critical value of the bending rigidity. This transition point is a different tricritical state, accessible to renormalization-group treatments and numerics. The fractal exponent v* is then 0 in the crumpled phase, 1 in the flat phase, and at the crumpling

273

2 Polymerized membranes, a review

disordere~ d membrane

0.8

9 THEORY ESTIMATES:

s

NP: Nelson-Pehli A t : A,.:.,r,.:,v,U Lut.,~,';,~-v .t L') .-?.,,r.~,,..... n D: 0-,,,o t-! 1 ,~ e4par,~,~.n SCSA Le ['k,u:;J, ~ . J

~,

SCSA

/ k NUMERICAL SIMULATION:

0.7 0.65

meC~anne[D

1

Le,bter Ma~.] ~. .) At,,br,~rr, .:..i 31 3 Gu,n.-).,e, :,i 4 L,130*~kV Ab, ~r,.tm o K.~mu,3 Batjrn~anr'~" ; Gomppe~- Kroll Morse 9 el al 9: i n p;.t.~.:t,+ Gr=..~t 1 i Z r ~ r , g - O a , , : ~'l~..d Zr,2={~g O.~,,5 K,~..d

%

16Z I

0.6

2

EXPERIMENTS: 1 C.Schmidl et al RBC

0.55 0.5

rl

O,,~,O,'l

87

88

89

90

91

,

,

92

93

TIME (years)

,

94 95 96

Fig. 7 Estimates of the roughness-exponent ~" as a function of time. (Courtesy of P. Le Doussal, with kind permission; figure by P. Le Doussal and L. Radzihovsky.) transition is given by the 1/d-estimate (David and Guitter, 1988; Paczuski and Kardar, 1989) 1 * 1 (2.56) Vc -d' which agrees with numerical values in d = 3 (Kantor and Nelson, 1987a,b). See also Nelson et al. (1989); Aronovitz et al. (1989); Guitter et al. ( 1989); Bouchaud and Bouchaud (1988); Harnish and Wheater (1991); Jegerlehner and Petersson ( 1994); Kawanishi et al. ( 1994); Baig et al. (1994). Also see Aronovitz et al. (1991) for a study of the membrane elasticity at low temperatures and Guitter (1990) for a stack of membranes. 2.5

E x p e r i m e n t s on t e t h e r e d m e m b r a n e s

Fig. 8 Image of a red blood cell (left) and the underlying spectrin network (right) (Liu et al., 1987" Falk and Speth, 1999) (from Kleinig and Sitte, 1999).

274

K.J. Wiese

Few experiments have been realized up to now. The most promising are: (i) The spectrin network of red blood cells (Fig. 8) forms a natural membrane, easily accessible experimentally (Elsgaeter et al., 1986; Schmidt et al., 1993). The inconvenience of this system is the large intrinsic bending rigidity which first has to be reduced. No experiment showing a crumpled phase has been done. In the flat phase, one finds an anomalous roughness exponent ( of about (flat ~ 0.6 (Schmidt et al., 1993), as discussed at the end of the preceding subsection. (ii) Two-dimensional networks of polymers (Stupp et al., 1993) seem to be promising. However, experimental measurements are missing. Recently, Rehage and co-workers have succeeded in producing sufficiently highly polymerized membranes (Rehage et al., 1997) and experiments to find the fractal phase are planned (Rehage, personal communication). (iii) Molybdene disulphide (MoS2) can be produced in extremely pure form. The experiments which we know of (Chianelli et al., 1979) find it in a strongly folded phase. (iv) Graphite oxide (Fig. 9): for this material, experiments have been realized. Graphite is a layered material, and only very weak (van der Waals) forces exist between different layers. One therefore may cut out a piece of such a layer. By an exothermic reaction of graphite with some oxidant (the principle of black powder), one obtains a sample which consists of pieces of a single layer of graphite, decorated with oxygen atoms at its border. One expects that these membranes have a very small bending rigidity. The first experiments undertaken by Hwa et al. (1991) have shown such a crumpled phase with a fractal dimension near to the Flory results (df = 2.5) besides a collapsed and a flat phase. This was achieved by varying the concentration of H + of the dispersion. In later experiments by Spector et al. (1994) this intermediate phase was no longer observed. The interpretation of these experiments is, however, not unambiguous. Extrapolating the lightscattering data of Hwa et al. ( 1991) reproduced in Fig. 10 predicts a fractal dimension of df -- 2.4, whereas the very similar data of Spector et al. (1994) lead to df --- 2.3. However, based on a technique where the sample is frozen ultrafast, then cut into thin samples and analysed via transmission electron microscopy, the latter authors were unable to see fractal objects and therefore concluded on the absence of a fractal phase. This debate certainly deserves further clarification. For more details see Wiese (1996a). In summary: the experimental situation is not very transparent. Let us still mention another very amusing class of experiments. Crunching a thin aluminium foil in the attempt to form a ball (Gomes and Vasconcelos,

2


275

Fig. 9 Image of a graphite membrane taken by a transmission electron microscope (Hwa et al., 1991). The linear dimension is about 1/zm. I00

. . . . . . . .

lo'

|

q)

q24

~" 1os

I0

S(cl)

,.

,

~ i ,,,1

~ IO

q (pLm "I )

A

i

i

i ,, IO0

~o~ lo o

lo' q ~=,-')

1o2

Fig. 10 Static structure factor of graphite oxide membranes in alkalic solution as function of the wave vector q obtained from light-scattering in the visible domain. (Taken from Hwa et al. ( 1991 ) (left) and Spector et al. (1994) (fight).

1988; Kantor et al., 1988), also allows one to measure a fractal dimension, which turns out to be very close to the Flory result of equation (2.15). This result is easily

276

K. J. Wiese

2.0~~ 1.5 1.0 0.5~ 0.0

"~

\

O

-" -0.5

"~

-1.0

,

~

-1.5 -2.0

-

9

9

log 2 ( l / L) = 89 (German DinA size) Fig. 11 Result of crunching a sheet of paper of linear size L to a ball of diameter R. This leads to a fractal dimension of df = 2.4, equivalent to v* = 0.82.

reproduced on a table-top experiment with paper (see Fig. 11). However, since crunching aluminium foil is certainly a nonequilibrium process, this may be a coincidence.

2.6

Numericalsimulations of self-avoiding membranes

In this section we review existing numerical simulations of tethered membranes. If not stated otherwise, these are membranes (D = 2) embedded into three dimensions. The first simulations for self-avoiding membranes were performed for very small systems (121 beads) by Kantor et al. (1986, 1987). They obtained v* -0.80 _-t-0.05 in agreement with the Flory approximation. Here, as in most of the simulations, self-avoidance is effective between the beads (of finite size) of the network. (For a visualization, see Fig. 5.) As we discussed in Section 2.4, phantom membranes show a crumpling transition induced by bending rigidity. Shortly after this had been established numerically (Kantor and Nelson, 1987a,b), an attempt was made to study this transition in the presence of self-avoidance (Plischke and Boal, 1988" Abraham et al., 1989; Ho and BaumgS.rtner, 1989" Petsche and Grest, 1994; Dovertsky et al., 1998). The transition has completely disappeared and the membranes were always found fiat for any (positive) value of the bending rigidity. A simple explanation due to Abraham and Nelson (1990a) goes as follows: the simulated model consists of beads (of finite size) and tethers linking the beads together. The tether length is

2


277

chosen such that the beads cannot penetrate through the holes left in-between.

278

K.J. Wiese

There thus exists a maximal angle smaller than Jr, by which the membranes can be folded. Then, the range of possible configurations is restricted and is reinterpreted as an effective bending rigidity. This bending rigidity was claimed responsible for the fiat phase, following the scenario of the crumpling transition of a phantom membrane, induced by bending rigidity. The question therefore arises, whether the fiat phase is an artifact of the simulations, or whether it is generic. Let us mention two simulations in this context: the first is due to Kantor and Kremer (1993). They studied the usual bead-andtether model, but restricted self-avoidance on the membrane to a finite distance l. Since now the interaction is local, one can study the crumpling transition induced by the bending rigidity tr. For cr > trc a flat phase is found, whereas for cr < Crc the membrane is found in a crumpled state. Taking now the limit of large l, the value of the critical bending rigidity crc scales to 0. They then concluded that this indicates that the fiat phase persists down to crc = 0. It would be nice to have more extensive simulations available than the 169 to 331 beads studied there. In another simulation, Liu and Plischke (1992) have found an intermediate fractal phase by adding long-range attraction, and then adjusting the temperature. This intermediate phase was found for some range of temperature and membranes of up to 817 particles. In a similar simulation, Grest and Petsche (1994) were also able to find this intermediate phase, but only for a specific value of the temperature. This is not surprising from the renormalization-group point of view: long-range forces are in general relevant interactions, such that a fine-tuning is necessary to reach the critical point. Let us also mention another trick used in Grest and Petsche (1994): they rendered the membrane much more flexible by adding additional beads between the nodes of the lattice forming the membrane. A similar idea is to dilute the membrane by randomly cutting off links (Grest and Murat, 1990; Plischke and Fourcade, 1991). This attempt was not very fruitful: the flat phase persisted up to the percolation threshold. The best numerical realization of tethered membranes is obtained by imposing self-avoidance not between beads but between the plaquettes forming the membrane. The first such simulation was carried out by BaumgS.rtner (1991) and BaumgS.rtner and Renz (1992), who indeed found the fractal phase. Within a very similar simulation, Kroll and Gompper (1993) were not able to confirm these conclusions. A repetition of these simulations with larger systems as those studied there (up to 496 plaquettes) would be very much welcome to clarify the situation. Other interesting simulations are for membranes in a four, five, six and eightdimensional space. Grest (1991 ) found that membranes are flat in dimensions d = 4, but crumpled and swollen in larger dimensions. Complementary simulations by Barsky and Plischke (1994) confirm this conclusion. These simulations are in agreement with the value of v* predicted by the Gaussian variational ansatz, War = 2 D / d (see Section 7.4), and larger than the two-loop results (see Fig. 20,

2


279

Section 7.6). It remains to mention simulations on a Sierpinsky gasket with fractal dimension of about 1.585 and spectral dimension of about 1.356 (Levinson, 1991). As in the case of polymers, the results for d = 3 are in agreement with the Flory approximation in equation (2.15). Also, the folding transition of a membrane has been studied numerically (Abraham and Kardar, 1991 ). Let us also mention studies of tethered membranes in confined geometries (Leibler and Maggs, 1989; Gompper and Kroll, 1991 a,b), of boundary effects (Gompper and Kroll, 1992), with negative bending rigidity (Mori and Komura, 1996), of dynamics (van Vliet, 1994), and a couple of short reviews about the simulational aspects of tethered membranes (Gompper and Kroll, 1997; Kroll and Gompper, 1997).

2.7

Membranes with intrinsic disorder

A lot of publications have been devoted to the treatment of tethered (phantom) membranes with intrinsic disorder, including two-dimensional gels (Nelson and Radzihovsky, 1991; Radzihovsky and Nelson, 1991; Bensimon et al., 1992, 1993; Kantor, 1992; Morse et al., 1992a,b; Morse and Lubensky, 1992; Nelson and Radzihovsky, 1992; Radzihovsky and Le Doussal, 1992; Le Doussal and Radzihovsky, 1993; Mori and Wadati, 1994a,b; Barri~re, 1995; Mori, 1995; Park and Kwon, 1996; Mori, 1996a,b). Let us give a brief summary of the main ideas, following the first publications (Radzihovsky and Nelson, 1991; Morse et al., 1992a,b; Morse and Lubensky, 1992). Two kinds of disorder can be added. Since we are interested in the stability of the flat phase to such disorder, we study the Hamiltonian of a membrane in an expansion about a flat configuration, generalizing (2.23). We consider the general case of a D-dimensional membrane embedded in a d-dimensional space, such that .

r(x) -- ~

(x~+u~(x)) h j (x)

'

(2.57)

where u(x) ~ II~D describes the D in-membrane (stretching) modes and h(x) ]I~d - D the fluctuations in the d - D transverse directions. The full Hamiltonian

then reads in generalization of (2.23),

7-[[u, h] = f dDx ~(Ah) 2 -k- 1 [2/2uZt~ + ku2• + aoe/~(x)uat~(x) + ?(x)Ah(x), ~ (2.58) where we recall the definition of the deformation matrix 1 1 -. -. 1 uor -- -~(Oau~ + Oflua) + -~(Ooth)(O~h) + -~(Oaur, l(O~ur,).

(2.59)

280

K.J. Wiese

cr~ (x) is a quenched random stress field, or variation of the metric. Microscopically it is due to different tether lengths in the spring and bead model of F~. 5. ?(x) is a quenched random curvature field, favouring the mean curvature Ah(x), and breaking the reflection symmetry between the two sides of the membrane. It may be caused by a local difference in the chemical composition between the two sides of the membrane. The correlations are short ranged, of the form

a~(x)a•

= [AxS~t38•

+ 2A~(8~•

+ 8~aS/3•

Ci (x)cJ (X t) = A ~ i j ~ D(x -- Xt).

- x') (2.60)

To study the renormalization group flow, the model is replicated, and the disorder averages are taken. This leads to an effective Hamiltonian similar to the pure model, but now with couplings between different replicas. One can then parallel the calculations of the pure model. The outcome is that at finite temperature, the long-wavelength properties of the membrane are unchanged. New physics emerges at or very near to zero temperature, characterized by a new nontrivial fixed point. Membranes with nonzero random spontaneous curvature are found in a flat phase with nontrivial critical exponents, analogous to the flat phase of the pure model at nonzero temperature (Morse et al., 1992a,b; Morse and Lubensky, 1992). This fixed point is accessible within an e-expansion. Membranes with disorder in the metric are more difficult to access, since the fixed point lies outside the perturbatively accessible domain (Nelson and Radzihovsky, 1991; Radzihovsky and Nelson, 1991). 3

3.1

Field-theoretical t r e a t m e n t of tethered membranes

Definition of the model, observables, and perturbation expansion

We start from the continuous model for a D-dimensional flexible polymerized membrane introduced in Aronovitz and Lubensky (1988) and Kardar and Nelson (1987). This model is a simple extension of the well-known Edwards model for continuous chains. The membrane fluctuates in d-dimensional space. Points in the membrane are labelled by coordinates x 6 IR~ and the configuration of the membrane in physical space is described by the field r : x 6 IR~ > r(x) IRd, i.e. from now on we use the notation r instead of F. In Section 2.3 we had discussed that at high temperatures the free energy for a configuration is given by the (properly rescaled) Hamiltonian 7-/[r] = 2 -Z D f x

21 (Vr(x))2 q- bZb # e f f ~ d ( r ( x x

y

) - r (y)).

(3.1)

2


281

The so-called renormalization factors Z and Zb have the form Z = 1 + O(b) and Zb -- 1 + O(b); they will be explained later. The reader may safely set both to 1 for the moment. The integral fx runs over D-dimensional space and V is the usual gradient operator. The normalizations are

f

.

f

1 = So

dO x

So

2 re~

F (D/2)

(3.2)

and

~a (r(x) - r(y)) = (4re)a/26 a (r(x) - r(y)).

(3.3)

The latter term is normally used in Fourier representation ~d ( r ( x ) -- r ( y ) )

=

f

e ip[r(x)-r(y)]

(3.4)

p where the normalization of fp is given by

f

= re-d~2

f

dd p

(3.5)

a -d/2.

(3.6)

p

to have

f e p2a p

All normalizations are chosen in order to simplify the calculations, but are unimportant for the general understanding. (They are collected in Appendix A.) is an internal momentum scale, such that/~x is dimensionless. It is introduced to render the coupling b dimensionless. The first term in the Hamiltonian is a Gaussian elastic energy which is known to describe the free 'phantom' surface. The interaction term corresponds (for b > 0) to a weak repulsive interaction upon contact. The expectation values of physical observables are obtained by performing the average over all field configurations r(x) with the Boltzmann weight e -~[r]. This average can not be calculated exactly, but one can expand about the configurations of a phantom, i.e. noninteracting surface. Such a perturbation theory is constructed by performing the series expansion in powers of the coupling constant b. This expansion suffers from ultraviolet (UV) divergences which have to be removed by renormalization and which are treated by dimensional regularization, i.e. analytical continuation in D and d. A physical UV cut-off could be introduced instead, but would render the calculations more complicated. Long-range infrared (IR) divergences also appear. They can be eliminated by using a finite membrane, or by studying translationally invariant

282

K.J. Wiese

serf-avoidance relevant 1.5

D

i serf-avoidance irrelevant 0.5

0

'

0

'

'

!

.

.

.

.

t

5

.

.

10

.

.

t

'

'

15

'

20

Fig. 12 The critical curve e(D, d) = 0. The dashed line corresponds to the standard polymer perturbation theory, critical in d = 4. observables, whose perturbative expansion is also IR-finite in the thermodynamic limit (infinite membrane). Such observables are 'neutral' products of vertex operators N

N

(.9 = l'-I eikar(xa)' a=l

~ ka = O. a=l

(3.7)

An example is given at the end of Section 3.3. Let us now analyse the theory by power counting. We use internal units # ~l/x, and note [X]x = 1, and [/z] x = - [/x]~ = - 1 . The dimension of the field and of the coupling constant are:

2-D

v := [rlx = ------~-,

e := [buC]~z = 2 0 -

yd.

(3.8)

In the sense of Wilson and Kogut (1974) the interaction is relevant for e > 0 (see Fig. 12). Perturbation theory is then expected to be UV-finite except for subtractions associated to relevant operators. We shall come back to this point later. For clarity, we represent graphically the different interaction terms which have to be considered. The local operators are 1 = 1,

l-(Vr(x))2 = + . 2

(3.9)

(3.10)

2


2113

The bilocal operator, the dipole, is

~d(r(x)--r(y)) = =

(3.1 l)

The expectation value of an observable is

f 79[r]O[r]e 7"t[r] (O[r])b =

(3.12)

f T)[r]e-7-t[rl

Perturbatively, all expectation values are taken with respect to the free theory:

jf. D[r]O[r]e-r~ f,- 89 (O[r]) o = f D[rle-~

.

(3.13)

--)~),

(3.14)

f,- 89

A typical term in the expansion of (3.12) is

(-bZblZe)n f f . . . f f

(0=

=...

=

where the integral runs over the positions of all dipole endpoints.

3.2 Locality of divergences In this section, we show that all divergences are short distance divergences. Note that even for massless theories and in the absence of IR-divergences, this is not trivial. Divergences could also appear, when some of the distances involved become equal, or multiple of each other. A simple counterexample is the integral

I

of lal - Ibl

I"

, where a and b are two of the distances involved.

That divergences only occur at short distances (i.e. when at least one of the distances involved tends to 0), is a consequence of Schoenberg's (1937) theorem. Here, we present a proof, based on the equivalence with electrostatics. We first state that with our choice of normalizations (see Appendix A.), the free correlation function C (Xl, x2)

C(xl,x2)

]210-- IXl - x 2 12-D (2~)0 p2 ( 1 - e ip(xl-x2))

:= ~1 ( 1~ [ r ( x l ) - r ( x 2 ) =-- ( 2 -

D)SD

(3.15)

284

K.J. Wiese

is the Coulomb potential in D dimensions. Furthermore, the interaction part of the Hamiltonian 7-[ is reminiscent of a dipole, and can be written as

~int -- bZbl~ f f 6d(r(xl) -- r(x2)) Xl x2

=bZou~fffe ik[r(x')-r(x2)l,

(3.16)

Xl x2 k

where k may be seen as a d-component (vector) charge. The next step is to analyse the divergences appearing in the perturbative calculation of expectation values of observables. To simplify the calculations, we focus on the normalized partition function Z

1 Z e-7t = ( e-7"ti"t ) = ~ 0 All configurations 0"

(3.17)

To exhibit the similarity to Coulomb systems, consider the second-order term

ffffff

1 (7"/2nt)0 = 2 ' ~ " ~ ' ~

(elk [r(xl ) -r(x2)]eip[r(y])-r(y2 )] )0

x l x2 yj Y2 k p

=

;'> ffffffe c xl x2 Yl Y2 k p

Ec = k2C(xl - x2) + p2C(yl - Y2) + k p [ C ( x l - Y2) -k- C ( x 2 - Yl) - C ( x l - Yl) - C ( x 2 - Y2)],

(3.18) where Ec is the Coulomb energy of a configuration of dipoles with charges +k, and + p , respectively. More generally, for any number of dipoles (and even for any Gaussian measure) we have

(e i F~ kir(xi) )0 = e-~c'

1

Ec -- -~ Z. . (kir(xi)kjr(xj))o.

(3.19)

t,J

Since Y~i ki = O, the latter can be rewritten with the help of the usual correlation function 1

C(x - Y) -- ~-~ ([r(x) -r(y)]2)o as 1 gc =

Z

4d t,J ..

kikj ([r(xi) - r(xj)]2)o.

(3.20)

2


285

As for any configuration of dipoles, specified by their coordinates and charges, the total charge is zero, the Coulomb energy is bounded from below, i.e. Ec > 0.

(3.21)

Formally, this is proven by the following line of equalities (remember that D < 2) 1

Ec -- -~ Z

(kir(xi)kjr(xj))o

9 . l,j

=

(2 - D)SD 2

f

d Op

1 io,x

(2n.)o Z. . k i k j -p2 -e"

=

(2 - D)So f 2

dO P

,-xj)

"12

l,J

1

(27r)0 p2 Z . kie'pXi

>_ O.

(3.22)

1

The last inequality is again due to the global charge neutrality, which ensures convergence of the integral for small p. Hence, Ec vanishes, if and only if the charge density vanishes everywhere. This implies that e -Ec < 1,

(3.23)

and the equality is obtained for vanishing charge density. Noting Ec = Y~i,j kikjQij, (3.22) even states that as long as xi ~: xj for all i # j, Qij is a nondegenerate form on the space of ki with Y~i ki - - O. This implies that integrating e -ec as in (3.18) over all ki with ~--~iki - - 0 gives a finite result, as long as some of the xi coalesce. Consequently, divergences in the integration over xi can only appear when at least some of the distances vanish, as stated above. This does of course not rule out IR-divergences. We will see later that they are absent in translationally invariant observables. An explicit example is given at the end of the next section; for a proof see David et al. (1997).

3.3

More about perturbation theory

Let us apply the above observation to evaluating the integrals in (3.18); this will give an intuitive idea of the kind of counterterms needed to cancel the UVdivergences, as will be made formal later. The basic idea is to look for classes of configurations which are similar. The integral over the parameter which indexes such configurations is the product of a divergent factor, and a 'representative' operator. For the case of two dipoles, one with charge k and the other with charge

286

K.J. Wiese

p - k, and approaching its endpoints (as indicated by the dashed lines below), one only sees a single dipole with charge p from far away, i.e. k ', ' ~ - ' ~ ' , ' -k .~ p = p - k ".-'~-.-~--'" - p + k

r

X e -k2(Isl2-~

(3.24)

The second factor on the r.h.s, contains the dominant part of the Coulomb energy Ec -- k2(Isl 2-o + Itl 2-D) of the interaction between the two dipoles; s and t are the distances between the contracted (approached) ends. The integral over k is now factorized, and we obtain

f e-k2(l~12-~176

(is12-O +

It12-~ -d/2

(3.25)

k

Finally integrating over p in (3.24) gives back the &-interaction -with ( ~ I'----'), where we define the coefficient as C~III

--

= ) = (Isl 2-D +

Itl2-D) -d/2 .

-- multiplied

(3.26)

The notation, which will be explained later, represents a scalar product or projection of a singular configuration of two dipoles onto a single dipole. Equation (3.26) contains the dominant UV-divergence upon approaching the endpoints; this will be made formal later. As an example of an expectation value, use in (3.7) the observable 69 = e ik[r(s)-r(t)], which is the generating function for the moments of [r(s) - r(t)]; the series up to first order in b reads (remember that Zb = 1 + O(b)) (0) b = e-k2C(s-t) { 1 +

blz C

xff[l-exp(lk2[C(s-x)+C(t-y)-C(s-y)-C(t-x)]a)]c(x-y) x y

x C ( x - y ) -a/2 + O(b 2) }.

(3.27)

Note that the integral over x and y is IR-convergent, but UV-divergent at e _< 0: there is a singularity for Ix - Yl --~ 0. This is a general feature of such expectation values. The purpose of the rest of this section is to introduce the basic tools to handle these divergences. For the example of (3.27), this is verified in Exercise 6 (Appendix H).

2

287


3.4 Operator product expansion (OPE), a pedagogical example Throughout this review, we will use the techniques of normal ordering and operator product expansion to analyse the short-distance behavior of the theory. Since their technical simplicity is as little recognized as their one-to-one correspondence to standard Feynman graphs, we shall give here a pedagogical derivation of the two-loop result for the exponent r/in standard scalar r before discussing the case of a membrane in the next section. Complementary material can be found in Cardy (1996). Readers familiar with the procedure can continue with Section 3.5. Define the renormalized q~4-Hamiltonian as 7-/ =

d

Z2 f

:(re(x))

z

2 -+- bZblZ"

fr 9

(x)'.

(3.28)

x

The integration measure is normalized as

f = ~1

f

ddx,

7rd/2 Sd = 2 F(d/2-------~,

(3.29)

where Sd is the surface of the d-dimensional unit sphere. This is done in order to obtain for the free expectation values (denoted by subscript '0')

C(x - y ) " - (r162

0 = Ix - yl 2-d.

(3.30)

Note the similarity and difference between the definitions in (3.15) and (3.30); the difference results from the 0-mode, which has to be subtracted in the case of polymers and membranes (D < 2), but not of the r (d > 2). The dimensional regularization parameter E is E -- 4 - d,

(3.31)

and/z is the renormalization (subtraction) scale. Note the difference from (3.8), where we use e instead of E. The renormalization Z-factors, introduced to render the theory finite, start with 1, and higher-order terms in b will be added to cancel the divergences. The dots ':' indicate the normal-order procedure. We define the normal order of an operator O as :(,.9: = ( 9 - all tadpole-like diagrams constructed from (.9.

(3.32)

In other words, by normal-ordering an operator, we just subtract all selfcontractions. Let us give some examples: :q~2(X): -" r

-- C(O) 1,

:r

- 6C(0) :r

-----r

- 3 C 2 ( 0 ) 1.

(3.33)

288

K.J. Wiese

Note that on the right-hand side all subtracted terms are normal-ordered. One can of course recursively replace them, which for "4)4(x)" e.g. leads to :~4(X)" = ~4(X) -- 6C(O)q~2(x) + 3C2(0) 1.

(3.34)

In the dimensional regularization scheme, these relations are much simplified through the rule that C(0) = 0. Note also that the normal-order prescription is associative. Normal ordering is a powerful tool to organize the perturbation expansion. Let us show this by proceeding to the real calculation. We want to study the short-distance behaviour of two operators :47(x): and :4~4(y): in an OPE. To this aim we first normal-order the product of the two interactions: :~bn(x)::~n(y):--:~b4(x)~bn(y):

+16 :~b3(x)~b3(y): C ( x -

y)

+72:4~2(x)4~2(y): C2(x - y) +96:4'(x)4~(y): C3(x - y) +24 1 C 4 ( x - y).

(3.35)

It is now essential that the normal-ordered product of two operators is free of

divergences when these operators are approached; the divergences are contained in the factors of powers of C(x - y). For instance at leading order, the first term in (3.35) becomes : ~ 4 ( x )qbn ( y ) : = : ~ 8 (Z ) " -+- . . . , (3.36) where z = (x + y ) / 2 . Let us now consider the perturbation expansion of the expectation value of an observable O (O) b := ~

D [4)] e-7-tO = e -bzhu' f:4~4(x):o/c~ '10

(3.37)

where (" " ")0 denotes the free expectation value, and we retain only diagrams that are connected to points in the observable O. The term quadratic in b contains (setting all Z-factors equal to 1 for the moment)

b2~2"ff :4)4(x)4)4(y)~ x

Observe now that

ff

x y

(3.38)

y

"~4(X)"~4(Y)"

(3.39)

2

289


possesses short-distance divergences according to (3.35). More explicitly, the first two terms, "~4(x)~n(y)" and 16 "t~3(x)~3(y) 9C(x - y) are free of divergences when Ix - Y l ~ 0. The third one is upon integration over x and y

v2ff :q~2(x)q~2(y)" C2(x-y)-72Af"q~4(z)" + finite, x

y

(3.40)

z

where -I

A--

f C2(t)= f t

tdttd x t

2(2-d) = 1~ # -~ ,

(3.41)

0

and # - l is the IR-cutoff. It is very important to note that the integral over C 2 (x y) is localized at x - y - 0. This means that for any smooth function f (x, y)

ffc2 x_y)f(x , y) ----- u,f f(z, z) + 0(60),

(3.42)

E

x

y

z

y) becomes in the limit of ~ ~ 0 a distribution

or more formally that C 2 ( x -

C2(x - y) - lz-~ Sd&d(x -- y) + 0(60).

(3.43)

E

This explains why in (3.40) we could simply replace "4~2(x)4~2(y) 9by "4~4(z) ". It is now easy to see that after introduction of a renormalization factor b Zb = 1 + 3 6 -

(3.44)

a second term of order b 2 will appear in the perturbation expansion, namely - 3 6 b2/zE

f

.t~4 (z) 9O,

(3.45)

E z

which will cancel the divergence of (3.40). This is the only renormalization necessary at one-loop order. Especially, no counterterm for fx 1 .(V4~(x))2. is necessary at leading order in b. However, it demands a renormalization at second order, arising from the term :4~(x)4~(y): C 3 (x - y).

(3.46)

As above, we now have to analyse the integral (t :-- x - y)

f c (t) "~(x)~(y)" . t

(3.47)

290

K.J. Wiese

Noting that

f C3( t ) -

f dtt

t dt3(2-d)--

/ dttt2~-2

(3.48)

t

the leading term is a relevant (quadratic) divergence. We therefore have to expand 4)(x) and 4)(y) up to second order

-

,(y-x

4~(x) =4~(z)+ x 2 YV4)(z) + 2

V

2

)2q~(z)+ O ()3) (x - y

(3.49)

to obtain [~

"r

="

l(x-Y

(z) + x - 2y v r

[~

(z) + y - xV~(z) + 2

= .q~ (Z)2.

)2

2 V r

-2

l(y-x 2

2

)2

(

(x-y)3

)]

(

V ~(z) + O (x-y)3

x

)]

.

- ~1 "[(x - y)V~(z)] 2 9+ ~1 "~(z)[(x - y)V] 2 q~(z)"

+O ( ( x - y)3).

(3.50)

With the help of (3.48), (3.47) becomes /z -I

f o

[

,2 "(V(~(Z))2 ,2 :~(z)A~(z)" +O (t 3) ]

dtt2~-2 :~(z) 2"

T

.2-2~ /2-2~ [ 1 ] = --2 + 2~ "q~(Z)2 9 :(V4)(z)) 2 " - "4)(z)A4)(z)" + finite. 2E 4d (3.511 The first term does not come with a pole in 1/E and in addition scales to 0 in the large L = 1//2 limit. It will thus be neglected. The remaining two terms are equivalent up to a total derivative, and thus (3.38) yields another divergent term 24b 2 dE

/1

~ "(Vq~(Z)) 2" O.

(3.52)

z

This is renormalized (cancelled) by setting Z

.

1

.

24(d - 2) b 2 . . d E

1-12-

b2 ~ finite. E

(3.53)

2

291


The last step is as usual to calculate the renormalization group functions/3(b) and o(b), quantifying the flow of the coupling b and the field 4) upon changing/z (Amit, 1984).* The result is /3(b) := #

+1

b -- - E b + 36b 2 + O(b 3)

(3.54)

In Z = 24b 2 + O(b3).

(3.55)

0

r/(b) : = / z o

Note that the/3-function has a nontrivial IR-stable fixed point (/3(b*) = 0) at b* = E/36 and that this is sufficient to get the exponent 11 up to order E2" ~2 O = rl(b*) = ~-~.

(3.56)

Finally, let us still note the equivalence of the OPE with standard Feynman diagrams. The first integral was

:4~2(x)4,Z(y)"f C e ( x - y ) = ~ .

(3.57)

x--y

Usually, this is written in momentum space as

f

1

1

(k + p)2 k-2"

(3.58)

The other diagram was

$(x)$(y)" 9 f

c3,x_y) =

. ~ ~,,,,,.__~

(3.59)

x--y

Note that if we parametrize the latter by the momentum p which is running through, then

~.....__.~

=

ffll

1 l p 2-2~ q2 q2 (ql + q2 + p)2 ~ ~ "

(3.60)

ql q2 The factor of p2 is the equivalent of the derivatives appearing in (3.51 ). *For membranes, a derivation of the renormalization group functions is given in Appendix D.

292

K.J. Wiese

3.5 Multilocal operator product expansion (MOPE) In Section 3.2, we showed that, for self-avoiding membranes, divergences only occur at short distances. The situation is thus similar to local field theories for which we discussed in the last section how the techniques of operator product expansion can be used to analyse the divergences. Our aim is now to generalize these techniques to the multilocal case (David et al., 1994, 1997). Intuitively, in the context of multilocal theories - by which we mean that the interaction depends on more than one p o i n t - we also expect multilocal operators to appear in such an operator product expansion, which therefore will be called 'multilocal operator product expansion' (MOPE). Its precise definition is the aim of this section, whereas we shall calculate some examples in the following one. We start our analysis by recalling the general form of a (local) operator product expansion of two scaling operators (1)a (Z + ~.X) and ~'B(z + ~.y) in a massless theory in the limit of ~. --+ 0: di)A(Z "k- ~.X)d~B(Z "~- ~.y) = Z

C i ( z , ~.x, ~.y)dPi(Z),

(3.61)

i

where Ci (z, ~.x, ~y) are homogeneous functions of k C i ( z , ~.x, ~.y) - ~.[~alx+[~B]x-[~i]xCi(z,x , y).

(3.62)

Here [~]x is the canonical dimension of the operator 9 in space units such that [x]x = 1, as obtained by naive power counting. If the theory is translationally invariant, Ci (z, x, y) is also independent of z, and we will suppose that this is the case, if not stated otherwise.* Also recall that this relation is to be understood as an operator identity, i.e. it holds inserted into any expectation value, as long as none of the other operators sits at the point z, to which the contraction is performed. An example for the multilocal theory is

(3.63) Let us explain the formula. We consider n dipoles (here n = 5) and we separate the 2n endpoints into m subsets (here m = 3) delimited by the dashed lines. The MOPE describes how the product of these n dipoles behaves when the points *Translation invariance is, e.g. broken when regarding systems with boundaries or initial time problems, see Section 8.4 and Diehl (1986) for a review. It is also broken when the underlying metric is not constant, see David et al. (1997) and De Witt (1984).

2

293


inside each of the m subsets are contracted towards a single point Z j. The result is a sum over multilocal operators dOi(zl . . . . . Zm), depending on the m points z l . . . . . Zm, of the form y ~ C i ( x l - Zl . . . . )(1)i ( Z l , Z2 . . . . .

(3.64)

Zm) ,

i

where the MOPE coefficients Ci ( X l - Z l . . . . ) depend only on the distances x t - z j inside each subset. This expansion is again valid as an operator identity, i.e. inserted into any expectation value and in the limit of small distances between contracted points. Again, no other operator should appear at the points z l . . . . . Zm, towards which the operators are contracted. As the Hamiltonian (3.1) does not contain a mass scale, the MOPE coefficients are as in (3.62) homogeneous functions of the relative positions between the contracted points, with the degree of homogeneity given by simple dimensional analysis. In the case considered here, where n dipoles are contracted to an operator ~i, this degree is simply - n v d - [r ]x. This means that Ci(k(Xl

- Zl) ....

) =

~.-n~d-[~PilxCi(xl

- - Zl . . . .

),

(3.65)

where [(1) i ]x is the canonical dimension of the operator (l) i and - d ( 2 - D ) / 2 is simply the canonical dimension of the dipole. In order to evaluate the associated singularity, one finally has to integrate over all relative distances inside each subset. This gives an additional scale factor with degree D ( 2 n - m). A singular configuration, such as in (3.63), will be UVdivergent if this degree of divergence 2-D D ( 2 n - m) - n ~ d

2

- [(1)i] x ,

(3.66)

is negative. It is superficially divergent if the degree is zero and convergent otherwise. The idea of renormalization, formalized in Section 3.8 and proven to work in Section 5, is to remove exactly these superficially divergent contributions recursively.

3.6

Evaluation of the MOPE coefficients

The MOPE therefore gives a convenient and powerful tool to calculate the dominant and all subdominant contributions from singular configurations. In this section, we explain how to calculate the MOPE coefficients on some explicit examples. These examples will turn out to be the necessary diagrams at one-loop order.

K.J. Wiese

294

In the following we shall use the notion of normal ordering introduced in Section 3.4. The first thing we use is that

:e ikr(x) : = e ikr(x).

(3.67)

Explicitly, tadpole-like contributions which are powers of

f

l

d~ p-~

(3.68)

are omitted. This is done via a finite part prescription (analytical continuation, dimensional regularization), valid for infinite membranes, for which the normalorder prescription is defined. Let us stress that this is a purely technical trick, which is not really necessary. However, adopting this notation, the derivation of the MOPE coefficients is much simplified, and we will henceforth stick to this convention. The suspicious reader may always check that the same results are obtained without this procedure. This is clear from the uniqueness of the finitepart prescription. The key formula for all further manipulations is

:eikr(x)"eipr(y)" = e kpC(x-y) :eikr(x)e ipr(y)" .

(3.69)

This can be proven as follows: consider the (flee) expectation value of any observable (.9 times the operators of (3.69). Then the the left- and right-hand sides of the above equation read

s

(O .eikr(x).:eipr(y). )O

7P~= ekpC(x-Y) (O .eikr(x)eipr(y).)O" First of all, for O

=

(:eikr(x)eipr(y)

1 and

1, the desired equality of E =

=

(:e ikr(x)':e ipr(y):)O

R holds, because

ekpC(x-Y)" N o w consider a nontrivial observable (.9, and contract all its fields r with e ikr(x) o r e ipr(y), before contracting any of the fields r (x) with r(y). The result is a product of correlation :)0

~-

functions between the points in (.9 and x or y, and these are equivalent for both E and 7r However, contracting an arbitrary number of times e ikr(x), leaves the exponential e ikr(x) invariant. Completing the contractions for E therefore yields a factor of ekpC(x-y), and the latter one also appears in R. Thus, the equality of E and R holds for all O and this proves (3.69). Now proceed to the first explicit example, the contraction of a single dipole with endpoints x and y:

x .... . y

----f "eikr(x'"e -ikr(y," . k

(3.70)

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295

This configuration may have divergences when x and y come close together. Let us stress that in contrast to r these divergences are not obtained as a finite sum of products of correlators: since C(x - y) = Ix - yl 2 - ~ the latter is always well-behaved at x = y. The singularity only appears when summing an infinite series of diagrams as we will do now. To this purpose, we first normal-order the two exponentials using (3.69)

f

:eik[r(x)-r(y)]. e-k21x-yl 2v

(3.71)

k Note that the operators e ikr(x) and e -ikr(y) are flee of divergences upon approaching each other, since no more contractions can be made. The divergence is captured in the factor e -k21x-yl2v. Therefore, we can expand the exponential 9e ik[r(x)-r(y)] 9for small x - y and consequently in powers of [r(x) - r(y)]. This expansion is

f{ 9 l+ik[r(x)-r(y)]-~

l (k [r(x ) - r (y)])2 + - - - } "e -kz Ix_yl2U

(3.72)

k

We truncated the expansion after the third term. It will turn out later that this is sufficient, since subsequent terms in the expansion are proportional to irrelevant operators for which the integral over the MOPE coefficient is UV-convergent. Due to the symmetry of the integration over k the term linear in k vanishes 9 Also due to symmetry, the next term can be simplified with the result /[1-~-

k2 d'[r(x)-r(y)]2"+''']e

-kzlx-yl2v

(3.73)

k

Finally, the integration over k can be performed. Recall that normalizations were chosen such that fk e-sk2 - s-d~2 to obtain

x

,[

-~" ( x - y ) V r

X_ y

9l x - y l - u ( d + 2 ) + . . . .

(3.74)

The second operator has a tensorial structure, which has to be taken into account in order to construct the subtraction operator 9 Using the shorthand notation a-4rg = 89 we can write this symbolically as

=(o ,),+(Q

~ + p ) a-~O + - - . ,

(3.75)

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K.J. Wiese

with the MOPE coefficients (in analogy to Feynman's bra and ket notation)

(:~, O--Ix.... ~+r

yl-vd

(3.76)

l

= - ~ ( x - y)a(x - y)~lx - yl -v~a+2).

(3.77)

As long as the angular average is taken (and this will be the case when integrating the MOPE coefficient to obtain the divergence), we can replace in (3.75) ,~+t~ by -4- "- 89 2 and (3.77) by +

= - ~ - ~ l x - yl

9

(3.78)

Next consider a real multilocal example of an operator-product expansion, namely the contraction of two dipoles towards a single dipole"

x-u/2X+U/2'.~'""'~7~~ ", y+V/2y_v/2--

f

eik[r(x+u/2)-r(y+v/2)]

k

f

eip[r(x-u/2)-r(y-v/2)]

P

(3.79) This has to be analysed for small u and v, in order to control the divergences in the latter distances. As above, we normal-order operators which are approached, yielding

eikr(x+u/2)eipr(x-u/2) = :eikr(x+u/2).:eipr(x-u/2)

:

= "eikr(x+u/2)e ipr(x-u/2)" e kpC(u).

(3.80)

A similar formula holds when approaching e -ikr(y+v/2) and e-ipr(y-v/2):

e-ikr(y+v/2)e-ipr(y-v/2)

= .e-ikr(y+v/2) ..e-ipr(y-v/2) : = .e-ikr(y+v/2)e-ipr(y-v/2).

ekpC(v).

(3.81)

Equation (3.79) then becomes

ff

:eikr(x+u/2)+ipr(x-u/2)::e-ikr(y+v/2)-ipr(y-v/2)"

e kp[c(u)+c(v)].

(3.82)

k p In order to keep things as simple as possible, let us first extract the leading contribution before analysing subleading corrections. This leading contribution is obtained when expanding the exponential operators (here exemplified for the second one) as

9e-ikr(y+v/Z)e -ipr(y-v/2) := "e -i(k+p)r(y) (1 + O(Vr))"

(3.83)

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297

and dropping terms of order Vr. This simplifies (3.82) to

ff

"ei(k+p)r(x)"e-i(k+p)r(Y)" ekp[C(u)+C(v)l

(3.84)

k p

In the next step, first k and then p are shifted: k

~ k-p,

then

p

> p+-.

k

(3.85)

2

The result is (dropping the normal ordering according to (3.67))

f

eik[r(x)-r(y)]

k

f e(~Ik2-p2)[C(u)+C(v)]

(3.86)

p

The factor of fk eik[r(x)-r(Y)] is again a g-distribution, and the leading term of the short distance expansion of (3.86). Derivatives of the g-distribution appear

t k2_p2

when expanding e(z )lC(u)+C(v)!in k 2" these are less relevant and only the first subleading term will be displayed for illustration:

f eik[r(x)-r(Y)] f e -p2[C(u)+C(v)] ( 1 + -~k2 [C(u) + C(v)] + . . . ) k p --

I:

~

o

+

(3.87) where in analogy to (3.75) and (3.77) "O"~"~"

(!: ~ : 1 :~-)

1

:, * = ~ [ C ( u ) + C ( v ) I

l-d~2

(3.88)

and .

~. =

~d(r(x)

-- r(y)),

" :: ~ ~---(--Ar)~ d(r(x) -- r(y)).

(3.89)

Let us mention here that the leading contribution proportional to the 3distribution will renormalize the coupling constant, and that the next-to-leading term is irrelevant and can be neglected. The same holds true for the additional term proportional to (Vr) which was dropped in (3.83). There is one more possible divergent contribution at the one-loop level, namely _- ~-"~. We now show that the leading term of its expansion, which is

298

K.J. Wiese

expected to be proportional to -

--, is trivial. To this aim consider

P

g~ J, -- [ :e ikr(u) ..e-ikr(x) ..eipr(y) .:e-ipr(z) : "-..7..-'

xyz

d

k,p -- f :e ikr(u) .:e-ikr(x)eipr(Y)e-ipr(z). e-p2C(y-z)ekp[C(x-z)-C(x-y)]. ,I

k,p (3.90) We want to study the contraction of x, y, and z, and look for all contributions which are proportional to

=

= --- f

:e ikr(u) ":e -ikr((x+y+z)/3)" .

(3.91)

The key observation is that in (3.90) the leading term is obtained by approximating e k p l c ( x - z ) - c ( x - y ) l ~ 1. All subsequent terms yield factors of k, which after integration over k give derivatives of the ~a-distribution. The result is that

(o This means that divergences of _

--

--)- (~ll)--0.

(3.92)

~@ are already taken into account by a proper

treatment of the divergences in @ , analysed in (3.75).

3.7

Strategy of renormalization

In the last two sections, we discussed how divergences occur, how their general structure is obtained by the MOPE, and how the MOPE coefficients are calculated. In the next step, the theory shall be renormalized. The basic idea is to identify the divergences through the MOPE, and then to introduce counterterms which subtract these divergences. These counterterms are nothing other than integrals over the MOPE coefficients, properly regularized, i.e. cut off. In order to properly understand this point, let us recall the two main strategies employed in renormalization: the first one subtracts divergences in correlation functions or equivalently vertex functions. This amounts to adding counterterms to the Hamiltonian which can be interpreted as a change of the parameters in this Hamiltonian. Calculating observables with this modified Hamiltonian leads to finite physical expectation values, but it is not evident that the integrals appearing in these calculations are convergent.

2


299

The other procedure is inspired by ideas employed in a formal proof of renormalizability, or more precisely when applying the R-operation to the perturbation expansion, as will be discussed in Section 5. It consists in adding to the Hamiltonian counterterms which are integrals, such that each integrand which appears in the perturbative expansion becomes an integrable function, and as a consequence the integrals and thus the perturbation expansion are finite. Of course, to finally obtain the critical exponents, the integral counterterms have to be reduced to numbers. However, we really want to think of them as integrals in the intermediate steps. The reason is the following: it is extremely difficult to calculate observables. However, this is not really necessary as long as one is only interested in renormalization. The above-mentioned procedure is then sufficient to ensure finiteness of any observable as long as there is no additional divergence when the dipole is contracted towards this observable. The latter situation would require a new counterterm, which is a proper renormalization of the observable itself. The procedure of considering whole integrals as counterterms is in the heart of our renormalization procedure, and the reader should bear this idea in mind throughout this review.

3.8

Renormalization at one-loop order

Let us continue the concrete example of the one-loop divergences, from which are obtained the scaling exponents to first order in the dimensional regularization parameter e. Explicitly, the model shall be renormalized through two renormalization group factors Z (renormalizing the field r) and Zb (renormalizing the coupling b). Recalling (3.1), this is 7-(Jr] = 2 - z D

-2 x

+ bZb

.

ff x

~d (r(x) _

r (y)),

(3.93)

y

where r and b are the renormalized field and renormalized dimensionless coupling constant, and # - L - l is the renormalization momentum scale. Let us start to eliminate the divergences in the case, where the endpoints (x, y) of a single dipole are contracted towards a point (taken here to be the centre-ofmass z = (x + y)/2). The MOPE is

x~y= (x~yll) l-q-(x~3,[u-+-~)a-+-~ -t-'".

(3.94)

The MOPE coefficients were obtained in the last section as (3.95)

300

K.J. Wiese

,..., y a + ~ (.or)

'

-- - - ~ ( x - y ) a ( x - y)~lx - yl -v(d+2)

(3.96)

We now have to distinguish between counterterms for relevant operators and those for marginal operators. The former can be defined by analytical continuation, while the latter require a subtraction scale. Indeed, the divergence proportional to 1 is given by the integral

l):f xo-'"x L

'(A

A-I

A -I 0):

f Ikl-~eik[r(x)-r(y)] d k

Ir(x) - r(y)l =-a

(3.120)

Then the most simple singular configurations which give rise to a renormalization of the interaction are those for which two interactions are contracted to a single one, as we have discussed in Section 3.5. We claim that their multilocal operator product expansion (MOPE), ~ i ~ ~ , does not contain a contribution proportional to - ~ --, but that the leading term is proportional to the short-range interaction - -. This is a consequence of the analytical structure of the long-range interaction" the contraction i,~__,~ is in complete analogy to (3.87) and with the same notations as there,

k

p

+subdominant terms.

(3.121)

2


305

In order to obtain a long-range term, a singularity at k + p = 0 is necessary. However, expression (3.121) is analytical at k + p = 0, and no long-range term is generated. This is easily generalized to any contraction towards - ~ - and hence to any order in perturbation theory. Let us now analyse the consequences. We want to study tethered membranes with long-range interactions, generalizing (3.1) or (3.93) to

7-/LR--2-D zf

+

+b/zS

x

ff x

~" • -"

(3.122)

y

Note that since in contrast to (3.1) the interaction is not renormalized, there is only one Z-factor in (3.122), namely for elasticity. This does, however, not mean that the/3-function is trivial. In analogy to (3.114), the relation between bare and renormalized coupling is bo = bZ(d-u)/21,z~, (3.123) where Z is as in (3.114) the renormalization of the field, and

6 = 2D-

v ( d - or).

(3.124)

The/~-function now reads

,8(b) -- bt + 1

b-

[ -3+

~ -2 d # Olz 0 ln Z]

(3.125)

bo

Using the fact that/z0/0/z In Z is nothing but - 2 times the anomalous dimension of the field, see (3.116), we make the replacement 0 / 2 7 In Z -- 2 ( v o#

v(b))

(3.126)

in (3.125). The result is /3(b) = - [2D - (d - ot)v(b)] b.

(3.127)

This fl-function has two zeros: for ~ < 0, the fixed point at b* = 0 is attractive. For S > 0 the nontrivial zero and fixed point of/3(b) is at b* > 0, implying the exponent identity 2D v* = v(b*) = ~ . (3.128)

d-or

Non-renormalization of the coupling thus allows one to obtain v* without calculating any diagram. Since this observation is quite generally useful, let us give a heuristic derivation of (3.128). We may then consider the formal derivation

306

K.J. Wiese

given above as a proof of the heuristic argument, and employ the latter confidently throughout this review. 'Power counting' for the dimension D of the interaction at a fixed point yields

D = 2 D - v*(d- or),

(3.129)

and this power counting gives the correct dimension of the operator, since the latter has no proper renormalization. Three different scenarios are now possible" if D < 0, then the associated coupling scales to 0, and the operator plays no role in the large scale limit. If D > 0, then the associated coupling grows under renormalization and we are not at an IR-fixed point; by definition this is not the situation considered here. The last possibility is that we are at an IR-fixed point, and this is (at least for one coupling) equivalent to D - 0. It again follows the exponent identity 2D v* = ~ . (3.130)

d - et

Also the crossover from short-range to long-range self-avoidance in a model with both couplings can be discussed in this framework. Following the line of arguments given above, long-range self-avoidance will scale to 0 and the shortrange fixed point is completely attractive as long as D, (3.129), evaluated with v* as obtained from short-range self-avoidance only, is negative. As a consequence always that interaction wins, which yields the larger value for v*. Physically, long-range forces play an important role for charged membranes, as discussed in Kantor and Kardar (1989).

4

4.1

Some useful tools and relation to polymer theory

Equation of motion and redundant operators

The equation of motion reflects the invariance of the functional integral under a global rescaling of the field r. This has important consequences. Consider the expectation value of an observable O in the free theory:

f D[r]Oe-r-~ f, + (0)o =

.

(4.1)

f D [ r l e - r ~ f, + We now perform a global rescaling of

r(x)

r(x)"

~, (1 +

x)r(x).

(4.2)

2


307

The expectation value of O, (4.1), remains unchanged. Expanding up to first order in x yields

(0)o_ jfD[r]O(l+~c[Olr)(1

2 2x_D fx 4- )

--

,

D[r]

1-2_

D

+

(4.3)

e

where [O]r is the canonical dimension of the operator O, measured in units of r, i.e. such that [r]r = 1. Calculating the difference of (4.1) and (4.3) gives

((.9f -'bconn-- v [O]r

(O) 0 ,

(4.4)

l0

where conn denotes the connected expectation value. For several operators we have

( / con. O102

-~-

= V ( [ O l ] r --F [ O 2 ] r ) ( O 1 0 2 ) 0 9

(4.5)

t0

A specific example is

("- f conn -vd "

4-

=

(.-

~") 0 -

(4.6)

l0

In the case of infinitely large membranes, these relations are equivalently valid for nonconnected expectation values. (To prove this, note that ( + ) 0 -- 0 by analytical continuation.) Let us try to understand (4.4) perturbatively. For this purpose, it will turn out to be convenient to integrate by parts the free Hamiltonian as

-2

(Vr(x))2 =

-~

r(x)(--A)r(x).

(4.7)

For simplicity, we consider infinite membranes, such that connected expectation values can be replaced by standard ones. For computational convenience further suppose that O[rl is a function of r(y), O[rl = Off(y)). Then

(O(r(y)) /x -+-x)0(= O(r(y))-~'ixr(x)(--A)r(x) )0

(4.8)

We now proceed according to the following strategy: first contract the field r(x) which is preceded by (--A) with any field in O(r(y)). This yields (for normalizations and conventions see Appendix A)

( r(x)~ D(x - y) O0(r(Y)) )o Or(y) = 2 - D ( r(y) O0(r(y)))o (4.9) 2 5r-~ "

= 2 - 2D f x -2l f x ( r(x)(-Ax)C(x - y) ~30(r(Y)))O Or(y)

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K.J. Wiese

Since (9 is a homogeneous function in r, then

O0(r(y)) r(y) ~ = [O]r O(r(y)), Or(y) the operator is reproduced and we recover the equation of motion (4.4). Note that in this argumentation, it is irrelevant how the second field r of + is finally contracted. In the case of several operators, the field r(x) which is preceded by - A can be contracted with any of these operators, and one recovers (4.5). Note also that without partially integrating the free action, no &D-distribution is obtained and it is impossible to assign + to one of the points with which its fields are contracted: the integral is delocalized. This is a subtle point which was ingeniously avoided up to now. To understand this point remember that the renormalization of the coupling and by this means the fl-function (3.115), not only contains the direct term (~v~--1"----'), but also a term 2-D

In Exercise 3 (Appendix H) the reader can show that the above arguments can be used to obtain this term directly. We now turn to another concept, which is also a consequence of reparametrization invariance, namely redundant operators, as introduced by Wegner (1986). Consider the path integral

f T)[r]e -~[rl

(4.10)

with the Hamiltonian of the interacting theory 7-([r] = 2 - D

fx

+ + bZblZe

f fy

x'-

:y

(4.1 1)

and make a change of variables

r(x)

~ r(x) + x(x)J~[rl.

(4.12)

.~'[r] is an arbitrary function of r(x), but may also involve fields r(y) at different points. (Explicitly, we think of .~'[r] = fl (r(x)) or .~'[r] = f2(r(x), r(y)) with x ~ y, where both fl (r) and f2(r, r') are functions of r and r, r' respectively. More general expressions for .Y'[r] including derivatives of r are possible, but shall not be considered here.) Of course, since this is a simple variable transformation, the path integral itself remains unchanged, even though formally new terms

2


309

are generated, t These newly generated terms contain no physical information, and are thus called redundant operators (Wegner, 1986). They are useful in relating apparently different operators. Let us extract the terms linear in x(u) ~ 6D(u -- x). Two contributions have to be taken into account: first, from the expansion of the exponential, one obtains a term ~7-/[r] -.T'[r]~. (4.13)

~r(x)

Second, the integration measure is changed, resulting in a term ~.T'[r] ~ .

(4.14)

~r(x)

Note that in the cases of.T[r] = fl (r(x)) or .T[r] = fz(r(x), r(y)) with functions fl and f2, this is equivalent to

037[r] ~D (0). (Or(x)) Combining (4.13) and (4.14), we obtain the redundant operator 37-/[r] R = .T[r ] ~

~r(x)

3.T[r] ~ .

6r(x)

(4.15)

The second term just subtracts the contraction of .T[r] with the variation of the (free) quadratic part of the Hamiltonian 67-lo[r]/3r(x) = - ( 2 - D) - 1 A r ( x ) at the same point. (This is the same structure as encountered within different discretization prescriptions in dynamic theories, see, e.g. Janssen, 1992.) Since 3U[r]/3r(x) ~ 3D(o) = f d~ is zero by analytical continuation, we will drop it in the following. Let us now explore some of the consequences of the above construction. First set .T'[r] := 1, yielding the redundant operator - or Dyson-Schwinger equation of motion in the terminology of elementary particle physics (Itzykson and Zuber, 1985): 7~ = 0 with = 67-[[r]Z6r(x) = 2-----~-~(-A)r(x) + bZb# e f f

(2ik)e ik[r(x)-'(y)].

(4.16)

Another example is obtained by choosing .T[r] := r(x), yielding the redundant operator R = 2 _D ~ r ( x ) ( - A ) r ( x )

+ bZblz e

[2ikr(x)]e ik[r(x)-r(y)] .

(4.17)

*Also note that the inclusion of an observable O[r(z)] is possible in the path integral, but leads to additional contact terms for z = x, and at other points on which .T[r] depends.

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K.J. Wiese

4.2

Analytical continuation of the measure

We now define the explicit form for the integration measure in non-integer dimension D using the general formalism of distance geometry (Blumenthal, 1953" David et al., 1993b). The general problem is to integrate a function f (xl . . . . . XN), which is invariant under Euclidean displacements (and therefore depends only on the N ( N 1)/2 relative distances Ixi - xjl between these points) over the N - 1 first points (the last point is fixed, using translational invariance) in IR~ for noninteger D. In order to define the integration, let us take D > N - 1 and integer. For i < N we denote by Yi = xi X N the i th distance vector and by ya its ath component ( a = 1. . . . . D). The integral over Yl is simple" using rotation invariance, we fix Yl to have only the first (a = 1) component nonzero. The measure becomes with the normalizations as listed in Appendix A. -

i,

fio

'

, =

-

~D

sl(sl) ~

Yl = (Yl, 0 . . . . .

,

0).

(4.18)

We now fix Y2 to have only a = 1 and a = 2 as nonzero components. The integral over Y2 consists of the integration along the direction fixed by Yl and the integration in the orthogonal space ~ o - 1 : =

2

'f

~D

dDy2=

s~ ? ~D

oo

dy~

So"

dy2(y2) D-2

Y2 "- (Y:~, Y22, 0 . . . . . 0).

(4.19)

For the jth point, one proceeds recursively to integrate first over the hyperplane defined by yl . . . . . y j-1 and then the orthogonal complement: = i

1-L SD

f

dDyj=

oH/_ /o dy~

So

.

dyj(yj) D-j

~

'

J 0 . . . . , 0). yj = (yJ . . . . . Yj, The final result for an integral over all configurations of N points is

-ls,

H j=l

i

__ S D - I . . . SD-N+2 -Sff~-2

l(Mf_ /o dyj

j=l

c~

dyJ(yj) D-j

)

(4.20)

.

(4.21)

This expression for the measure, now written in terms of the N (N - 1) variables (2 y j , c a n be analytically continued to noninteger D. For D < N - 2 this measure a is not integrable when some of the y j ~ O. For D not integer, the integration

2

311


is defined through the standard finite-part prescription. This means that the measure (4.21) becomes a distribution. Integer D can be recovered by taking the limit of D to the integer value. Let us make this explicit on the example of N = 3 points. The measure is then

SD-I SD

j0 dy~(Yl )O-1

dy~

f0 dy2(y2)o-2.

(4.22)

It is well defined and integrable for D > I. For D = 1 the integral over y2 diverges logarithmically at y2 __+ 0, but this singularity is cancelled by the zero of So-l and the measure becomes 1 2

/o f oo dy~

+ ~ dy~ ~

/o

~ dy~3(y 2)

~

dyl

dy2

(4.23)

~

thus it reduces to the measure for two points on a line. (The factor of 88=

""(1~ 2

is due to our definition of the measure (A.2).) For 0 < D < 1 the integral over y2 diverges at y2 ~ 0, but this divergence is treated by a finite part prescription. For integrals over N > 3 points, a finite part prescription is already necessary for D < 2. This is the case of the two-loop calculations (see Section 6).

4.3

IR regulator, conformal mapping, extraction of the residue, and its universality

For the simple case of the MOPE coefficients (3.103) and (3.109), the residues could easily be calculated directly. In more complicated situations, however, it is useful to employ a more formal procedure to extract the residue, which is presented now; first on the example of the one-loop counterterms, then in a more formal setting.' Note from (3.103) that

1 1 L e.

. . . .

(4.24)

2De

(For the normalization of the measure, see Appendix A.) The residue can most easily be extracted by applying LO/19Lto (4.24). This yields

L

0_~ 0. In the limit of e ~ 0, convergence is guaranteed, as long as there remains at least one factor of (fit)~. However this is guaranteed, since we have seen above that the spanning tree, when restricted to a subdiagram, comes along with a factor of (1 - Tp~). This completes the proof. We can finally state the most general version of the above theorem. 59. Theorem: General criterion for renormalizability. A statistical field theory is perturbatively renormalizable, if

2

335


(i) the theory is renormalizable by power counting, (ii) divergences are short-range, i.e. no divergences appear at finite distances,

(iii)

the dilation operators defined above commute,

(iv) there exists a multilocal operator product expansion, which describes these divergences, (v) the divergences of the multilocal operator product expansion must not have an accumulation point at dimension zero. Especially, after subtracting them, the integrand has to be convergent when the distances are contracted. 60. Remark: Absence of an accumulation point at dimension zero. Note that in proof 58 we have seen that a factor of (~x)a with 3 > 0 is necessary to ensure UV-convergence. This is not the case if (v) of the above theorem is violated. 61. Remark: Observables which demand a proper renormalization. The above considerations have to be modified in the case of observables (_9 which demand a proper renormalization. In that case, the observable points have to be added to X, and the subtraction operator contains all diagrams which can be constructed from this enlarged set of points, with the exception of those contributions, that involve contractions of the observable points themselves. Note that in the case when (.9 does not demand a proper renormalization, the MOPE coefficients of the contraction towards points of O factorize, and the subtraction operator contains no new terms.

5.3 Some examples In this section we give some illustrative examples of the abstract construction presented in the last section, such that the reader can convince himself that the prescription actually works. 62. Example: Forest construction. Let us consider, as an example for the forest construction, the divergence when contracting two three-point interactions (see Section 9) as

This contraction has several subdivergences, which have to be subtracted. One of these is described by the forest .~

~

'"

"-.O."

,

;., 9 .e"

'O. i 9 .. .'9 9 ".e:

"*

*9 : "

",e" ",e"

I

9

9

(59)

336

K.J. Wiese 9

9

It is important to note that 9 9

could not be added to the forest, since it consists .O-

of one and only one connected molecule (see definition 5), and its intersection .9 9

with o 9 9 is nonempty, nor is it included or includes the latter one. 9 o.

Note, that in the case of

o9 o .o

o.

o.

splits into two connected components .o.

9 -O

and

O

.,. which appear as -o.

individual elements. A possible forest would be

e

1.~~

o.

.o.

,

o.

,

.o- I

.0,

.O

oi

"O.

.

..O.

Let us finally construct the equivalence class of nests, as used in proof 21, for of (5.9). It consists of three elements, namely

921 ~ "

r

o 9 ..o.-o ~ 9 9

e 9 -.o.-O ~ 9e

..e..o-. -,e.o 9 .o-

9

ioo .o ....j 9 e -< 9 9

923 ~

9 e 9 o

:.o:o 9o.

I"

Since in the R-operation r c o m e s and 922, the contribution to R of r

~

-< e : i e 9 e

:e:o 9~

~

-
0

D.-2.2-v

(7.19)

i.e. with (7.18) for d < 5.

(7.20)

For D > 0 we expect that the fractal exponent vc* at the crumpling transition is different with or without self-avoidance. One can now argue that at the crumpling transition and in the presence of selfavoidance, bending rigidity enhances the exponent v*, but does not intervene in a proper renormalization of self-avoidance (Wiese unpublished, 1997). Following the arguments of Section 3.9, this would imply that renormalization of selfavoidance is driven by renormalization of the field, such that v* at the crumpling transition and in the presence of self-avoidance would be given by the variational estimate 2D * -~ Vc+SAd"

(7.21)

Values near to this result were obtained in numerical simulations by Grest (1991) in d = 4, 5, 6, 8 and by Barsky and Plischke (1994) in d = 4, 5. This suggests that their membranes are at or near to the crumpling transition.

358

K.J. Wiese Other critical exponents, stability of the fixed point and boundaries

8.1

Correction to scaling exponent to

The two-loop calculations presented in Section 6 allow us in principle to compute other scaling exponents for self-avoiding tethered membranes. The first exponent is the so-called correction to scaling exponent to which governs the corrections to the large L scaling behaviour. It is known that this exponent is given by the slope of the/~-function at the IR fixed point b* (Amit, 1984)

0 to = -~,6(b)lb=b..

(8.1)

Its e-expansion is given by t to = e + 2 e (CI -+- (72 -+- C3) -+- 2De (Jc"1 -+- .~'2 q- .~'3) 82 .+_ 0(63) (l -k- .A) 2

(8.2)

with

:l

1

F ( 2_--~~) 2

(8.3)

(For the definition of Cl . . . . . C3 and ~'l . . . . . Y'3, see (6.14) to (6.16) and (6.19) to (6.21).) Since for membranes, the term of order e 2 is much larger than the order e term, no numerical value can be extracted (Wiese and David, 1997).

8.2

Contact exponents

Another scaling exponent is the so-called bulk contact exponent 02 (Wiese and David, 1997). For a general introduction to contact exponents for polymers and membranes we refer to Duplantier (1989a). The contact exponent 02 is related to the probability of finding two fixed points Xl and x2 inside the membrane at a relative distance r = I~1 in external d-dimensional space

P(r; xl, x2) = (~d (r -- [r(xl) -- r(x2)])) .

(8.4)

For a large membrane, P is expected to take the scaling form

P(r; Xl, x2)

=

R I / F(r/RI2),

*Note that there are some misprints in Wiese and David (1997).

(8.5)

2

359


where R12 is the mean distance between xl and x2. R22 = ~1 ( [ r ( x l ) - r

(X2)]2).

(8.6)

The contact exponent 02 is given by the small r behaviour of the scaling function F ) ~ F (R--~2) ~ ( r~12

when

r ~ 0.

(8.7)

02 is related to the scaling dimension o912 of the two-membrane contact operator

312(Xl, X2) =

~d(rl(Xl) -- r2(x2))

(8.8)

in the model of two independent self-avoiding membranes, ./~1 and ./~2- This model is described by the Hamiltonian

16M12(Vrl(Xl))2

7-[ = 2 - D

-k- ~

2~M2~(Vr2(x2))

+bZ~ (fxl6MlfyleM! ~d(rl(Xl) -- rl(Yl)) +fxzeMz fYzeMz~d(r2(x2)--r2(Y2)))

+2tZ'"fxfy,M,

(8.9)

~M~ ~ d ( r l ( X l ) - - r 2 ( Y 2 ) ) "

It is renormalized by the same factors for r and b as the single-membrane model, but with an additional renormalization for the intermembrane coupling to - t Z t ( b , t ) Z ( b ) d / 2 # ~. T h e new counterterm Zt contains the same divergent diagrams as those which contribute to Zb, but with different numerical factors. In particular, when t -- b, Z t ( b , t = b) = Z b ( b ) , so that the symmetric two-membrane model reduces to the single membrane model. As a consequence of this formalism, one can define a new RG-function, fit, which measures the dimension of 612 defined in (8.8) as ~t(b, t) "-- #

i9 I 0

t = -t

e + / 3 ( b ) [ ~ In Zt(b, t ) + d ~ In Z(b)] 1 + t o In Zt

,

(8.10)

calculate the RG-flow in the (b, t) plane and check that (b, t) = (b*, b*) is the IR-stable fixed point which governs the scaling behaviour of a large membrane. Insertions of f x ~ m ~ fyz~M2 ~a(rl (Xl) - r2(Y2)) are generated by varying the intermembrane coupling t from its fixed-point value t* and have dimension

O912:=

0

-~-i~t]b=t=b,.

(8.1 1)

360

K.J. Wiese

In order to obtain O2, recall that (i) (8.8) does not contain integrations over x and y at variance with membrane-membrane self-avoidance, (ii) P(r; Xl, x2) R12(d+~ , and (iii) Rl2 ~ Ix - - yl v*, to get the final result 1

02 = ~

(8.12)

[O912 q- 2D - v ' d ] .

can now compare Zb, given in (6.6),* and Zt:

We

( A ( ' A + 8 9 - ~ "A )b 2 + Zb(b) = 1 + ---Ab e + e2 4 D- e + - e l + C 2 + C 3 ... Z t ( b ' t ) = 1 + m et

+

~

+ CI

t2 +

27-2

4 D e ~-C2 +C3

bt + . . . .

(8.13) with .A given in (8.3). The final result to order e 2 reads 4 |

.,4 e

+4 e ((3 + .A)CI + 2(C2 + C3) - 4(~'1 + .~'2 + .~'3)DA) e2 + O(e 3) (2 - D)(1 + .,4)3 (8.14) As in the case of the correction to scaling exponent w, no reliable estimate for can be extracted from the e-expansion (Wiese and David, 1997). Let us also mention that if one or both of the points Xl, x2 in (8.4) are on the boundary of the membrane, the contact exponent changes. This is first of all due to the reduced available domain of integration for operators approaching ~d(r -- [r(xl) -- r(x2)]), and second to the change of the propagator itself, as discussed in Section 8.4 (Duplantier, 1987, 1989a; David et al., 1994). |

8.3

Number of configurations: the exponent y

In principle, one can also calculate the partition function of the membrane, which is defined as (David et al., 1994, 1997) Z(b) = f

D [ r ] ~ d ( r ( O ) ) e -7"lIr].

(8.15)

The factor of $d (r(0)) is included in order to eliminate the 0-mode of the path integral over r, which in dimensional regularization would lead to a factor of 0. *Note that there are some misprints in Wiese and David (1997).

362

K.J. Wiese

where self-avoidance is only affective in some part of the membrane, from now on simply called membrane .A4. The most beautiful way to treat this was proposed by Duplantier (1987, 1989a) who introduces the characteristic function of the membrane ~(q) -- j,~ dDx eiqx, (8.19) with the help of which, the self-energy of the membrane, namely the integral

l'--fxeM fye~ (~a(r(x)-r(Y)))~

(8.20)

can be written as (A is a geometric well-behaved prefactor)

I=a

f dDq ~(q)~(--q)lq[ va-D.

(8.21)

The analysis in Duplantier (1987, 1989a) shows that there are additional contributions for odd integer values of D for hyperellipsoids, and for any integer values of D for a hypertorus. There is another source of terms violating (8.17), which is already present for non-self-avoiding membranes and to which we turn now. In the latter case, the partition function can be calculated exactly, using methods developed in conformal field theory, which involves the ~'-function regularization. The result for Y0, is (Duplantier, 1987, 1989a; David et al., 1997)

-vd -vd+~

for D not integer

2

yo-l=

-ud+ X-_ 2 8.4

for D = 1 with X = 0 for a closed chain, and g = 1 for an open one

(8.22)

for D = 2 with X the Euler characteristics of the manifold.

Boundaries

Up to now, we only considered infinitely large membranes. An important question, not only theoretically but also experimentally, is how boundaries influence the bulk critical behavior. This is the more important since all available simulations deal with relatively small systems, which may be completely dominated by boundary effects. There are several things which can be done. Consider first a membrane, obtained by cutting an infinite membrane along the line x_L = 0. We then use a coordinate system (Fig. 21 ), in which x_L measures the distance from the boundary

2


363

Fig. 21 Coordinate system for the boundary of a membrane. and xll is a (D - 1)-dimensional vector parallel to the boundary. The correlation function is as usual obtained by inverting the Laplacian on this half-membrane with Neumann boundary conditions (Diehl, 1986). Note that these conditions have to be fulfilled by B(x, y) "- (r(x)r(y))0, but are not satisfied by C(x, y) l ([r(x) - r(y) ]2 )o" Explicitly, this is -

0

Ox•

B(x, Y)I

x• =0

= 0.

(8.23)

The solution to this equation is the usual bulk correlator plus its image part:

B(x, y) = Bbulk(X, y) -k- Bbulk(X, .Y),

(8.24)

where ~ is the mirror image of y with respect to the boundary. Denoting y -(Y• YlI), this is ,~ - ( - y • YlI). For the correlator C(x, y), this reads explicitly 1 2-D

= Ix - yl 2-~ + [(xll- y,)2 + (~ + y•

_!

~

+

2 The first term on the r.h.s, is the usual bulk correlator. Due to H61der's inequality (Dieudonn6, 1971), and for xii - Yll = 0 the additional terms satisfy (x• + y•

~0_

1

- ~ ( 12x•

>0

D + 12y• 12-D) /

-- 0

for

1,

D = 1, D 1, C ( x , y) > Ix - yl 2 - ~ The case D < 1 is more subtle since when integrating over f dxl and f d~ the latter is only defined by analytical continuation. For a polymer, no additional contribution appears. This is understood by observing in analogy to a random walk that the correlator between two points x and y only depends on what happens between these two points. Let us now proceed to self-avoiding membranes. Our interest is to understand what happens when a dipole approaches the boundary. Due to power counting, the most relevant boundary operator is

=f

d~

1.

(8.27)

As already mentioned in the last subsection, this operator is marginal at D = 1, and relevant for D > l, necessitating a (multiplicative) renormalization only in D = I. In the language of q~4-theory, this renormalization shows up in the scaling exponent 17. All other boundary operators are irrelevant at e = 0.

9

9.1

The tricritical point

Introduction

In many experiments, one encounters tricritical behaviour. In the context of polymers, the tricritical point exists due to a competition between long-range attractive interactions, e.g. van der Waals forces and hard-core repulsion. At high temperature, repulsion dominates and the polymer is swollen. Lowering the temperature, finally attractive forces will collapse the polymer into a compact state. For a single long polymer, the transition between these two states occurs at the O-point, which represents a different multicritical state for the polymer (De Gennes, 1975) (and des Cloizeaux and Jannink (1990) for a general review). A similar transition is expected for membranes. As shown in Wiese and David (1995), two regimes can be distinguished, depending on the dimension D of the membrane and the dimension d of embedding space. For polymers and for membranes with D close to l, one expects an effective three-body repulsive interaction to be relevant to describe the O-point close to the upper critical dimension dc = 3D/(2 - D). For larger D, a modified two-body interaction, repulsive at short range, but attractive at larger range, is relevant to describe the O-point close to the upper-critical dimension, now given by d~ - 2(3D - 2)/(2 - D). The crossover between these two interactions occurs at D -- 4/3, d = 6. While the modified two-body interaction can be treated analytically at one-loop order, this is impossible for the three-body interaction. Numerical methods to calculate


2

2.0

l

1

1.5

t-

,

i

,

l

'-

365

i

6u = 0

(c) governed by . . . .

•h=0 '(b) governed by .,~ C~

j

1.0

(a) Gaussian 0.5 E u --0

0.0

~ 0

'

I

,

2

I

,

4

1 6

8

10

Fig. 22 Critical dimensions for various operators (solid lines) and phase diagram. The phase separatrices are the fat lines. the diagrams involved have been developed in Wiese and David (1995). We will focus here on the crossover between the two-body and the three-body interactions which can be studied via a 'double e-expansion' about the critical point D - 4/3, d = 6. As a result it is shown that depending on D and d, the O-point is described either by: (a) a Gaussian fixed point, (b) the three-body repulsive interaction, (c) the modified two-body interaction. The three corresponding domains in the twodimensional (d, D) plane are depicted in Fig. 22. The fat lines are the separatrices between these domains. The fat dashed line is a linear extrapolation of the oneloop result. This line separates the domains (b) and (c) indicating that the modified two-body interaction should be relevant to describe two-dimensional membranes at the O-point, independent of the dimension d.

9.2 Double ~-expansion The leading operator studied in the previous sections was the two-body interaction

ff xy

(9.1) xy

By fine-tuning its coupling such that the renormalized two-body interaction vanishes, one reaches the O-point separating the swollen from the collapsed phase.

366

K.J. Wiese

In the RG-analysis, the next-to-leading operators then control the physical behaviour. Analysing the canonical dimensions of all subleading operators, one finds two candidates: The bilocal operator ( - - A r ) g d ( r ( x ) -- r ( y ) )

=

(9.2)

~--~--~,

and the trilocal operator (r(x) - r(y))~ a ((x) - r(z))

=

(9.3)

~ . e / -,~

The Hamiltonian one has to study is therefore 7"~u'b--2-

-+- "~- uZulAeu

D

x

fff.L,

-~- bZblAeb

x y z

ff

:~ --' (9.4)

~"

x y

with the same normalizations as in the previous sections (see also Appendix A). Both couplings have canonical dimension (in momentum units) eb := [blzeb]u -- 2 D - 2 v ( d + 2 )

eu "= [ u u e " ] u = 3 D - 2 v d ,

=- 3 D - 2 - v d .

(9.5) They become relevant at the Gaussian fixed point when eu or eb respectively become positive. We have drawn the critical curves (eb = 0 and eu = 0) in Fig. 22. We see that at D = 4/3 the two operators . . . . and ~ interchange their roles. Below 4/3, _ ~ is more relevant, above 4/3, - --. At D = 4/3 and d = 6 both operators are marginal" we have the interesting situation of a system with two coupling constants. In contrast to standard perturbation theory, where in first order of the coupling /f,.~

I

/

constant the divergences are single poles, the leading singularity o f / ~ i ) ]+It" isa double pole, due to the sequence of divergent contractions

This prevents us from performing the renormalization in the standard way. Let us look at the problem from another point of view. As the modified two-point interaction . . . . renormalizes the elastic energy 89 2, perturbations in this operator have to be controlled by a small coupling b as is done in the Hamiltonian (9.4). On the other hand, the three-point interaction ~ renormalizes the modified two-point interaction . . . . via the contraction ,

((i~z>----.[~. ~;--)~. ~ - - .

(9.6)

2


367

Therefore there has to be a small parameter controlling the ratio of f l , , and . . . . . These demands are satisfied by replacing the three-point coupling constant u by bg. The Hamiltonian becomes o-

fff.

-+- + b g Z b Z g # e b + e g

7"~g,b--2--D x

+ bZbbeb

x y z

ff

-u

= '~ ="

x y

(9.7) Perturbation theory in b and g is now performed, by counting orders in b and g equivalently. They have canonical dimensions

eb "= [blxeb]u = 3 D - 2 -

vd,

eg "= [g#e*]u = 2 -

vd,

(9.8)

which will lead to poles in eb and eg in the perturbation expansion. Note that this situation with two completely independent couplings b and g and independent canonical dimensions eb and eg is quite unusual in standard perturbation theory. Here it is due to the fact that both the inner dimension D and the dimension of embedding space d can be chosen independently. The perturbative expansion will be performed in the vicinity of the critical point de = 6 and Dc = 4/3. The theory will be finite in terms of the renormalized quantities r(x) = z-l/Zro(x)

b = Z - d / z - 1 Z b I ~ - e b bo

(9.9)

g = z-d/Z+ 1Zg I #-eg go, where the renormalization factors have up to first order in b and g the form Z=

l+pg+q

b -

6g

6b

with constants p and q. We draw attention to the important point that pole terms in eb are always proportional to b as should be evident from dimensional arguments. The same is true for eg and g. It is also important to note that this would not be the case for the parametrization (9.7), nor if one there replaces u0 by u 2, as one might be tempted to do. In order to explicitly perform the calculations, we study as in Section 3.1 expectation values of an IR-finite observable 69. The following diagrams contribute to first order in g and b at D -- 4/3 (for more details cf. Wiese and David, 1995): -+-'"

({~)1--

= -1

(9.10)

b

:: =l = 1 b

(9.11)

368

K.J. Wiese

1

(9.12)

(9.13)

They determine the renormalization factors at one-loop order: Z = l+-

b

(9.14)

Eb

7b Zg = 1 + 43 gEg + _ _ 2eo

(9.15)

Z o = 1-

(9.16)

-3 g - ~ - - 4 Eg

Eb

As usual, we define the renormalization group fl-functions of the two couplings b and g as their variation with respect to the renormalization scale/z at fixed bare parameters:

fib(b, g) = U

b

(9.17)

g.

(9.18)

0

fig(b, g) = U

+1 0

Inserting the definitions of b and g, we get two coupled linear equations in fib and fig, which can be solved after some algebra. They lead to the fl-functions at one-loop order: fib(b, g) = - - e b b -

3 -~bg + 5b 2

11 3g 2 ~g(b, g) = - e g g + --~-bg + -~ .

(9.19) (9.20)

The scaling function of the field v(b, g) becomes 1 0 v(b, g) - v - - I ~ lnZ 2 011. 1 1 =-+-b. 3 2

(9.21)

369


A~./~

Cb _______

B

/

/12 / / c

D - 4/3

b

H d-6

G

D --~

t/D 0

~

f ~b

D

b

f

/

,ZI{

Fig. 23 The different domains in the d-D-plane (middle). diagrams are drawn around the central diagram.

9.3

~

o .,~------.-_

The corresponding flow

Results and discussion

The system of equations (9.19), (9.20) determines four fixed points in the (g, b) plane. The physical couplings must correspond to a repulsive interaction at short distance, hence to the domain (b > 0, g > 0). One of the fixed points is IR-attractive, one IR-repulsive and the other two have one attractive and one repulsive direction. For special values of the parameters eb and eg, fixed points

370

K.J. Wiese

may coincide. Passing through these special values describes the transition from one fixed point to another, resulting in an eventual nonanalyticity of the critical exponent v(b, g). We first list the different critical points visualized in Fig. 23. (PI) The Gaussian fixed point bc = 0 and gc = 0: it is stable for ea < 0 and eg < 0 . (P2) The fixed point bc - 0 and gc -- 4eg describes also a trivial theory, although gc has a nontrivial value. Indeed, regarding the Hamiltonian (9.7), we see that both interactions are renormalized to 0. Also the critical exponent v(b, g) equals that of the free (Gaussian) theory. The stability condition is eg > 0 and eb + eg < O. (P3) The fixed point bc = 89 and gc = 0" for this nontrivial fixed point only the modified two-point interaction plays a role. It is stable for eb > 0 and l leb > lOeg. (P4) The fixed point bc = 2(Eb d- eg) and gc -- ~ ( - 1 lt?b d- 10eg) is the most interesting one. Both couplings flow to a finite nonzero value. This point is stable for eb d- eg > 0 and l leb < lOeg. It corresponds to the fixed point for the case of a three-point interaction only in the limit of D -~ 4/3 from below. We will explain that in more detail below. Let us discuss the graphics of Fig. 23: We can distinguish eight different regions in the (d, D) plane around the critical point (dc = 6, Dc = 4/3), named A to H. The separating lines are: (1) eg = 0 separating D,E and A,H; (2) eo + eg = 0 between E,F and A,B; (3) eb = 0 separating F,G and B,C; (4) 1 leb = lOeg between C,D and G,H. The flow graphs in Fig. 23 correspond to these regions A to H, starting with region H in the upper left comer. Coming back to the general situation depicted in Fig. 23, the flows are such that: (1) In regions C, D and E, the Gaussian fixed point Pl or the pseudo-Gaussian fixed point P2 are IR-stable. The modified two-point and three-point interactions are irrelevant and the large-distance properties of the manifold at the (-)-point are those of a free Gaussian manifold.

2


371

(2) In regions A, B and H, the fixed point P3, described by the modified twopoint interaction only, is IR-stable. The three-point interaction is irrelevant and the modified two-point Hamiltonian (9.7) with g _-- 0, also discussed in Exercise 4, is sufficient to describe the large-distance properties of the manifold at the O-point through an eb-expansion. (3) Finally, in regions F and G the fixed point P4, which contains a mixture of three-point and modified two-point interactions, is IR-stable. As discussed in Wiese and David (1995), this fixed point corresponds to the limit D 4/3 for the three-point Hamiltonian, i.e. setting b - 0 in (9.4). Therefore the pure three-point Hamiltonian is sufficient to describe the O-point in an eg expansion. If one extrapolates these one-loop results, one obtains the picture already summarized in Fig. 22 for the O-point as a function of the external dimension of space d and of the internal dimension of the membrane D: the (d, D) plane is separated into three regions: (1) For D < 2 and d sufficiently large, both the three-point interaction and the modified two-point interaction are irrelevant. The O-point is described by the Gaussian model. (2) For d < dc - 3D/(2 - D) and D sufficiently small, the three-point interaction is more relevant than the modified two-point interaction and governs the O-point. (3) For d < d~ = 2(3D - 2)/(2 - D) and D sufficiently large, the modified two-point interaction is more relevant than the three-point interaction and governs the O-point. At one-loop order, the separatrix between these two domains is given by line number 4. (1 leb = 10eg, with eb and eg given by (9.8)), i.e. by the line d = 1 0 8 D - 138.

(9.22)

Thus, if we trust this picture far from the critical point (d = 6, D = 4/3), we expect that for two-dimensional membranes (D -- 2), the modified two-point interaction will always be the most relevant one to describe the O-point, even for d < 6. One also checks that the modified two-point interaction is less relevant than the standard three-point interaction to describe polymers (D = 1) in two dimensions (d = 2) at the O-point. Finally, let us stress that the analysis of the relevance of the two interaction terms leads to results drastically different from naive power counting or approximate schemes. Naive power counting predicts a separating line given by d =

4 2-D

(9.23)

372

K.J. Wiese

and that for D -- 2 the three-body interaction is always more relevant than the modified two-body interaction. Flory-type arguments give a separatrix d = 3D + 2,

(9.24)

while a Gaussian variational approximation leads to d = 6.

(9.25)

Both approximations predict that for D = 2 the three-body interaction is relevant for low dimensions d (d < 8 and d < 6 respectively).

10

Variants

10.1 Unbinding transition An interesting and much simpler variant of the self-avoiding membrane model (3.1) is a non-self-avoiding (phantom) membrane attracted to a fixed point

l i' ~(XTr(x))2 _ bZbl~e i

7-/pin[r] = 2 -- D

x

~d(r(x)),

(10.1)

x

where now

e=D-vd,

2-D v= ~ . 2

(10.2)

This was originally considered as a toy model in Duplantier (1989b) and David et al. (1993a,b), where it is used to develop the methods for the proof of perturbative renormalizability of the self-avoiding case. In recent time, it has also found a more axiomatic treatment in Cassandro and Mitter (1994) and Mitter and Scopolla (2000), where it is proven that there is a true fixed point close to the perturbatively obtained one. The model also appears in the context of wetting, reviewed in Forgas et al. ( 1991 ). The model (10.1) only necessitates one renormalization, namely for b (Duplantier, 1989b; David et al., 1993a,b). The elastic energy is not renormalized. Physically this is understood from the observation that ( 10.1 ) has also to describe the membrane far away from the binding point, and there clearly no renormalization of the elastic energy is required. Consider now renormalization. First note that the only divergences Come from approaching the 3-interactions. The leading term is (denoting 9 := ~d (r(x)))

(:::,.i:,:::,1.)-t

,x- y)]-d/2

(10.3)

2


leading with b0 = b Z b I x ~ and Zb = 1 - ~ ( : : ~(b)

"= Ix

+l

'( ::': I l

b = -eb-

,, ' 9

- :/

o

373

.... I')r to the one-loop/7-function

+ O(b 3)

2

'

(:/:/il.iil}l~l = 1.

(10.4>

The non-trivial fixed point lies at e < 0 and is repulsive. The binding of the membrane is described by the derivative w of the ~-function at the nontrivial fixed point b* with ~(b*) - 0:

d

w := d-~f(b)

I

-- e.

(10.5)

b=b*

This means that near to the critical point Ib - b*l ~ Ixbo=ftro]fV[o]fV[r

O e-J[r~176

(11.5)

where the normalization is such that (1)bo -- 1. The derivation and interpretation of (11.4) is simple: the path integral over ?0(x, t) enforces the Langevin equation (11.1) to be satisfied, and the term proportional to ~.2 reproduces the noise (11.2). Since J is quadratic in ~, the corresponding integral can still be performed, leading to

if f[

,.7"[ro, ?o] = 2 - D

x

?0(x, t)

(;

,7-/ ) _

t)2]

0(x, t) + ~.o(2 - D) 6r0--~] t)

t

(11.6) Introducing now renormalized fields and couplings ?o = v/-}?, ;Co = ~.Zx and b and r as in (3.114) yields J[r,;]--

2_ D x

+)~blz e

l

Z~,Zb

ssf

~

-_.

(11.7)

xyt

The symbols for the local operators are in the same spirit as in (3.10) and (3.11) defined as = ~(x, t ) i ( x , t),

~

= ~(x, t ) ( - A ) r ( x ,

t),

~

= ~(x, t) 2,

(11.8)

where a wiggly line always indicates a response field. The interaction is 9= 2 [ 7(x, t)(ik)e ik[r(x't)-r(y't)]. L/ k

(11.9)

380

K.J. Wiese

These notations are collected in Appendix B. Note that in (11.7), the response field ? could also be integrated over and thus eliminated. This is sometimes done (Zinn-Justin, 1989; Janssen, 1992). However, there are two disadvantages of this procedure: first of all, this would generate a term quadratic in &7-t/&r(x, t), rendering the analysis of divergences rather tedious. Second, the field ?(x, t) has an immediate physical meaning. To see this, add in (11.1) a force F(x, t). This yields an additional term proportional to fx,t F(x, t)F(x, t) to the dynamic action (11.7). The response of the field r(x, t) to a small applied force F(y, t') therefore is

(r(x, t)F(y, t')}0.

(11.10)

It is called a response function. Perturbation theory is now performed by expanding about the Gaussian theory. We use the free propagator (response function) R and correlator C in position space

C(x, t) :=

' 0 at least for small 3. Since the hydrodynamic interaction is long-range, as discussed in Section 3.9, it is not renormalized. Accordingly, the exponent identity in (11.41) is replaced by 2 [F]f + (2 - d)[r]f + 2 0 [x] + [M]f = 0. (11.53)

2 Polymerizedmembranes, a review

389

Together with (11.40) this yields for the dynamical exponent z*

Z~-d.

(11.54)

The stability condition for the fixed point 17 -- 0 is therefore < ( d - 2 ) ( v * - v).

(11.55)

At one-loop order, the separating line is e a = ( d - 2)~.

(11.56)

Numerical evaluation yields the thin line separating the regions with z* = d and z* = 2 4- D/v* in Fig. 25. There is, however, a priori no reason to trust this estimate for membranes, i.e. s = 4. We know, however, that in any dimension the Flory estimate VFlory = (2 + D)/(2 + d) is quite a good approximation for v* in the fractal phase, for polymers as well as for membranes (David and Wiese, 1996; Wiese and David, 1997). Inserting this relation we obtain for the separatrix d = 2(D + 1).

(11.57)

In Fig. 25, this is the fat line between the regions with z* -- d and z* = 2 4- D/v*. Let us stress that we only use the Flory approximation to estimate v*, but not any of the systematically wrong assumptions which have to be used to derive it. Another possibility to get (11.57) is to require that the value of z* is continuous on the phase separation line. The equivalence of the results obtained by the two methods is a consequence of the general structure of the renormalization group. We can also give a rigorous bound for the phase separation line. As v* < 1, hydrodynamics is always relevant for d < D + 2. 12

(11.58)

Disorder and nonconserved forces

In this section, we want to study the dynamics of polymers and D-dimensional elastic manifolds (0 < D < 2) diffusing and convected in a static random flow. The velocity pattern of the flow ~(r) is constant in time and leads to convection of the polymer in addition to diffusion. We are interested in the general case of a nonpotential flow: the extreme example is the hydrodynamic divergenceless flow, with V 9~(r) = 0, but mixtures of potential and divergenceless flows are also considered. See Fig. 26 for a visualization. This is a generalization of the Rouse and Zimm dynamics discussed in the last section. This study is interesting for several reasons: technically, disorder can be treated with the help of the same tools as self-avoidance: the nonlocal term is nothing but

390

K.J. Wiese 9"

~..... "

~

'

'

/

"

..-/b/,~..c_i

.... ;

=...,

/,

'W

,,~..,~,:;..,~ ... \'

/

'" .:

;:''~

',

.,.

.X.' r: i

/

-.:--.-..',:::

e ","

-

"

,s.",//..... "

.-

-..'.~

, 'X', \ '

,""~"

.....

-.. \

, ,.' , . . ' :.

,!

i....

..

..,', .- ,,".,'..:,"--"-.'::.:.-.--*;,. ,."L ! ':

'..:/':~,,'./~;9',,,'" " ",-="...--""q~' ,.',,,,~.:,~.4_:.;j. :,_~ ::~:~..x.i:' ,,-:. /,/~Ill"

//-

, i,,

//"1'~

'/'.,",,'/-:;~< "'

/.-

/.-~.-

-~ ("u/t~

~.,/',4.~4-

.

"

7,

', ' '

:..

.. ,,, .

-":"', %

A .

,

. .......

,,

:

'..,

".~

"

" '"'

.- - .

;

.w

" ," .

...?,.'-~_ ,. :,,'""

.

9'

" '" - / ~ / . , ' .

:k-'~

~,,,":,~-',,.

.-/

.,.t ; '

-',"

~I'

~h'

; "~" I i,' ,,~ :/ :' I ' ~ " " '

' ;'~-;:~xtti ",~;:'"

/

y-" /

"'\ .

-\

",..~,,,; ~ ;,

--"~"-' 0, i.e. 4+2D

d < dc(D) = 2-------~ .

(12.16)

As we have done for self-avoiding membranes, we want to renormalize the model within an e-expansion. To this aim, renormalized fields and ~. are introduced by setting ro = q/-Z r

F0 = v / ~ F

(12.17)

~.0 = Z x ~ .

It is more complicated to introduce renormalized couplings gL and gT. Set

i

_--,,v,,J(pi~gL.+. pTjgT),~.211,e - i.w,.e.

_-v,~J(pLJg~ + pTJg T) ~.2.

(12.18) This equation is to be understood such that quantities on the l.h.s, are renormalized, and those on the r.h.s, are bare. The noninteracting (gL = gT = 0) theory is the same as in (11.6), such that also the free response R (x, t) and correlation functions C (x, t) are the same as in (11.12) and (11.11).

12.3

Fluctuation-dissipation theorem and Fokker-Planck equation

Before embarking on the analysis of divergences, we have to clarify an important point related to the fluctuation-dissipation theorem (FDT). The latter states that as

2

395


long as all forces in (12.1) can be derived from a potential, then the full correlation and response function are related by Janssen (1992)

|

-~ [r(x, t) - r(O,

0112

--

Z)~ r(x, t)?(O, O) .

)~

(12.19)

This relation is violated in the presence of nonpotential forces, for our model in the case of gT # 0. Also note that in the case of purely potential disorder, the equation of motion (12.1) can be recast in the form (11.1). The latter implies that trajectories of the Langevin equation sweep out configuration space (as long as the dynamics is ergodic) and that the probability to find the membrane in a given configuration r(x) is given by e -7t[r~x)]. Equal time expectation values (the 'statics') are simply obtained by studying the partition function with the weight e -~[r~x)], as was done in Sections 3 to 9. This is proven by going from the Langevin equation (12.1) to the (functional) Fokker-Planck equation (suppressing all indices '0' for bare quantities)

d

1 d 79[r(x), t] ~. dt

8

Z ~r i(x) { (Ari (x) -4- (2 - D)Fi[r(x)]) 79[r(x), t]} i=l + ( 2 - D)SD .

d~l ( ~r i8(x) )2 79[r (x ) , t ] .

(12.20)

Supposing that Fi[r(x)] can be derived from a potential, 8

Fi[r(x)] -- - ~ V [ r ( x ) ] , 8ri(x) then (12.20) can be rewritten as 1 d P [ r ( x ) t] - (2

)~ dt

D)SD i ~ l "

8r i (x)

7:'Jr(x), t]

8r i (x)

7-/[r(x)]

+Sri(x)79[r(x),t]

, (12.21)

where 7-/[r(x)]- f

2 - 1 D + + V[r(x)].

Assuming that equilibrium is reached (d/dt79[r(x), t] - 0) the solution of (12.21) reads 7~[r(x), t] ~ e -7-t[rCx)l, (12.22)

396

K.J. Wiese

which should be demonstrated. The most important consequence of the above demonstration is that in the case of purely potential disorder, no nonpotential (transversal) disorder can be generated.

12.4 Divergences associated with local operators We now analyse the model, using the techniques of the multilocal operator product expansion (MOPE) as explained in Sections 3.5 and 3.6. In order to simplify notations, we shall in the following suppress the factor of k, i.e. set

kt ~ t.

(12.23)

This is not problematic, as k always appears with time. At the end of the calculations one has to replace t by kt which is necessary in order to get the renormalization factors correct. The first class of divergences stems from configurations where the two endpoints of the interaction are approached. The interaction is

L

gL

T

gT = f

~i (X, t)eik[r(x't)-r(y't')]r j (y, t')

k

x (PLJ(k)g L + PyJ(k)gT).

(12.24)

In order to extract the divergences for small x - y and t - t', the first possibility is not to contract any response field. We then start by normal-ordering the r.h.s. of (12.24). Within dimensional regularization, "eikr(x't): = e ikr(x't) and we can use the identity (analogous to (3.69))

.eikr(x,t)..e-ikr(y,t'). = .eikr(x,t)e-ikr(y,t'). e-k2C(x-y,t-t').

(12.25)

Expanding the normal-ordered vertex operators on the r.h.s, for small x - y and t - t', the leading contribution is

I e -k2C(x-y't-t'),

(12.26)

yielding the first term in the short-distance expansion of (12.24) (for the normalization of the k-integral cf. (A.5) ff.)"

e k2C(x-y't-t') "ri (x, t)rJ(y, t')" (PLJ(k)g L -b PTJ (k)g T) k 2'

,.,2 )2( gT ( 1---~l) + g -c') t,,,2 ~ C(x-y,t-

2

397


+ subleading terms

-

~)C(x-

y, t - t') -d/2 at- subleading terms. (12.27)

The second contribution to the normal-ordered product of (12.24) is obtained upon contracting one response field. Due to causality, this must be the field with the smaller time argument. For simplicity, let us take t > 0 and put y = t' - 0. By the same procedure as above, we obtain the contribution:

f

.~i (X, t)e ik[r(x't)-r(O'O)] " R(x, t)(ik)Je -k2C(x't) (PLj (k)g L + P;J (k)gT).

k

(12.28) The next step is to expand :eik[r(x't)-r(O'O)l: about (x, t). (It is important to expand about (x, t) as otherwise 7(x, t) has to be expanded, too.) This expansion is

(

:e ik[r(x't)-r(O'~

1

= 1 + (ik) l t i"l (x, t ) + ( x V ) r I (x, t ) - - ~ ( x V ) 2 r I (x, t) +subleading terms.

)

(l 2.29)

Upon inserting (12.29) into (12.28) and integration over k, only terms even in k survive. We can also neglect the term linear in x, which is odd under space reflection. The remaining terms are

f

9?i(x,t)

k

(

til(x,t)-

-j(xV)Zrt(x,t) '

)

9 (ik)t(ik)JR(x,t)e-kZc(x,t)

x (PLj (k)g L + PTj (k)g T)

---- -1R(x'2

t)C(x, t) -d/2.1

t~

+ ~

~

gL.

(12.30)

For the contribution proportional to , ~ , we have retained from the tensor operator f(x, t)(xV)2r(x, t) only the diagonal contribution i f ( x , t)x2(A)r(x, t), which is sufficient at one-loop order. For the subtleties associated with the insertion of this operator at the two-loop level cf. Section 6. Using the perturbative FDT, (11.15), this can still be simplified to

O_.O_C(x,t)-d/2 Ot

~-

+ ~x. ~

) gL.

(12.31)

Equations (12.27) and (12.31) contain all possible divergent terms in the shortdistance expansion of (12.24) and all terms which have to be taken into account in one-loop order. Notably, due to causality, no term independent of ? appears.

398

12.5

K.J. Wiese

Renormalization of disorder (divergences associated with bilocal operators)

In analogy to (3.79), there are also UV-divergent configurations associated with bilocal operators, which renormalize the disorder, and which are depicted in Fig. 28. Up to permutations of the two interaction vertices, there are two possibilities to order their endpoints in time, namely Dl and D2 in Fig. 28. We first calculate Dl, starting from

ri(y,t)

f

(gTPTJ(k)-k-gLPLJ(k))eik[r(y't)-r(x'O)]rJ(x,

O)

k X ~l(yt, t - - c r ) f ( g T p l m ( p )

+ gLpLm(p))eip[r(y"t-a)-r(x"-r)]rm(xt,-'t').

p (12.32) For small x - x', y - y', r and or, with r, a > O, there is one contribution for the renormalization of the interaction. First, due to causality, ~l (y,, t - or) and ?m(x',-r) have to be contracted with a correlator field in order to obtain two response fields at the end. Then the short-distance expansion for nearby vertex operators reads:

eikr(y,t)eipr(y', t-tr) = .eikr(y,t)..eipr(y',t-a) : = :eikr(y,t)eipr(y',t-tr) 9 ekpC(y-y',tr) .~ .ei(k+p)r(y,t). ekpC(y-y',cr) = ei(k+p)r(Y ,t) ekpC(y-y',cr) (12.33) where the first and last equality are due to analytical continuation (see Section 3.6). Analogously, we find for the other pair of points

e-ikr(x'~ -ipr(x''-r) ,~ e -i(k+p)r(x'O) e kpC(x-x''r).

(12.34)

This yields up to subleading terms:

ff

~i (y, t)Tj (x,

O)ei(k+p)Ir(y't)-r(x'O)]

k p • (gTPTJ(k) -~-gLpi~(k))(gTelTm(p)-~-gLeLm(p)) x ( i k ) t ( - i k ) m R ( y - y', c r ) R ( x - x', r ) e kp[c(y-y''tr)+c(x-x''r)l

(12.35) In the next step, first k and second p are shifted: k k

~ k-p,

P

~ P+-Z.

z

(12.36)

2

399


t(

t/'\

l--O"

time

t--t7

time

o(

/

0k

--l"

Fig. 28

The diagrams

l"

D1 (left) and D2 (fight).

The result is

fri(y,t)FJ(x,O)eik[r(y't)-r(x'O)][f(gTPTJ(P--k/2)+gLeLJ(P--k/2)) k p (gTp~m(p + k/2) + gLplm(p + k / 2 ) ) ( k / 2 - p)l(k/2 - p)m R(y - y', a)R(x - x', r)e (k2/4-p2)(C(y-y''e)+c(x-x''r))

J .

(12.37)

To compute the correction proportional to the disorder, the expression in the rectangular brackets is expanded for small k. As the integral has a well-defined limit for k ~ 0, convergent for d > 2, no term of the form k ik j /k 2 can be generated. The leading term of the above expansion is then (the p-integral being defined in (A.5) ff.)

_,,,,,,y f ( g T ( , i y _ p - 2 p i p y ) + g L p - 2 p i p y ) g L p 2 P xe -p2(C(y-y''a)+C(x-x''r)) R(y - y', a)R(x - x', r)

~_ ~v~ x d2 =',,,,,~

=v~ ( g T ( 1 _ l ) ff_g Ll (C(y - y' , o) + C(x - x'

--,~ ( g T ( 1 - - d ) + g C l

c

r)) -d/2-1 C(y d ) gL

y

,

o)C(x

- x

, r)

4OO

K. J. Wiese

2 3 3 ( C ( y - y', or) + C ( x - x', r)) -d/2+l . d-23r 3o

(12.38) Note also that we have used the (perturbative) FDT, (11.15), for the first transformation. The second possible way to do the contraction (see D2 in Fig. 28) is performed similarly. The leading term is -1

~

)2

--d(gL

2

3

3

d-2OrOcr

( C ( y - y ,l c r ) + C ( x - x ,t r ) )

-d/2+l

.

(12.39) Note that there are two other possible contractions, which can be obtained from Dl and D2 by replacing r and tr with - r and - t r respectively. Together they add up to _~gLgT

d-

4

2

(1_

1)3 3 (C(y-y',~r)+C(x-x' d ~r-r ~a-a

' r))-d/2+l

.

(12.40) This result is remarkable in several respects: first, the contribution to the renormalization of disorder from the disorder-disorder contraction is isotropic. We will see that this stabilizes the isotropic fixed point. Second, there is no divergent contribution in the purely transversal or purely longitudinal case at one-loop order. (Note, however, that there are finite contributions in the transversal case, see appendix C of Wiese and Le Doussal, 1999).

12.6

The residues

The dimensional regularization parameter is e = 2 + D-

yd.

(12.41)

We now follow the general procedure outlined in Section 3.8. In the context of dynamical critical phenomena, we both have to put a cut-off L on the space integration and a cut-off ~.L 2 (where we did set ,k = 1) onto the time integration. Note also that we have chosen normalizations and notations as in the rest of this article, which differ from those in Wiese and Le Doussal (1999). The term proportional to ~ is L

= f0

L2

0

2

401


1.35

1.3

1.25

1.2

1.15

I.i

J

1.05 I

0.5

1

.

,

1.5

2

D

Fig. 29 The function I(D).

L

L2

fdxx~

t)-d/2

X

0

0 oo

-- ~

dt C(I, t) - a / 2 + finite,

(12.42)

S 0

where in addition d has been set to dc, defined by e ( D , dc) - O. The residue is oo

f dt

t,

0 =: I (D).

(12.43)

We did not find a closed form to calculate this. Using the approximation for the correlator 2-D

C(1 t) ,~, ' F(-~)

Itl +

4

'

(12.44)

402

K.J. Wiese

which is exact* for t = 0 and t --+ cx~, we obtain for the integral 1 (D) 2

1 r(_~)r ~ i(n).

(12.45)

I (D) = 2D

Were the approximation in (12.44) exact, I ( D ) would equal 1. This is not the case, and the value for i ( D ) obtained from numerical integration is plotted in Fig. 29. Proceeding equivalently, the diagram correcting the friction coefficient is calculated as oo

....

~

=-~

dt C ( l , t ) - d / 2 0

1

(12.46)

= --I(D).

d We obtain the interesting relation

which can be used to simplify the RG calculations. It is a reflection of the FDT, valid in perturbation theory, and in the longitudinal case in the full theory. The elasticity of the membrane is renormalized through L

L2

0

0 L

L2

=

xD

dt

--C(x, Ot

2dD o

t) -d/2

o L

_

=

1 [ d X x D + 2 C ( x , 0) -d/2 -~- finite 2dD J x o 1 Le

t- finite.

2dD

(12.48)

e

The residue is therefore given by "~

l=

- 2d----D"

'

(12.49)

'*Note that this approximation is better than the one used in Wiese and Le Doussal (1999), implying

that [(D) is closer to 1.

2


403

Finally, the diagram correcting the disorder is evaluated as follows:

=),.:=

=f~xOf~y~ fd. 4 L

L

L2

L2

(, ,)

d-2

0

0

0

0

0 0 (C(x, r) 4- C(y, O')) -d/2+l Or 0or

4(1 ')fdxx~176 L

m x

_

d-2

L

d

(C(x, O) + C(y, 0)) -d/2§ 4- finite terms

By y

0

0

oo

4 d-2

( 1 - d1 )f0

d X x ~ (C(x, O) + C(1 0)) -a/2+l ~ + finite terms x ' e

"

4(l-l)f-~ (x)

d

--

2

-d

XD

(

X 2-D

) -d/2+lLe

4- 1

4- finite terms 8

0

=

4(1

-

d - 2

1)1 d

F2(2_-~DD)L e 2 - D 1-'(22_-~DD) e

~- finite terms.

(12.50)

The residue is given by the expression l -,

-

~

~

4(1_ .-2

1) 1 F2(2_~DD) a (2-,,~ ~(~~

(12.51)

The factor involving the F-functions is familiar to the renormalization of the interaction for self-avoiding membranes (see (3.111)). 12.7 Results and discussion

In this section we analyse the general renormalization group flow given by the two /~-functions for longitudinal and transversal disorder. We identify the fixed points and compute the critical exponents at these fixed points. To present conveniently the analysis below we introduce the following notation for the three independent coefficients, computed in Section 12.6 as

. (12.52)

404

K.J. Wiese

C:= -d(2-

D)(~,,

"~'+-l"

The explicit expressions for these coefficients are given above in (12.43), (12.51 ) and (12.49). They are non-negative. We are now in a position to define the el-functions, quantifying the flow of the renormalized theory upon a variation of the renormalization scale, through r

gT) := #

(12.53) 0

r

gT) := #

(12.54) 0

In terms of the three coefficients .,4, B and C, they read: /3L(gL gT) _

2C(gL) 2 _egL + -21 ( ( 1l- ) -~ (d + 2)A - 13)gLgT - d -2---d-

flT(gL, gT)=--egT---~

1(

(12.55)

13-k- d -d 2 C) gTgL + ( l - d ) d +2 2 "A(gT)2" (12.56)

Let us now discuss possible observables. As in (3.116) and (11.43), we study the roughness exponent v* and dynamic exponent z*, which are defined as ([r (x, t) - r (x', t)]2)

~

IX

--

X' 12v",

([r(x,

t) - r(x, t')] 2) ~ It - t'[ 2/z*. (12.57)

We assume scaling behaviour ([r(x, t) - r(0, 0)12) ~ Ixl 2~* -n(t/xa*)

(12.58)

with 6" - v'z*. Let us note that in the literature (and also in the original article (Wiese and Le Doussal, 1999)) different conventions are chosen; they are obtained by replacing v* ~ ~*, z* ~ l/v* and 6" ~ z*. The drift velocity under a small additional applied force f in (12.1) is

v ~ {r(x, t))/t,

(12.59)

v ~ f~*

(12.60)

and scales as at small f , with ~.

=

6" -- v*

2 - v* +/3"

> 1,

(12.61)

2


405

gL

v

Fig. 30 RG-flow diagram for SR-disorder. The physics is controlled by the fixed point I* at gT = gL.

indicating trapping of the membrane by the flow./~* is the anomalous dimension of the elasticity of the membrane. For a derivation, see Wiese and Le Doussal (1999). The expressions for the exponents then read to lowest order:

v(gL, g T ) =

2

+ -~-

1 -- ~

, A - 2---~C

1

(12.62)

a(gL, gT) _ 2 + ~(,A - C)g L

(12.63)

/~(gL ' gT) _ _ g L 1d C "

(12.64)

Note that in the end the exponents will depend on the amplitudes only through their ratios B / A and C/A which are universal. For an isotropic manifold, we find from the RG-equations that the RG-flow is as depicted in Fig. 30, with the following fixed points:

(1) Gaussian fixed point. The Gaussian fixed point at gL = gT = 0 is completely unstable for d e=2+D-~(2-D)

>0.

(2) Potential disorder. The line gT = 0 is preserved under renormalization, and we find a flow towards strong coupling. This problem describes the dynamics of an isotropic manifold in a long-range correlated random potential (short-range correlated force). The statics of this problem has been much

K. J. Wiese

406

studied and is indeed expected to be described by strong disorder. (For examples see Cates and Ball, 1988; Nattermann and Renz, 1989; Machta and Kirkpatrick, 1990; Le Doussal and Machta, 1991; B latter et al., 1994; Ebert, 1996). (3) I s o t r o p i c d i s o r d e r f i x e d p o i n t . Note by taking the difference of (12.55) and (12.56) that the line gT = gL is preserved by the flow; we thus find an isotropic fixed point at gL = gT = g , . =

+ O(e2). (12.65)

2ed

(d - 1) (d + 2).,4 - d B - (d - 2)C We have checked numerically that the denominator of (12.65) is always positive, which is necessary for this fixed point to be stable and to be in the physical domain. We have also checked numerically that this fixed point is completely attractive, and its domain of attraction covers all perturbative situations except the potential case gT = 0. It also controls the line gL = 0 (except for D = 0) and describes the large-scale behaviour of an isotropic manifold in a random short-range force flow, see appendix C of Wiese and Le Doussal (1999). The critical exponents at this fixed point, v* and 3*, defined in (12.57) and (12.58), are with the same diagrams v* -- v(g*, g,) = 2 - D I ((d - 1) .A - C) e + O(82). 2 (d - 1) (d + 2).,4 - d13 - (d - 2)C (12.66) The first coefficient in the numerator, .,4, is positive as before, since it arises from the upward corrections to the temperature. However, the elasticity is also renormalized upwards (the polymer tends to shrink to take advantage of favourable regions). This produces the second coefficient - C , which is negative. The competition between the two opposite effects finally gives a positive sum and the membrane is stretched. Also note that the e-correction vanishes like l / d for D --+ 2. A similar formula is valid for the exponent 3: 3* = 3(g*, g*) = 2 +

2 ( . A - C)e (d - 1) (d + 2).,4 - d 1 3 -

( d - 2)C

+ O(e2). (12.67)

The e-correction is always positive, but vanishes like l i d 2 for D ~ 2. The elasticity is also renormalized upwards and gives rise to a nontrivial exponent/3*" [:3* = f l ( g * , g*)

=

-2Ce +0(82). (d - 1) (d + 2),,4 - d B - (d - 2)C

(12.68)


2

407

Table 1 Results for isotropic polymers and membranes at the isotropic fixed point, SRdisorder. Results for v are obtained from the extrapolation of vd, and are the most reliable ones. One observes a nice plateau in extrapolations for 1/z, but no corrections to the Flory approximation can be deduced. The exponents a and ~ do not allow for direct extrapolations, but are always corrected upwards. Results for fl are significant for polymers. For membranes only a bound seems to emerge from extrapolations.

Polymer

Membrane

d

v*

2 3 4 3 4 6 8 20

-~ 1 0.8 0.67 0.8 0.68 0.5 0.4 0.2

3" > > > > > > > >

2 2 2 2 2 2 2 2

1 --

fl*

0.5 0.40 0.33 0.40 0.33 0.25 0.20 0.09

-0.08 -0.06 -0.03 -0.2...0 -0.2... 0 -0.2...0 -0.2...0 -0.2...0

Z*

~* > > > > > > > >

1 1 1 1 1 1 1 1

The other exponents can be obtained as 1/z* - v*/a* and q~* = (a* v * ) / ( 2 - v* + fl*). One notes that in the limit D ~ 0 one recovers the results for the particle (Fisher et al., 1985; Bouchaud et al., 1987; Honkonen and Karjalainen, 1988) for 1/z* and ~*. For a polymer, D = 1, the disorder becomes relevant below d = 6 and setting D -- 1, we find that the above results yield v* = 0.5 + 0.130792e,

3" = 2 + 0.03996e,

fl* = - 0 . 0 1 5 4 4 6 e (12.69)

with e = 89 0.0369373e. e = 1.5) that v* = 0.76, 6"

- d), as well as l * / z = 1/4 + 0.060401e and 4~* = 1 + The most naive extrapolation to d = 3 would be (setting v* = 0.70, 3" = 2.06 and/3* = - 0 . 0 2 3 and in d = 2 that = 2.08 and 13" = - 0 . 0 3 1 .

One can try to obtain more reliable estimates for these critical exponents from expressions (12.66), (12.67) and (12.68) for polymers (D = 1) and membranes (D = 2) in three or two dimensions by optimizing on the expansion point. This is a tedious task since ~ is rather big. The numerical values obtained by the methods detailed in Section 7 are not very precise, as we could not find a combination of the exponents and D or d, which in suitable extrapolation variables builds up a nice plateau. Different extrapolation schemes yielded strongly varying results. Some indicative values obtained by this methods are summarized in Table 1.

K. J. Wiese

408

Since v* seems to increase rapidly as d decreases, an interesting question is whether there is a dimension dt below which the polymer will be fully stretched (v* = 1). The result of Table 1 seems to indicate that dt could be around 2. Our calculations are not precise enough to decide on whether the polymer is already overstretched or not in d = 2 but that would be an interesting point for numerical simulations. (4) Transversal disorder fired point. The transversal fixed point (gL = 0) is at gL = 0,

gT =

2e

+ O (e2),

(12.70)

where the diagram is given in (12.52). It is unstable towards perturbations of gL. For the critical exponents v* and 3", we recover the Flory result at one-loop order: v*

--

2-D 2

3" = 2

t-

e d+2

+ O(s2).

+

O(e 2)

(12.71) (12.72)

As was discussed for the directed case, this result was also found to be true for the particle (D = 0) to all orders, but for D > 0 finite nontransversal terms are generated in perturbation theory which will drive the system towards the isotropic fixed point discussed above (see appendix C of Wiese and Le Doussal, 1999).

12.8

Long-range correlated disorder and crossover from shortrange to long-range correlated disorder

It is equivalently interesting to study long-range disorder. We have discussed in Section 3.9 that LR-disorder can be treated on the same footing as short-range disorder, with one important difference: there are no disorder corrections, i.e. ( ~ ['~"~ ~)L = 0. We have discussed that under these circumstances there are additional exact relations among the scaling exponents, which sometimes even allow to determine them. Since under renormalization long-range correlated disorder always generates a short-range correlated one, short-range disorder may be relevant in situations where power-counting indicates that long-range disorder dominates. As discussed in Wiese and Le Doussal (1999), the crossover is very complicated and contains some unexpected features. The most striking one is that under certain circumstances, the canonical dimension of an irrelevant subdominant operator serves as expansion parameter instead of the canonical dimension of the leading operator, as is usually the case. Due to a lack of space, we refer the interested reader to Wiese and Le Doussal (1999).

409


0 (N)-field theory

4, no solution for the instanton with finite action exists. It is interesting to give a physical interpretation of the instanton for the Edwards model, since this interpretation is the same for membranes with D # 1. Let us first recall the standard interpretation of the instanton for the LGW model with action (14.37), i.e. negative q~n-coupling. The classical false vacuum ~ ( r ) = 0 is separated from the true vacua qJ(r) = 7:cx~ by a finite barrier. The instanton solution ~0 describes a metastable droplet of true vacuum (with ~0(r) # 0 inside the droplet) in the false vacuum, which is on the verge to nucleate. Indeed, if the droplet is slightly larger, the positive surface energy dominates and the droplet shrinks and finally vanishes, while if it is slightly smaller, the negative volume energy dominates and the droplet expands. Consider the energy density $[ V ] given by (14.17). It corresponds to the total free energy of a polymer globule trapped in the potential well V (r) < 0, where this effective potential results from the attractive two-point interaction between elements of the polymer (since we are at negative coupling, b < 0). To see how ,S varies with the average radius of gyration of the polymer, it is convenient to

2


443

consider the following scale transformation on V: 2D

V(r) ~

Vx(r) = Xzzt~ V(~.r).

(14.41)

Simple dimensional analysis shows that under (14.41) s

f

~-c-o g[V],

V 2 __, ~r:7~

(14.42)

(here D = 1 and e = 2 - d/2). As long as e < D, and for large ~., i.e. when shrinking the polymer globule, it is the first term s < 0 which dominates and the total free energy S becomes large and negative; while for small X, i.e. when expanding the globule, it is the second term on the r.h.s, of (14.17) which is larger than 0, and which dominates. Thus in this mean-field picture, i.e. neglecting thermal fluctuations around the instanton, large globules tend to expand, while small globules tend to collapse. This has a simple physical interpretation: the polymer trapped in its own potential is subject to two opposite forces, (i) attractive forces between its elements which would like to make the polymer collapse, (ii) entropic repulsion which exerts a pressure on the well and would like to expand the polymer (until it becomes a free random walk). What our calculation implies is the simple fact that for large radius (i.e. small X) entropic repulsion dominates, while at small radius (large X) attraction dominates and the polymer collapses. Thus the instanton solution describes a polymer with attractive interactions on the verge of collapsing into its dense (and most stable) phase; this is similar to the instanton in the LGW theory which describes a bubble of true vacuum on the verge to nucleate and to destroy the false vacuum.

14.3 Gaussian variational calculation For D # 1 (and in general for 0 < D < 2 noninteger) we know of no exact method to calculate the instanton. A simple and natural approximation is the variational method, i.e. the Hartree approximation. To evaluate the free energy density g[ V] of the free, i.e. noninteracting membrane in a potential V, and described by the Hamiltonian

7-Iv =

f (' dDx

~(Vr) 2 + V(r)

)

,

(14.43)

we introduce the trial Gaussian Hamiltonian

7-[var =

if' dDx

f

=

dDk

(

dDy ~ r(x) K (x - y ) r ( y )

1

yo 5

(14.44)

444

K.J. Wiese

where- denotes the Fourier transform. The free energy for the trial Hamiltonian is

~;1 In [f 7~[r]e-~"a~]

Ga,= = ~

(2n,)o In

K(k)/k 2 ,

(14.45)

D[r]

and the factor of l / k 2 comes from the normalization of the measure taken such that ,5'[V = 0] = 0. V is the total volume of the membrane. The Hartree-Fock approximation amounts to replacing g[ V] by the best variational estimate s V] 1

,5'[V] _< Gar[V] = Gar + ~

( ~ v - 7"/var)vat.

(14.46)

{ )var denotes the average with respect to the trial Hamiltonian 7"(var and one must look for the trial Hamiltonian 7"/var (i.e. the kernel K) which minimizes 8var[V]. Denote by f' (p) the Fourier transform of the potential V (r). Since the variational Hamiltonian is Gaussian, it is easy to compute the second term on the r.h.s, of (14.46), V -l (7"/v - 7"(var)var in the infinite volume limit:

1[

(V(r(0))var + ~ =

f

f d~ K(x)(r(x)r(O))var]

(Vr(0))Z)var -

ddp ~, (p) (eipr(0))var + (27r)d

f (~-'~TD d~ k2-k(k)(~(k)~(-k)) vat

ddp (/(p) [p2fd~ 1 ] dfdOk( k 2 ,) _f (27r) d ~- (27r)D/~(k) 2 (27r)D exp --

+

_

K(k)

(14.47) Combining (14.17), (14.45) and (14.47), we finally obtain the variational estimate for the total energy of the instanton

Svar[V]- Cvar[V] + ~ =

f(

driP (2rr)a

+7

r V(r) 2

9 ( p ) exp

(2Jr) o

In

both

(p2fa~ 1) -~- (~-~b Kik) + ~V(p)(,'(-p) --

4

/((k)

1 .

(14.48)

We now extremize (14.48) with respect to K (variational approximation) and with respect to V (to obtain the instanton solution). Extremizing with respect to

2


k (k) yields the equation K (k) = k 2

1

ddp p2 ~,(p) exp -(2rr) d T

(2yr) D K(k)

,

(14.49)

which implies that the variational Hamiltonian depends just on a mass mvar" 2 r. /~ (k) = k 2 + mva

(14.50)

Extremizing (14.48) with respect to f' (p) gives -var [ p2 ] Vinst (p) -- - exp - - ~ - A

(14.51)

with

f

dDk

A --

1

(2rr) ~

D-2 1-'(1 -- ~ ) = mvar (47r)D/2 .

(14.52)

F is Euler's Gamma function. Thus, in the variational approximation the instanton potential is Gaussian. Inserting (14.51) into (14.49) yields the self-consistent equation for mvar d ap p2 vat - -~1 f (2rr )d e- p2 A _ 1 (47r)-d/2 A-l-d~2

m2

(14.53)

We finally get in terms of D, e and d = 2(2D - e)/(2 - D) I

mvar-~/~

[2I"(-~)l+~lwzT-e.

(14.54)

The final result for A reads 1

A - - 4rr

-2

D-2

F (~~ _/ ~" )\" ryzT- 27TzT-~.

(14.55)

We can now insert these results into (14.48), and after straightforward calculations get the variational instanton action D

Sinst = Svar[ ViVstl -

1- ~

2 1-' L ~

.

(14.56)

The corresponding variational estimate for the large order constant C defined by (14.23) is d

1/cvar = 2 F ( L ~ )

v .

(14.57)

As claimed in the previous subsection, although intermediate results are singular at e = D, the final result is regular for all e > 0. We shall discuss the physical significance of these results in the next subsection.

446

K.J. Wiese

14.4 Discussion of the variational result 14.4.1

D = 1

It is interesting to compare the variational estimate with the exact result for polymers, i.e. for the case D = 1. Let us consider the LGW instanton action, as given by (14.37). It is equal to the inverse of the large-order constant C. In Fig. 39 we have plotted the variational result for I/C TM, as given by (14.57) and the exact result for I/C obtained by numerical solution, as a function of 0 < d < 4. First, we note that always C > C TM, (14.58) as expected from the variational inequality E < '5'vat. This implies that the variational method gives an underestimate of the large orders. Second, the variational estimate becomes good for small d, and exact for d 0. This is not unexpected, since in that limit the membrane M has no inner degrees of freedom, and the functional integration over V (r) reduces to a simple integration over V e 1R. Since this integral is Gaussian, the variational method becomes exact. Finally, the variational estimate for C is regular when d ~ 4, and then equals 1/(2rr2); this is 50% smaller than the exact result 3/(4rr2). Thus the variational method is only qualitatively correct when e = 0. This is not so surprising, since the limit e ~ 0 is somewhat peculiar. Indeed when d - 4 the ground state energy E0 in the equation (14.30) for the wave function ~0 is equal to 0. Then the most general solution to (14.30) (for d = 4 and E0 = 0) is 2r0 9 0(r) = r~ + r 2 '

(14.59)

with r0 an arbitrary scale (the size of the instanton), r0 is fixed by the normalization condition (14.31) which cannot be fulfilled at d = 4 for finite r0. In fact a more careful analysis of the rotationally invariant solutions of (14.30) and ( 14.31 ) (see Appendix A of David and Wiese, 1998) shows that as d ~ 4, E0 should scale as E0 ~ 4 - d and that for 0 < 4 - d 2 and becomes large as D' ~ 2, while for D' < 1.6, r/opt(D') < 2 and becomes small as D' -~ 0. In the first regime (D' -~ 2) we thus expect that the power series in e will behave like a convergent series, up to some quite large order r/opt. In the second regime (D' small), we expect that the power series in e will be divergent from the very first terms. This is in agreement with the calculations at second order in David and Wiese (1996) and Wiese and David (1997). For large d, the two-loop results for v* can neatly be resummed, and the stability of the various resummation procedures and extrapolation schemes analysed in David and Wiese (1996) and Wiese and David (1997) is good. The final estimates are close to the prediction of a variational approximation 4/d for v*. For smaller values of d stability is less good, but in all cases the reliable extrapolations are obtained for values of the extrapolation dimension D' ,~ 1.6 or larger. It is not possible to resum safely the two-loop results if one starts the e-expansions from D' _< 1.5. Thus it seems that our rough estimates for the largeorder behaviour may explain some general features of the calculation at second order, and corroborate the results of the estimates of David and Wiese (1996) and Wiese and David (1997).

2

nopt


449

6

0.5

1

1.5

2

D'

Fig. 38 Optimal order nopt(D t) for the e-expansion of a membrane as function of the extrapolation (dimension) parameter D t, as obtained from the variational estimate for the large orders.

14.4.3

Limit D --+ 2

Of course these arguments are valid if the variational approximation for the instanton action stays (at least qualitatively) correct in the limit D ~ 2. First let us note that, although (14.56) and (14.57) give estimates for Sinst and C which are singular when D ~ 2, our variational formula for nopt is much less singular, since according to (14.66) it behaves as 1 16e -4• nopt ( D ) ~ e ( 2 - D) 2

as D ~ 2

'

e fixed

'

(14.67)

with y = 0.577216, the Euler constant. If we use as in Section 3 the usual coupling b0, which according to (14.4) is related to b by 2( b=b0s2

4rr ) d/2 (2-D)SD

(14.68)

as expansion parameter instead of g - bL ~, the large-order constant C in (14.7) and (14.23) is

Cbo__C$2 [(2-D)SD] -d/2 -~

4zr

'

(14.69)

450

K.J. Wiese

which in the variational approximation reads d

Cb~ - -S2D ~ [ (72 - D4rr ) S D=F ( 2 - D ) ~2 ]

~ const ( 2 - D ) 2

2-~

as D ~ 2, e fixed, (14.70) with const = zr2(rr e-2Y) 2-e/2. Therefore in this normalization also the singularities as D ---, 2 are simply algebraic. The same remark holds for the 'second virial coefficient' (Duplantier, 1987) z, defined as (2 - D)So ]d/2

z=

4zr

b L e.

(14.71)

14.5 Beyond the variational approximation and l/d corrections The variational result for the instanton-action (14.56) can be used as a basis for a systematic expansion in lid (David and Wiese, 1998). A straightforward 1~dexpansion, however, is inconsistent with the fact that the variational method is exact for d = 0. In David and Wiese (1998), the following improved 1/dcorrection was obtained"

S - Svar D

Svar

D(2D - e) sin Jr2o (2 + D - e)(D - e) Jr • fo ~ d p p a-I [ In ( 1 - 2 + D 4- e

2(p)

+

2+D-e 4

2(p)], (14.72)

where J(p) =

f0'

dx l + x ( l - x ) p

1

2 ~-2 .

(14.73)

Remember that as discussed in Section 14.1, this result is only true for e positive, i.e. d < dc(D). In the remainder, we shall focus on the case D = 1, for which we can most easily test (14.72). In D = 1, 2(p) is exactly given by 4

2(p) = 4 + p2"

(14.74)

Equation (14.72) is then integrated (using the residue calculus) with the result

Svar

D--I

(d + 2)(d - 2) ~ g

} § 28

~

.

(14.75)

2

451


20 o ~

15

o ~

9

9

9

j

J

10

5

1

2

3

4

Fig. 39 The inverse of the large-order constant 1/C for the Edwards model (D = 1) as a function of the bulk dimension d. The dotted curve is the variational estimate (14.57), the dashed curve the estimate from (14.76), the continuous curve the exact result.

The large-order estimate is finally obtained as

C - l (D = 1) ~ 2 7r d/2

1 + d + 2

-2 + g -

+ ....

(14.76)

which is plotted in Fig. 39. We see that this corrects 50% of the deviation of the variational result from the exact result in d - 4, and is even better in lower dimensions.

15

Conclusions

In this review, we gave an overview of techniques which allow one to generalize the concept of local field theories to multilocal ones. The most prominent example of such theories are self-avoiding polymers and membranes. The same techniques apply to dynamical problems and even to the motion of a extended elastic objects in the presence of quenched disorder.

452

K.J. Wiese

Some less settled topics have not been studied here, but certainly deserve further consideration. The most urgent of these questions is why simulations generically see a flat phase. In Wiese and David (1997) it had been argued that the pure self-avoidance fixed point should become unstable for space dimension d < dl, with art .~ 3.8. A solution of this problem first demands an identification of the mechanism which destabilizes the pure self-avoidance fixed point (Wiese and Shpot, in preparation), and second a treatment of the full problem. Most promising seems to be the route via the functional renormalization group approach (Wiese, work in progress). A thorough theoretical understanding would also help to design experimental tests. It would certainly also be promising to work directly at D = 2, instead of calculating at D < 2 and then continuing analytically to D = 2. However, all attempts to use methods adopted to two dimensions, e.g. conformal field theory, have failed so far. We hope that also this route will be explored in the future. In conclusion: we have described very powerful methods to treat nonlocal interacting systems, and we hope that this review will help to make these techniques profitable to a broader audience.

Acknowledgements This work would never have been accomplished without the inspiration by Franqois David, Pierre Le Doussal and Mehran Kardar, who directly collaborated in the subject matters of this review. I am very much indebted to them. I also have learned much from numerous discussions with Edouard Br6zin, Hans Werner Diehl, Jean-Michel Drouffe, Gerhard Gompper, Gary Grest, Emmanuel Guitter, Terry Hwa, Jaques Magnen, Stefan Kehrein, Lothar Sch~ifer, Mykola Shpot, Jean Zinn-Justin and Jean-Bernard Zuber, and I am very grateful to all of them. The article has much profited from comments and proof-reading by Franqois David, Hans Werner Diehl, Johannes Hager, Pierre Le Doussal, Mehran Kardar, Stefan Miiller, Henryk Pinnow, Martin Smock and especially by Andrea Ostendorf and Mykola Shpot and they all deserve my warmest thanks. Last not least, I am most grateful for the constant and very generous support from Hans Wemer Diehl, which gave me the opportunity to do this work.

Appendix A: Normalizations

We use peculiar normalizations in order to simplify the calculations. First of all, we normalize the integration measure of the internal space as = -~D

dDx'

(A.l)


4,53

where 2 yrD/2 SD=

(A.2)

F(D/2)

is the surface of the D-dimensional unit ball. This provides fx I x l e - O |

--Ixl)

=

-

ILe

(A.3)

9

Consequently, the f-distribution in x-space is defined such that (A.4)

fy f (y)~D(x -- y) = f (x).

The f-distribution in the embedding space is normalized according to ~d(r(x) -- r(y)) = (47r)d/2fd(r(x) -- r(y)) = fp e ip[r(x)-r(y)l

(A.5)

with fp -- 7r -a/2 f dd P

(A.6)

fp e-P a = a-d~ 2

(A.7)

in order to have

Using for the free Hamiltonian 7-[0 = 2 -

1

D

f ~l ( V r ( x ) )2

(A.8)

yields the correlator (for a derivation see Appendix G) C ij (x - y) :=

1

-~[r i (x) -- r j (y)

]2)0= 6 ij C(x

- y)

(A.9)

with C(x - y) - - I x - yl 2-D

(A. 10)

It satisfies the Laplace equation (see Appendix G) A C ( x - y) -- ( 2 - D)~D(x -- y)

= ( 2 - D)SD6D(x -- y).

(A.I I)

454

K.J. Wiese

Appendix B" List of symbols and notations used in the main text Throughout the text, we abbreviate the operators encountered by the following symbols: 1=1

1

-+- = ~ ( V r ) 2

--

-- = $CI(r(x) - r(y))

" ': .~ = (--Ar)g d(r(x) - r(y)) (--Ar)n~ d(r(x)

--(2n);~ = =

-

r(y))

-- • - - - LR-interaction

~~..

=

(r(x) - r(y))~ a (r(x)

/" ( Z ) ) .

In dynamic theories, we use the following graphical symbols:

~4- = r(x, t ) ( - - A x ) r ( x , t) = ~(x, t)i'(x, t) -- ~(x, t) 2 . = 2~(x, t) fk(ik) eiktr(x't)-r(y't)]

_-v~J = f ~i (X, t)eik[r(x't)-r(y't')]r j (y, t')

i~v~

k

=f

t)eiklr(x't)-r(y't')]r(y,

t')

k

L

f

PLj (k)? i (x,

T

f

P;J (k)r i (x, t)eiklr(x't)-r(y't')]r j (y, t').

t)eik[r(x't)-r(y't')]r j (y,

t')

When two endpoints are approached, we denote this by a dashed line. For instance, ~d (r(x) -- r(y)) for small x - y is denoted as

~.

x .... . y

,

(B.1)


455

and the arguments x and y will be dropped, whenever confusion is impossible. This contraction has a MOPE, which is denoted by

An expression like \(~ -~- . . I + ) i s a MOPE coefficient, i.e. a function o f x - y. The I

/

notation is chosen in the spirit of Feynman's bra and ket notation. Furthermore, we denote a 'diagram'

.

.

.

.

.

yIl.

(G.4)

We conclude I-(2-D)

1/(0)f fo l dr rl-o

< ( 2 - D)

--

fo ~

+(2-D)

/;

< ( 2 - D)(rl 89

dr ~ l f ( r ) rl_o dr

rl I f ( r ) - f ( 0 ) l rl-o

+ r/ 89

So for all f E C ~ there is the following diagram:

f dx ( - A g q ( x ) ) f ( x )

f(o)

--

f dx g r l ( x ) ( - A ) f ( x )

f dx g(x)(-A)f (x)

proving (G. 1).

Appendix H: Exercises with solutions

Exercise 1" Example of the MOPE Calculate the MOPE coefficient ((@i.: I'---')"

f(0)l

.

Polymerized membranes, a review X2 A I

" " ,.

C

/

f"'" ,

,"

', I

z

/

/

.= Y2 d

X3

,-b

Y3 - .e

w

x]

Yl Fig. 40 The distances and the points in (H. 1).

Solution: Start from xl ~

e. Yl x2:

[fpfq

: Y2 x3~

= Y3

9eikr(xl). :eipr(x2)..eiqr(x3)..e-ikr(yl).

:e-ipr(y2)..e-iqr(y3)..

(H.1) These dipoles shall be contracted like , ~.-.-'--~,

.... '

,

~

,

(H 2)

'

We therefore use the OPE for the points x l, x2 and x3, supposing the differences between these points become small: .eikr(xl)..eipr(x2)..eiqr(x3).

=

:eikr(xl)+ipr(x2)+iqr(x3). ekP a2V+kq b2V+ pq c 2v

(H.3) The new variables for the distances between the points are given in Fig. 40. An analogous relation is valid for yl, y2 and Y3. In order to retain only the most important contribution, we expand "eikr(xl)+ipr(x2)+iqr(x3):=:e i(k+p+q)r((xl+x2+x3)/3)

(1 +

O(Vr))"

(H.4)

and neglect the contributions of order O(Vr) because they are proportional to irrelevant operators. After a shift in the integration variable q,

q

>q-k-p,

equation (H. 1) becomes q .eiqr((xj+x2+x3)/3). :e-iqr((yl+Y2+Y3)/3) : xfkLekp(a2V+d2~')+k(q-k-p)(b2V+e2~')+P(q-k-p)(c2~'+f2v)

(H.5)

464

K.J. Wiese

The integral over q yields the g-distribution plus higher derivatives of this distribution. The latter are irrelevant operators and can be neglected. As they come from the expansion of this exponential factor in q, we only have to retain the last factor, evaluated at q = 0. This gives

f

fp ekp(a2V+d2V)_k(k+p)(b2V+e2V)_p(k+p)(c2V+f2v)

_ (b2V+ e2V)(c2V+ f2v)

_ 41

(b2V+ e2V + c2V + f2v _ a2V _ d2V (H.6)

This can still be factorized as is known from ancient Heron:

""

I

) [1 ( v/a2u d2v V/b2u e2v

)

•

)

•

2v)

x(v/a2V+d2V+v/b2V+e2V-V/C2V+f2v)] -d/2 Exercise 2'

(H.7)

Impurity-likeinteractions

Show that there is no term proportional to ~d (r (0)) in the MOPE of n dipoles with ~a (r (0)). Show also that this implies that the full dimension of ~a (r (0)) is

-v*d.

Solution:

The MOPE of ~d (r(0)) =

fk eikr(O)with n g-interactions is

fkeikr(O) fPl eiplir(yi)-r(z"l"'fp, eipn[r(yn)-r(z")] = fk fpl "" fP. :eikr(O)eiPl[r(Yl)-r(zl)l"""eipn[r(yn)-r(zn)l"eZ~=' kpj[C(yj)-C(zJ)]

piPj[C(yi-Yj)+C(zi-zj)-C(yi-zj)-C(yj-zi)]. The leading term proportional to ~d (r(0)) = fk eikr(O)is x e 89Zin--IZj=I

(H.8)

fk eikr(O'fPl "" "fP. e89~-'~:i~'7=iPipj[C(yi-yj)+C(zi-zj)-C(yi-zj)-C(yJ-zi)]" (H.9)


465

Note that from eZ~ =~ kpj[C(yj)-C(zj)l only the leading term 1 has to be taken, since subleading terms generate derivatives of the g-distribution 6d(r(O)). However, (H.9) is nothing but the MOPE 1) • ~d (r(0)), n dipoles

and the divergence is subtracted by the counterterm proportional to the volume, or equivalently by normalizing expectation values by the full partition function. Therefore, the only renormalization to 6a(r(O)) comes from the anomalous dimension of the field r, leading to the above-stated dimension of - v * d . Exercise 3: Equation of motion Show that the factor of Z a/2 which intervenes in the renormalization of the coupling bo = blz EZoZ a/2 and thus in the renormalization group fl-function can also be understood with the help of the equation of motion or redundant operators. Solution: Two kinds of counterterms are needed: counterterms proportional to - - (taken care of in the renormalization factor Zo), and counterterms proportional to + (taken care of in the renormalization factor Z). The latter appear in the form 2- D

-+- x

(H.10)

and using the equation of motion can be interpreted as

d2 z b b l Z e f x f y x ,

ey

(U.11)

Exponentiation then leads to

e-yf-i3 fx * x -blaeZb

f x f v. x "----" v

E q n . o f m o t i o n ), e

-Y~TY fx + x -blzeZbZd/2 fx fv. x'----" y (H.12)

as stated above. The same is achieved by using the concept of redundant operators. Exercise 4" Tricritical point with modified two-point interaction Study the tricritical point, which appears when one demands that the renormalized coupling proportional to self-avoidance, o --, vanishes. Show that for D > 4/3, the next-to-leading operator dominating the tricritical point, is . . . . . Determine the MOPE coefficients to leading order, following the lines of Section 3. Show that the/~-function has an IR-stable fixed point and calculate the anomalous scaling exponent v*. This problem was first discussed in Wiese and David (1995) and David et al. (1997).

466

K. J. Wiese

Solution: Let us start from the bare Hamiltonian

'ix'

7-/[r0] = 2 -

D

2- D

2 (vr0(x))2 + bo -+- + b0

f /y

--

(--Ar)$d(ro(x)- ro(y))

_.

(H.13)

The canonical dimension of the coupling constant b0 is !

(H.14)

e :=[bo]u=3D-2-vd

and the model has UV-divergences, i.e. poles in e', for e' = 0. As in Section 3, two renormalizations are needed, namely

r(x) = z-l/2ro(x) b = Z - d / 2 - l Z b I lz-e'bo.

(H.15) (H.16)

At one-loop order, the counterterms are in analogy to (3.112) and (3.113) Z=

1

e'(2-D)

(H.17)

-~ , + O ( b 2 ) ' ....

b 'i'~"S/

(H.18)

However, note that the leading term of the MOPE of . . . . is proportional to -- --, such that (scale-dependent) fine-tuning is necessary in order to stay at the tricritical point. J

..

i

t

The residues ( ~ 1+)o, and ( ~ i ~ : 1--------)~, are analytical functions of D for 0 < D < 2 (Wiese and David, 1995) 1

(Q4+l: 2 ~ ~

o), ,00_02-8

(6)2

(H.19)

\2-01 The fl-function and the scaling dimension v(b) of the field r are as in (3.115) and (3.116) defined as

fl(b) : = ~

b, bo

v(b) :=

g 2

2

lnZ. bo

(H.20)


467

The new 3-function has a nontrivial IR-fixed point b* > 0 for e' > 0 and the scaling dimension of the membrane at the O-point becomes*

IJ

-+- 0 (8'2) 1 I

2-D

+ O ( e '2)

1+ 10D- D2-8

.

(H.21)

F(2-~DD)2 +2D

Exercise 5: Consequences of the equation of motion Show that (Wiese and David, 1997) (H.22) I

2

-I

...

I

(H.23)

Solution"

Details are given in appendix C of Wiese and David (1997). The idea is to use the equation of motion (4.4) to write

(ff ff f~

="

ff ~ ~176 :)0

(H.24)

Identifying divergences proportional to o : on both sides and again using (4.4) then yields (H.22). The second identity (H.23) is proven along the same lines by starting from

tNote a misprint in (12.32) of Wiese and David (1997). Using the extrapolations for v*(d + 2) (expansion about the mean-field result VMF -- 2D/(2 + d)) or v*(d + 4) (expansion about the Flory result VFlory = (2 + D)/(4 + d)) in three dimensions yields results very comparable to the Flory estimate of v* - 0.57. This result is smaller than the geometrical bound of 2/3, indicating that the tricrital point may not be observable in experiments.

468

K.J. Wiese

Exercise 6: Finitenessof observables within the renormalizedmodel Show that within the renormalized model (3.93) and to first order in b, the expectation value of O = eik[r(s)-r(t)]as given for the bare model in (3.27) is UV- and IR-finite. Solution:

Two counter-terms are relevant at first order in the renormalized coupling b, namely A T / l , (3.99) and AT-/., (3.101). The third one-loop term, AT-/: _-, (3.101), will only show up at order b 2. Also note, that the first one, AT-/1, has already been taken into account in (3.27), due to the normalization introduced in (3.12). At first order in the renormalized coupling b, we thus only have to take care of the counterterm (3.101), with the full tensorial structure, resulting in

(O)b x y

( l k2 [C(s-x) + C(t- y)-C(s- y)-C(t-x)] 2) -exp

~

C(x-y)

+~ k2[(x-y)Vz(C(s-z)-C(t-z))]2C(x-y) O ( I x - y l +O(b2) },

< L)]

(H.26)

where z may either be chosen as x or y, or the symmetrized version may be used. The terms in the big square brackets exactly cancel the divergence for small x - y. Note that there is a possible IR divergence for Ix - Yl ~ oo. In the original term, this is regularized by the observable-positions s and t, which effectively cut off the integral at Ix - y l ~ Is - t l. In the counter-term, the IR cut-off is explicit.

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Index

Page numbers in italics indicate references to figures. Absorbing subset, 56-57 Actin filaments, 10 Algebraic decay, 63, 67, 68, 74, 99 Aluminium foil, crumpled, fractal dimension, 275-276 Anharmonicity of elastic terms, 376 Anisotropy cubic, 430, 431,433 tethered membranes, 375 Annihilation process, 6, 49-50, 72, 195, 210 see also diffusion limited pair annihilation (DLPA); pair annihilation; random walk, annihilating Annihilation--creation process, 6, 26-27, 84 Annihilation-exclusion process, 173 Annihilation-fusion process, 206 Anticommutation relations, 175, 177, 200, 202 Anticorrelations, 196-197 Antiferromagnetism, 32, 156 Antiperiodic boundary conditions, 178, 179 Approximation methods, 5 see also specific methods, e.g. saddle point approximation ASEP see asymmetric simple exclusion process (ASEP) Asymmetric simple exclusion process (ASEP) applications, 52 boundary conditions, 11,44 empty interval probabilities, 237-238 exact solution, 125-135 finite size behaviour, 111-114 generalizations, 159 master equation, 114 nearest-neighbour jumps, 43 randomness, 160

on a ring, 110--114, 160 self-duality, 104-110 self-enantiodromy, 108-110 shocks, 12, 117-123 stationary distributions, 117 Asymptotic behaviour ASEP, 112, 113 nonequilibrium systems, 53-71 particle systems, 59-69 scattering amplitude, 101 Autocorrelation function, 99, 385 13-function derivation, 416, 446 fixed point, 302, 347, 357, 373,464, 466 incomplete, 142 one loop order, 367, 417,419, 447 renormalization, 307, 402,403,446, 464 zeros, 304 Band collapse, 215 Baxterization, 50 Bead and tether model, 276-277 Bending rigidity crumpling transition, 276-277 entropic, 258 flat phase, 268, 271,272 fluid membranes, 261-262, 265-267 self-avoiding membranes, 356 spectrin in red blood cells, 274 Bessel functions, 64, 98, 121, 169, 170 Bethe, Hans, 32, 34 Bethe ansatz equations (BAE), 40, 110, 111-112 Bethe ansatz methods anisotropic Heisenberg chain, 114-117

482

asymmetric simple exclusion process (ASEP), 111 coordinate, 90 correlations, 51, 99-100 current fluctuation, 158-159 formalism, 24 history, 34 infinite system, 114-117 integrability, 52, 79 polymers, 220, 223 solution, 35-4 1, 84-88 typical results, 5-6 wave function, 97, 99-1 t30, 101 Bethe wave function (~p), 86, 101, 111, 166, 175 Bilinear property, 212-213 Bilocal operator, 365,397-399 Binding centre, 372 Binomial coefficients, 107, 129, 130 Biological systems fluid membranes, 257 growth, 3, 163-164 Biopolymerization, 10-12, 43, 104, 148, 220--222 Blockage, 160 Boitzmann weights, 225,227, 377, 431 Borel methods, 260, 349, 446 Boundary conditions antiperiodic, 178, 179, 213 and bulk state, 125 exclusion interaction, 86, 114-115 free fermion system, 213 Glauber dynamics, 213 Neumann, 362 nonequilibrium systems, 103 open, 89, 125-126, 211-212, 226 periodic, 38-40, 63, 110-114, 178, 179, 182, 194, 213 polymers, 220 reflecting, 66, 67, 104, 105, 111 two-particle, 93 types, 44 wave function, 37 Boundary densities, 96, 151 Boundary effects critical behaviour, 361-363 phase transitions, 52, 103, 104, 126, 143-154, 157 relaxational behaviour, 98 reservoir-coupled systems, 88-89 shock distributions, 120 Boundary fields, 44, 97-98

Index

Boundary operators, 360, 363 Boundary terms, master equation, 91 Bra-ket notation, 24, 295,454 Branching processes, 47, 150-151,152, 172 Branching ratio, 206 Branching-fusion process, 184, 185 Brownian motion, 9, 23, 65, 411,439 Bubbles, 431 Bulk contact exponent, 357-359 correlator, 362 density, 125-128, 138, 139 dynamics, deterministic, 136, 160 equation, 66 Hamiltonian, 90 hopping rate, 145 phases, 126 scheme O(N) system, 425 state, and boundary conditions, 125 Burgers equation, lattice realization, 104 Canham-Helfrich Hamiltonian, 261-262 Canonical dimension, 365,366, 407,456, 465 Canonical distributions, 62, 108 Cells see biological systems Characteristic function, 321,361 Chemical annihilation, 194, 199 Chemical annihilation-creation reaction, 6, 49 Chemical potential, 156, 373, 412, 413, 414, 421,423,428,457 Chemical reactions, diffusion limitation, 6, 194-197, 222-223 Coagulation, 47 Coalescence, shocks, 47, 117-123, 150-151, 152

Coexistence line, 134 N-Coloured membranes, 408-434 Combinatorial factor, 420 Commutation relations, 141, 177 Commuting transfer matrices, 31, 39 Compact phase, 256, 259 Complex systems modelling, 6-13 in nature, 3-4 Conditional probabilities, 67, 79, 84-85, 86-88 Configuration space, 394 Confining tube, 8, 10, 217 Conformal field theory, 451 Conformal mapping of sectors, 311-313 Connected components and molecules, 319, 320

Conservation laws, 31, 32, 35, 99

Index

Conservation of probability, 92-93, 181, 183, 189 Contact exponent, bulk, 357-359 Continuous and discrete time processes, 13-16, 28, 30 Contraction dipoles, 293-297, 300-301,352, 353, 360, 382, 462 divergent, 365 interaction, 383-384 Convection, 388 Correlation/correlators additive, 313-314 amplitude, 143 Bethe ansatz method, 99-100 calculation, 197-198, 209, 214 canonical dimension, 456 and canonical distributions, 108 decoupling, 222 density, 108 derivation, 452 divergences, 297 driving effects, 203 energy-energy, 257,408 equal time, 140 equations of motion, 164-165 exponential decay, 168 higher order, 96, 164 Laplace transformed, 315-316 large-scale/limit behaviour, 88, 199, 456 lower order, 165 multipoint, 141 q~4 model, 316 and pair annihilation, 195 and perturbations, 99 polymers, 315, 412 random force field, 390-391 and response function, 383,394 static, 377 stationary, 140-141 three point, 117 time derivative, 380 time-delayed, 58, 59, 141 time-dependent, 6, 25, 74, 84, 91, 136, 141, 142-143, 174 two-point, 117, 140, 141, 166 Coulomb energy, 283, 285,380-381 Coulomb potential, 283 Counterterms, 298, 299, 34 I, 382, 383 Coupling constant, 302, 342, 366, 435 canonical dimension, 365,366, 465

483

renormalization, 354, 382, 420, 447 physical interpretation, 368 renormalized, 301,302, 393 Critical behaviour boundary effect, 361-363 modelling, 4 N-component spins, 256 and nonleading terms, 259 reaction-diffusion systems, 70-71 renormalization group, 5 tethered membranes, 259 Critical dimensions, phase separators, 364 Critical exponents, 351, 421,426 Critical points, 369, 370 see also tricritical point Critical theory, 428 Crossover, 66, 116, 197, 199, 218, 305, 407-408 Crumpled phase, 265,275 Crumpled swollen phase, 257-258, 259 Crumpling transition, 263,265-267, 273, 276-277, 356 Cumulant function, 204-205 Curie temperature, 32 Current fluctuation, Bethe ansatz methods, 158-159 maxima and minima, 111, 144, 149-154, 161 maximal, 133-134, 143, 144, 146, 155, 157 nonvanishing, 59 stationary, 66, 104 Current-density relation exact results, 148, 150 and parallel updating, 137-138 and phase diagram, 144, 153 point of inflection, 125 stationary, 111 traffic models, 155-156, 157 distribution, 452, 463 interactions, 371,372 Decoagulation, 47 see also branching Decoupling correlators, 222-223 equations of motion, 164-167 Defects, in polymers, 8, 9 Deformation energy, 268 matrix, 267-268, 278 Deformed harmonic oscillator algebra, 97 Density

484

asymptotic behaviour, 53 correlation function, 93 decay, 72, 141, 171, 195, 198, 205 expectation value, 168 finite, 101, 117 fluctuations, 167-169, 203-208 functional theories, 230 matrix renormalization method (DMRG), 71 nonconstant, 161 perturbation, local, 59 profile constituent profiles, 140, 143 diffusion limited pair annihilation (DLPA), 209 exact results, 100, 128-134, 132 field induced oscillations, 167-169 Gaussian distribution, 67 lattice derivative, 130-131 relaxation, 97 shape, 134, 143 single particle, 64 stationary, 126, 128, 148-149 TASEP, 139, 149 thermodynamic limit, 131-134 time evolution, 62, 99, 117-118, 122 relaxation, 59--61, 97-99, 165-166, 167, 169, 185 stationary, 137, 168 time dependent, 208 Detailed balance, 57-59, 62, 103, 104-105, 110 Determinant representation, 158, 159 Diffusion coefficient, 23, 82-83, 124, 148, 169 collective, 116 constant, 60, 116, 149, 197 discrete-time dynamics, 97 in inhomogeneous media, 42 limitation, 4 master equation, 63 perturbation, 124 propagator, probability conserving, 314 of reconstituting dimers, 101 shocks, 117-123 tracer, 44-45 Diffusionless random sequential absorption process, 167 Diffusion-limited annihilation, 49-50, 195 Diffusion-limited branching-fusion, 74, 167 Diffusion-limited chemical reactions, 6, 194-197, 222-223

Index

Diffusion-limited fusion, 163, 164, 168-169, 181 Diffusion-limited pair annihilation (DLPA), 49-50, 193-210 density, 205,209 and DLFPA, 77 dynamics, 193-211 and Glauber dynamics, 74, 163 Hamiltonian, 73, 181 local properties, 208-210 Diffusion-limited pair annihilation-creation, 74, 176 Diffusive mixing/spreading, 61, 149, 176, 208, 259 Dilation operator, 321 Dimensional regularization parameter, 399 Dipoles approaching boundary, 363 bilocal operators, 282 coincidence, 381 contraction, 293-294, 295,297, 300--301, 352,353, 360, 382, 462 Coulomb energy, 283-285 MOPE approach, 291-292, 418 replacement, 299 Dirac sea, 40 Dirac-Hilbert space notation, 19 Discontinuities, 435,437 Discrete time systems, 52, 69-70, 97 Disorder averaging over, 259 correction, 402 correlation, 388, 392 --disorder contraction, 399 environment, 77, 388-408 interaction, time ordered, 392 intrinsic, 278-279 isotropic, 389 nonpotential (transverse), 388, 389, 395,407 quenched, 259, 389 renormalization, 397-399 static, 389 topological, 42 Dispersion integral, 435 Dispersion relation, 67, 115 Displacement vector, 268 Distance geometry, 309 Divergences bilocai operators, 397-399 local, 282-284, 380-381,395-396 overlapping, 324

Index

DLPA s e e diffusion-limited pair annihilation (DLPA) DNA dynamics, 215-220 gel electrophoresis, 8 relaxation, 217-219 reptation, 10 structure, 375 Domain of attraction, large, 389 growth, 47, 389 statistics, 214 wall in ASEP, 105, 117 diffusion, 149, 171 driving, 185, 187 duality, 74, 79, 81, 163, 176, 179, 188-191,210-211 dynamics, 134-136 fluctuations, 141-143, 148-149 hopping, 188 motion, 145 random walk, 122, 135, 143, 148 as shock, 144-145 and spin-spin interaction, 36, 47 stability, 124-125 types, 146 Double E-expansion, 364-368 Double poles, two-loop diagrams, 345-346 Drift diffusion-limited annihilation, 195 distance, 149 and hard-core interaction, 104 velocity, 23, 60, 116, 146-147, 215,216, 217, 403 Driving process, effects, 195, 199, 203 Droplets boundaries at criticality, 429-430 energy cost, 431 instanton behaviour, 441-442 nature, 260 spin, 431 Duality, domain wall, 74, 79, 81, 163, 176, 179, 188-191,210-211 Dynamic matrix ansatz, 89-95, 99, 101 Dynamical equations, 5 Dynamical exponent, 65, 158, 385,387, 389, 403 Dynamical scaling, 63-65 Dynamical structure function, 59-61 Dynamics stochastic, 3-4, 14

485

ultraslow, 389 Dyson-Schwinger equation, 69-70, 166, 308 -expansion divergences, 360 double, 364-368 extrapolation, 349, 350, 429, 446-447 numerical predictions, 264 renormalization, 393, 439 technical convenience, 256 and variational estimates, 446--447 Wilson-Fisher, 446 Edwards model, 279, 314, 434, 436, 439-441, 450

Edwards-Wilkinson universality, 235 Effective field theory, 378 Effective Hamiltonian, 269, 270, 279, 436 Effective system, 6 Eigenstates, 31, 39, 90 Eigenvectors, 54, 90 Eight-vertex model, 5, 52, 160 Einstein relation, 215 Elastic energy, 371 Elastic manifolds s e e polymers Elastic terms, anharmonicity, 376 Elasticity, renormalization, 401,404, 405 Electric field, 104, 167, 217 Electron, motion, 167 Empty interval probabilities, 167, 173, 181, 184, 212, 237-238 Enantiodromy, 74-75, 8In, 82, 162-164, 173 Energy barriers, random, 42 --energy correlation function, 257, 408 expression, integral representation, 86 function, Ising model, 48 gap finite, 67-68 free fermion systems, 191-192, 193 limit behaviour, 69, 193 and relaxation time, 171 volume independent, 110, 111 Equations of motion calculations, 464 consequences, 305,466 decoupling, 164-167, 169 free fermion model, 184, 187, 213-214 Glauber dynamics, 187 and global rescaling, 305-307 for magnetization, 187 Pauli matrices, 182 solving, 141

486

for time-dependent operators, 198 for vacancy strings, 169 Equilibrium distribution, 54, 104-105, 117 systems, 4, 57-58 Equivalence, 72-74 Ergodicity, 54-57 Error function, 88 Euler /~-function, 301 F-function, 444 Exact exponent identities, 272 Exact solution dynamical equations, 5 equilibrium systems, 4 exclusion processes, 136, 148 many body systems, 30--31, 33, 34 nonequilibrium systems, 4 random sequential absorption process, 167 reaction-diffusion systems, 5 three state systems, 29 Excitons, 13, 49, 176, 196, 222-223 Exclusion process asymmetric, 43, 101, 104, 158 discrete time, 225-229 elementary move, 41, 42 exact results, 136, 148 generalized, 144 integrability, 50-51 interaction, 86, 114-115, 116 and interface dynamics, 231 k-step, 125 nonequilibrium, 103 partial, 43, 51 partially asymmetric, 113 partially symmetric, 96-97 with particle injection and absorption, 79 quantum Hamiltonian formalism, 41-45 single species, 45-47 symmetric, 43, 79-102, 89, 101 traffic flow, 154-157 two-type, 51 universality, 199 see also asymmetric simple exclusion process (ASEP); symmetric simple exclusion process (SSEP) Expansion 1/d, 449-450 Flory approximation, 354-355 parameters, 348 see also perturbative expansion Expectation operators, 4

Index

Expectation values calculation, 204 connected and nonconnected, 306 decay, 66 density, 168 equal time, 394 experimental measurement, 22-23 observables, 280, 282, 293,305-307, 366-367, 378,467 quantitative behaviour, 193 spectral decomposition, 62 stationary, 54, 58 transformed systems, 72 Experimental realizations, 215-223,385 Exponential decay, 63, 65 Exponential relaxation, 67 Extrapolation, 349-351,406, 429, 430, 433, 447, 448 Fermi edge, 40 Fermion interaction quartics, 194 Fermionic operators, 175, 177 Fermions see free fermion approach Ferromagnetism, 32, 200-201,230 see also Ising model Feynman diagrams, 286, 290, 318 integrals, 256-257, 318,319, 408, 436 Field theoretical methods convenience, 256 and fluctuating lines, 408 multilocal, 257, 258 nonlocal, 260 O ( N ) model, 413,414 renormalization, 333-334, 392-393 tethered membranes, 279-305 Finite systems, 62-63, 65 First-order transition kinetics, 145 First-passage time, 75-77, 79, 84, 217-218 distribution, 83-84, 89 Fisher waves, 172 Fixed point cubic, 432, 433 Gaussian, 365,369, 404, 432 Heisenberg, 365, 369, 404, 432, 433 infrared (IR), 305,354, 357, 358, 368, 369-370, 426, 446, 457,464, 466 isotropic, 399, 405-407 perturbative, 389 pure self-avoidance, 451 renormalization group flow, 404--405 Flat phase

Index fluid membranes, 258 persistence, 277 self-avoiding membranes, 277, 356 in simulations, 451 stability, 263,272, 278 Flory approximations crumpled phase, 275 disordered system, 391 expansion, 348, 354-355,466n fractal phase, 388 one loop order, 407 phase separation, 371 self-avoidance, 266-267, 276, 278, 376 Flow diagrams, 368, 369, 403,404, 432 Fluctuating lines, and field theories, 408 Fluctuation-dissipation theorem, 62, 70, 159, 380, 383,389, 393-394 perturbative, 396, 399, 401 Fluctuations, 230--236, 272, 374 Fluid membranes, 256, 257-258, 261-262, 431 Fokker-Planck equation, 4, 65,394 Folded phase, molybdenum disulphide, 274 Forest construction, 264, 318, 320, 321,322, 334-335 Four loop calculations, 434 Four point correlator, 82 Four vertex model, 160, 225 Fourier transform methods density distribution, 116, 208 disorder model, 391 dynamic matric ansatz, 93, 94, 95 dynamical structure function, 59-60 fiat phase stability, 268 free fermion systems, 192 Gaussian variational approach, 442--443 longitudinal and transverse projectors, 454-456 master equation, 63 normalization, 280 self-duality, 82, 85 string expectation value, 183 Fractal dimension aluminium foil crumpling, 275-276 calculation, 264, 278, 356, 446 crumpled swollen phase, 257, 259 crumpling transition, 263,264 graphite oxide, 275 and renormalization, 385 Sierpinsky gasket, 264, 278 Fractal exponent, 273 Fractal phase, 77, 277, 356, 388 Free energy, 265, 266, 279, 374

487

density, 436, 439, 442 Free fermion approach boundary conditions, 211-212, 213 classification, 175, 178, 181-187 dynamical properties, 191-193 exact results, 197-200 formalism, 24, 175 Hamiltonian, 178, 180, 189, 194 interacting particle system, 199-200 as mathematical tool, 5, 51 nearest neighbour interaction, 191 one dimensional, 200 physical meaning, 193 relaxation times, 191-193 stationary states, 191 translation invariant, 175-176 Free 'phantom' surface, 280 Free propagator see response function Friction coefficient, 401 Fugacity, 408, 412 Fusion asymmetry, 183 branching process, 167, 169, 191 diffusion-limited, 163, 164, 168-169, 181 -pair annihilation process, 183-184 reactions, 72 F-function, 402, 444 Galilei transformation, 66, 199 Garden of Eden, 160 Gauge theories, and random surfaces, 256 Gaussian density profile, 67 disorder force, 392 elastic energy, 280 fields, 390, 413 fixed point, 365,369, 404, 432 noise, 377, 390, 392 phase, 386 random walk, 64-65 variational ansatz, 277, 352, 355, 371, 442--444 Gel electrophoresis, 8, 16, 159, 215-216 Gels, two-dimensional, 278 Ghost coordinates, 66, 87 Gibbs measure, 57 Glassy dynamics, 389 Glauber dynamics boundary conditions, 213 decoupling, 185 and DLPA, 74 enantiodromic, 164

488

equations of motion, 187 equilibrium system, 57, 191 free fermion systems, 176 Ising model, 47-49 nonfactorized state, 191 and pair annihilation-creation process, 51 spin flip, 13, 49-50, 76, 213, 214 stochastic rules, 188 zero temperature, 163, 193, 206-207, 213, 264 Global rescaling, and equations of motion, 305-307 Global symmetries, typical results, 5-6 Grand canonical distribution, 62, 80, 105, 108, 123 Grand canonical ensemble, 457,458 Graphite, 256, 274 Graphite oxide, 275 Gravitational field, 104 Green function, derivation, 459-461 Gregorshin theorem, 54 Ground state energy, 439 Hamiltonian anisotropic transverse X Y model, 6 Canham-Helfrich, 261-262 counterterms, 298, 299 diffusion-limited pair annihilation, 73 driven and undriven systems, 197 effective, 269, 270, 279, 436 equivalent, 72 exclusion process, 41-45 free, integration by parts, 306-307 free energy, 279 free fermion approach, 178, 180, 189, 194 Heisenberg, 34-35, 165 integral counterterms, 298 membrane, 278 modifed, 297 nonstochastic, 178-180 quantum see quantum Hamiltonian quantum spin model, 5, 33-34 renormalized, 341 rescaled, 436 static, 377 stochastic, 104, 178-180 trial Gaussian, 442--444 two-membrane model, 358 Hard core constraint, 84 interaction, 79, 96, 103-104 repulsion (site exclusion), 149-150, 155, 194, 363

Index

Harris criterion, 433,434 Hartree-Fock approximation, 442,443 Hecke algebra, 50, 63, 68, 188, 211,213 Heisenberg, Paul, 33 Heisenberg fixed point, 365,369, 404, 432, 433 Heisenberg quantum chain anisotropic, 11, 34-35, 43, 114-117, 165, 175 Hamiltonian, 34-35, 43, 83, 163, 165,216 integrability, 35, 101,220 isotropic, 111, 163,216 properties, 104 representation, 23, 33-35 stationary states, 80 SU(2) symmetry, 35 wave function, 97 zero-field, 43 Heisenberg quantum ferromagnet, 6, 10, 34-35, 41-42, 89 Helmhoitz fluctuation theory, 230 Hepp sectors, 318, 323,325-326 Hermitian, symmetric Hamiltonian, 58 Hexatic membranes, 264-265 High-density phase, 132, 133, 138, 139, 140, 142, 145, 146, 172 High-temperature expansion lsing model, 257,408,431 O ( N ) model, 260, 409-412, 421 Holder's inequality, 362 Homogenous factorized transformations, 179 Hopf--Cole transformation, 158 Hopping asymmetry, 104, 183, 197, 203, 212 biased, 66 matrices, 42, 43, 50, 91, 110, 158, 220 nearest neighbour, 84, 85 rate, 9, 19, 27, 57, 103, 145 time, 217 Hydrodynamic divergenceless flow, 388 Hydrodynamic interaction, 387 Hydrodynamic limit, 62, 124-125 Hydrodynamics, Zimm model, 259, 385-388 Hypergeometric function, 238 Impurity-like interactions, 463-464 Infinite membranes, 417, 418,456 Infinite systems Bethe ansatz, 114-117 density relaxation, 98-99 and dynamical scaling, 63-65 exponential decay, 65 hard-core interaction, 103-104

Index

late time behaviour, 159 Infinite time limit, 53-54 Infinite volume limit, 63, 64 Infrared (IR) convergence, 285,299 cutoff, 272, 321, 419 divergences, 280, 282, 284 finite observable, 281,366--367 fixed point, 305, 354, 357, 358, 368, 369-370, 426, 446, 457, 464, 466 Initial conditions, random, 202, 203 Injection rate, overfeeding, 135 lnstanton methods, 260, 434, 435-439, 441--442,443 Integrability checking, 52 concept, 4--6 consequences and practical applications, 31 and matrix representation, 50, 101 multi-species processes, 51 scattering amplitude, 101 Integrable systems algebraic properties, 50-51 nearest neighbour processes, 47 quantum spin models, 51 relaxation times, 63 stochastic processes, 6, 30-52 Integral counterterms, 298, 299, 341 Integral representation, energy expression, 86 Integration measure analytical continuation, 309-310 normalization, 286, 378, 379, 451-452 Interacting particle systems free fermion nature, 199-200 quantum Hamiltonian formalism, 41-50 stochastic dynamics, 3-4 Interactions attractive, 199, 441-442 contraction, 383-384 exclusion, 86, 114-115, 116 local, 68-69 repulsive, 84, 368, 435 on site, 6 time-ordered disorder, 392 see also long-range interactions; short-range interations Interface dynamics, 76, 231 fluctuations, 79 growth, 104, 158 Interparticle distribution function, 184 Ionic conductors, modelling, 159

IR see infrared Ising energy, 36 spin configuration, 32 Ising fixed point, 432, 433 Ising model, 32-35, 47 bond disorder, 433 energy function, 48 geomtrical description, 430-431 high-temperature expansion, 257, 408,431 Kawasaki dynamics, 68 kinetic, 76, 185, 186, 187 low-temperature expansion, 256, 408, 431 one-dimensional, 13, 154, 156, 162, 191 partition function, 431 planar, 230 random bond, 260, 430, 433 singularities, 430, 431 spin configuration, 33 two-phase, 231 Ito (prepoint) discretization, 377, 387, 392 Jordan-Wigner transformation, 175, 177-178 Kardar-Parisi-Zhang (KF-Z) equation, 52, 158 Kawasaki spin-flip dynamics, 43, 68, 108 Ladder operators, 20, 35-36, 107 Lain6 coefficients, 268 Landau-Ginzburg-Wilson theory, 440, 442, 445,446 Langevin approach, 4, 234, 235,236, 377-380, 385,390-391,394 Laplace equation, 412,422, 452 transform, 315 Large canonical ensemble, 80 Late time behaviour see asymptotic behaviour Lattice Brownian motion, 9 Burgers equation, 104 derivative, density profile, 130-131 diffusion, with absorbing boundary, 170 equation, solution, 173 gas collective velocity, 147 diffusive, 157-158 driven, 3, 103-161,171 hard core, 41-45 as model system, 8-9, 13 multi-species, 16, 97 one-dimensional, 8-10 particle number conservation, 55

490

shock, 124-125 gauge theory, 256, 408 random walk, 23, 82, 412 sawtooth configuration, 231,232, 235 sites, finite number of particles, 6 spacing, vanishing, 62 Laurent expansion, 299, 342, 343-344 Length scales, separate, 131, 149, 171 Lenz, W., 32 Lie algebra, 34, 158 Light cone, 141-142 scattering methods, 385 Lipid bilayers, 261,262 s e e a l s o fluid membranes Local current, time evolution, 158 Local divergences, 282-284, 380-381, 395-396 Local field theory, 317 Localization lengths, 145, 149, 173, 174 Long-range interactions competition, 363 correlated disorder, 404, 407--408 and fractal phase, 277 hydrodynamic, 387 in membranes, 272 nonrenormalization, 303-305 one-dimensional systems, 54 and phase transitions, 125 physical importance, 305 tethered membranes, 304-305 Longitudinal projectors, Fourier representation, 454-456 Low-density phase, 134, 135, 139, 140, 142, 145, 146, 171 Low-dimensional systems, 4 Low-temperature expansion, Ising model, 256, 408, 431 Macroscopic mechanisms, 6, 7 Magnetic susceptibility, 32 Magnetization, 5, 230, 233 Magnons, 36-37 Manifold closed compact, 421 compact, 422 free Gaussian, 435 generalized model, 430 integrals, 459-460 isotropic, 404 single noninteracting, 428 size, 427

Index

theory, 408-409 Many body systems conservation laws, 32 exact solutions, 30-31, 33, 34 master equation, 5 modelling, 3-4, 23-27 tensor basis, 23-26 Markov process, generator, 18 Markov property, 5, 17 Massive s e e bulk Master equation ASEP, 114 biased single-particle diffusion, 63 boundary terms, 91 and conditional probability, 84-85 determinant representation, 158 formulation, 4, 102 Markov property, 5 quantum Hamiltonian formulation, 17-29 stationary form, 59 symmetric simple exclusion process (SSEP), 10 three-particle, 117 two-particle, 85, 117 vector form, 21 Matrix, time-dependent, 91 Matrix algebra, 96, 101, 159, 160 Matrix product ansatz, 52, 90, 137, 159, 161 Matrix representation, 50, 53, 97 Maximal current, 133-134, 143, 144, 146, 155, 157 Maximal transport capacity, 135, 147 Mean field analysis continuum limit, 172 diffusion-limited pair annihilation, 195-197 domain wall dynamics, 135 limitations, 4 membranes, 265-267 pertubative expansion, 348-349 phase diagram, 126, 127 tubular phase, 375 and variational approximation, 354 Membranes of arbitrary dimension, 258 characteristic function, 361 N-coloured, 408--434 density, 422, 423,425,427 E-expansion, 256, 349 fluctuations, 267 Hamiltonian, 278 with intrinsic disorder, 278-279 local field theory, 317

Index

mean field description, 265-267 O ( N ) model generalization, 421-426 partition function, 359-361 properties, 261-279 self-avoiding, 319 self-energy, 361 tethered (polymerized), 262-265 tricritical point, 363 tubular phase, 374--377 two-loop calculations, 350 see also fluid membranes Mermin-Wagner theorem, 272 Messenger RNA see biopolymerization Minimal current phase, 144, 152, 153 Minimal sensitivity, 350, 354-355 Minimal subtraction (MS), 299, 340-342 Model A, 259, 377 Modelling, 3-4, 6-13, 43, 47, 104, 215-223, 258-261 Molecular dynamics algorithm, 377 Molybdenum disulphide, folded phase, 274 Momentum space formulation, 95,459 Monte Carlo methods equilibrium systems, 57 gel electrophoresis, 216 in modelling, 5 phase transitions, 151, 152, 153, 154 taking averages, 28 TASEP, 144 traffic models, 157 two-loop diagrams, 347 MOPE see multilocal operator product expansion (MOPE) Multi-time correlation functions, 74 Multilocal operator product expansion (MOPE) coefficients evaluation, 292-297, 372, 461-463,464 factorization, 334, 345-346, 372-373 integral, 341 Laurent series, 343-344 in renormalization, 298-299 residue extraction, 31 0-313 definition, 291-292 dipole contraction, 360 dipole divergence, 418 disordered system, 395-396 dynamic case, 381-385 important technique, 259 notation, 454 one-loop order, 298 in renormalization, 419 residue, 425

491

in Taylor expansion, 321-322 Zimm model, 387 Multiple occupancy, 41 Nagel-Schreckenberg model, 155, 157 Natural phenomena modelling, 3-4, 6-13, 43, 47, 104, 215-223, 258-260 universality, 256 see also biological systems; complex processes Nearest neighbour annihilation rate, 195 distances, ordering, 318 hopping, 84, 85 interaction, 191, 194 processes, one dimensional, 47 spin-spin correlation, 206 Nernst-Einstein relation, 215 Nest formulation, 318, 323, 324, 327, 328 Neutron scattering methods, 217, 385 Newton's equations of motion, 17, 19 Next nearest neighbour interactions, 156 Noise correlation, 386 function, 4 Gaussian, 377, 390, 392 Non-Abelian symmetry, 80, 82 Nonequilibrium systems asymptotic behaviour, 53-71 behaviour, 80-83 boundary conditions, 103 exact solution, 4 exclusion process, 103 maintaining, 89 modelling, 3, 104 phase transitions, 6 randomness, 4 statistical mechanics, 3 Nonergodic systems, 66 Noninteracting particles, 116-117 Nonlinear growth, 373 Nonlocal interactions, polymer, 259 Nonrenormalization, long-range interaction, 303-305 Non-self-avoiding membrane see phantom membrane Nonstochastic generators, 28 Normal ordering, 286-287, 293, 395,396 Normalization, 84, 294, 451-452 see also renormalization Notation, 239-241,453-454

492

Nuclear magnetic resonance experiments, 26 Number operators, 20 Numerical methods, 5, 15, 28, 113, 276-278, 363-364 see also Monte Carlo methods Occupancy, unrestricted, 29 O (N) model applications, 429-434 behaviour for large N, 427-429 field theoretical description, 256, 413,414 generalization to membranes, 421-426 high-temperature expansion, 260, 409-4 12 One-dimensional interfaces, fluctuation, 43 One-loop order, 292, 296, 298-303, 367, 381-385,407 On-site annihilation or interaction, 6 Operator product expansion (OPE), 259, 286-290, 462 Order, optimal, 446, 4 4 8 Overfeeding, 135, 147, 153 q~4_theory correlation function, 316 independent scaling exponent, 360, 363 Landau-Ginzburg-Wilson, 440 limit behaviour, 427 N-component, 316, 317 perturbation expansion, 256 in renormalization, 417 renormalized Hamiltonian, 286 scalar, 264 variational approximation, 354 Pair annihilation process and correlation, 195 diffusion-limited, 73, 163, 181,193-211 integrability, 50-51 rate, 222 on a ring, 194-195 see also annihilation process Pair annihilation-creation process, 51, 55-56, 176, 185, 188, 191 Pair annihilation-fusion process, 193 Pair creation process, 213-214 Pair exchange process, 80 Pair excitations, 201 Pair transition matrix, 46 Parallel updating, 28, 137, 156, 158, 160 Particle absorption, 88 -antiparticle excitations, 40 current, 61-62

Index

inward and outward, 122 nonvanishing, 110 stationary, 89, 110, 156 density, 5, 183 distributions, 80, 82 energies, 111 -hole symmetry, 9, 82, 126, 134, 139, 147, 155,227-228 injection, 88, 226 interaction, 150 number, 197 calculation, 203, 204 change, 197 conservation, 55, 61-62, 99, 105, 109, 125, 165 decrease, 211 distribution, 205 dynamical, 218 even and odd, 197, 198 fluctuations, 206 independent of bias, 203 parity operators, 177 reservoirs, 89, 98, 99 systems late time behaviour, 59-69 and spin models, 24, 27, 30 two-type, 27, 118, 121-122 see also interacting particle systems transport, boundary conditions, 44 Partition function equilibrium distribution, 106 free, 427 free Gaussian manifold, 435 high-temperature expansion, 409 Ising model, 32, 431 membrane, 359-361 polymer, 314-315,408 self-avoiding membrane, 436-437 singularities, 431 stationary distribution, 62 Path integrals see Feynman integral Pauli matrices boundary conditions, 178 commutation relations, 177 equations of motion, 182 and fermionic operators, 200, 202 in Heisenberg quantum formalism, 33, 41, 48, 106-107 notation, 20, 239 properties, 26 two-dimensional, 203 Peirls contour, 231

Index

Persistence distributions, 83-84 probabilities, 75-77 Perturbation anomalously small response, 389 and correlations, 99 diffusion, 61, 124 expansion critical exponents, 351-352 near critical point, 366 extrapolation, 347-348 O (N) model, 409-4 10, 412 $4-theory, 257 polymer Hamiltonian, 414 R-operation, 298 second-order terms, 288 self-avoiding membrane, 435 simplification, 313-314 variational method, 352-354 large order behaviour, 437-450 local, 68, 123-124, 209 spread and decay, 168, 209 stationary state, 59--60 theory, 2 8 1 , 2 8 4 - 2 8 5 , 3 7 9 , 380-381 analysis, 117 corrections, 270-273 divergent, 438 first-order, 439 fixed point, 389 large order results, 434-450 polymers, 372-374 renormalizability, 318-340, 371,446 Phantom membranes, 263,264, 276, 319, 356, 371,434, 436, 437 Phase angle, 43 diagrams, 138-139 and boundary densities, 151 construction, 153 and current--density relation, 144, 153 mean field approximation, 126, 127 and particle interactions, 150-151 self-avoiding membrane, 386 separator, 105, 132-133, 1 5 1 - 1 5 2 , 3 6 4 , 388 transitions boundary-induced, 44, 52, 103, 104, 126, 143-154, 157 and domain wall diffusion, 171 ferromagnetic systems, 32 field-induced, 169, 172 first-order, 147, 172 and long-range order, 125

493

nonequilibrium systems, 6 prediction, 131 second-order, 34, 135, 147 and shock dynamics, 127 surface, 230 symmetry breaking, 256 Physical phenomena see natural phenomena Pinning, 389 Plane wave ansatz, 36, 66-67, 183 Plaquette models, 256, 258, 277 Poles see scattering amplitude Polymerized membranes see tethered membranes Polymers branched configurations, 431 correlation function, 315 defects, 8, 9 directed, 158 Edwards model, 439--441 exact results, 218,445 exciton dynamics, 13, 176, 222-223 fully stretched, 407 globules, 442 modelling, 52 nonlocal interactions, 259 open and closed, 412, 413 perturbation theory, 372-374, 413 random flows, 390 renormalization, 412-419 reptation, 8-10 self-avoiding, 256 stiff and flexible, 375 theory, 305-317 tricritical point, 363 two-dimensional network, 274 two-loop calculations, 350 variational estimates, 445-449 see also tethered membranes Potential (longitudinal) disorder, 389, 391,395, 404-405 Potts model, 77, 214 Power law correction, 133 decay, 72, 196 divergence, 124 Prepoint discretization see Ito (prepoint) discretization Probabilistic approach, 5, 15, 17 Probability distribution master equation, 17-18 time-dependent, 90

494

vector, 20 Product measures, 25, 105, 110, 117, 128 Projection operator, 56 Protein synthesis see biopolymerization Pure annihilation process, 206 Pure fusion process, 213 Pure self-avoidance, fixed point, 451 Quantum algebra, 51,106n, 120, 158 Quantum chain see Heisenberg quantum chain; quantum spin representation Quantum field theory, 257, 408 Quantum Hamiltonian construction, 13-14, 26-27 DLPA, 49 first-passage time distribution, 83 formalism, 21-22, 80-81 Heisenberg ferromagnet, 10 integrable, 50 interacting particle systems, 41-50 master equation, 17-29 mathematical properties, 33-35 in modelling, 5 nonstochastic, 28 particle and spin systems, 30 Quantum numbers, 31, 39, 40, 41, 112-113 Quantum spin representation, 6, 14, 30, 45, 51, 72, 90, 97 Quasi-one-dimensional systems, 125 Quasi-stationarity, 79, 99-101 R-operation (subtraction operator), 298, 318, 319, 322, 323,328, 329, 336, 337 Radioactive labelling, 44--45 Radius of gyration, 391,457 Random force field, correlation, 390--391 Random sequential absorption, 49-50, 101, 167 Random sequential updating, 28 Random stress field, quenched, 279 Random surfaces, and gauge theories, 257 Random variables, in description of nonequilibrium systems, 4 Random walk annihilating, 175, 199-200 biased, 113, 122-123, 146, 148-149 in disordered media, 28 domain wall, 122-123, 135, 143, 148 Edwards model, 436 free, 436 Gaussian distribution, 64-65 lattice, 23, 82

Index

nearest neighbour moves, 18 shock, 173 single loop, 41 l two-class particles, 122 unbiased, 153, 172 Randomness ASEE 160 nonequilibrium systems, 4 traffic flow models, 154 Rate equation, 4, 5, 172, 196 Reaction rate, 196 Reaction-diffusion rates, alternating see spin flip rates Reaction-diffusion systems critical phenomena, 70-71 enantiodromy, 162-164 exact results, 5 experimental realizations, 215-223 Glauber dynamics, 188 integrability, 50--51 lattice gas modelling, 13 mean field approximation, 4 modelling, 3 with nearest neighbour interaction, 162-174 quantum spin representation, 72 rate equations, 4, 172 single-species, 45--47, 162-174 statistical approach, 3 Recursion relations, 127, 128, 129, 137, 160 Red blood cells, 256, 273, 274 Redundant operators, 307-308,464 Reflecting boundaries, 66, 67, 104, 105, 111 Reflecting principle, 66, 101 Relaxation boundary effects, 98 DNA, 217-219 exponential, 99 late time, 110 order parameter, 99 rate, biased, 171 times, 142, 170-171,218 finite systems, 62--63, 65--66 free fermion systems, 191-193 infinite systems, 65-66 polymers, 218 and system size, 63 Renormalizability theorem, 322 see also renormalization, criteria Renormalization criteria, 333-334, 381,387 disorder, 397-399

Index

dynamic, 381-385 factors see Z factors generalization to N colours, 420-421 group analysis, 377-378 approach, 13, 70071, 72, 104, 256 fl function, 446, 464 calculations, 206 critical phenomena, 5 equations, 392-393,456-457 factors, 298 flow, 402,403, 404, 431-432, 434 functions, 290, 302, 342-343,347, 358, 424, 439 polymers, 412-4 19 O ( N ) model, 414-418 one-group order, 298-303 perturbative, 318-340 self-avoidance, 356 strategy, 297-298 Reparametrization invariance, 458 Reptation, 8-10, 43, 217-219 Reptons, 9, 215,216, 217, 218 Reservoir coupled systems, 88-89, 100, 103, 125-126, 148, 153-154 see also particle reservoir Reshetikin relation, 52 Residue extraction, 310-313,399-402, 425, 449-450 Response fields, 378-379, 388 function, 379-380, 383,394 Resummation procedures, 256, 349, 446--447 Reverse Coleman-Weinberg mechanism, 260, 430, 431,432 Reversibility (time-reversal symmetry), 58-59 Rooted union and subtraction, 325 Rotational symmetry, breaking, 431 Roughness exponent, 271,272-273,403 Rouse model, 259, 377, 385,386, 387, 388 Rubenstein-Duke model, 215,216, 220 Saddle point methods, 436, 437,438 Scaling arguments, 377-378 autocorrelation function, 385 behaviour, 101,403,456 exponents, 65,357, 360, 439, 446, 457,464 see also ~-expansion function, 210, 347-348, 367,417 invariance, 72 limit, 446

495

radius of gyration, 391 relation, 428 and self-similarity, 3 theory, Widom, 230 time dependent correlation function, 142-143 variables, 87 Scattering amplitude, 86, 87, l0 l, I 1l, 115, 166 Schr6dinger equation, 5, 19, 21,315 Schwinger formulation, 318 Second virial coefficient, 449 Self-avoiding systems interaction, 314 membranes flat phase, 356 grand canonical ensemble, 458 large orders and instantons, 435-439 modelling, 264 numerical simulations, 276-278 and phantom membrane, 434 phases, 386 two-loop extrapolation, 355-356 physical information, 347-356 polymers, 256, 263 renormalization, 356 restricted, 277 summing over, 411 tethered membranes, 258-259 Self-duality, 81, 104-110 Self-enantiodromy, 74--75, 8 In, 82, 108-110, 113,211 Self-energy, 361 Self-similarity, 3 Semi-infinite systems, 98, 100 Shear and domain growth, 389 modulus, 267, 269, 272 Shock ASEP, 12 in biopolymerization, 12 branching and coalescence, 150~ 151,152 diffusion and coalescence, 117-123 distribution, 118, 119, 120, 121, 173 as domain wall, 144-145 dynamics, 6, 127 evolution, 195 formation, 104 multiple, 159-160 random walk, 173 stability, 124-125 structure, 52

496

in traffic models, 155 velocity, 118, 123, 143, 172 Short-range interactions correlations, 391,404, 407--408 disorder, 388, 406 and long-range order, 54 N-component spins, 255 repulsive, 256, 368 Sierpinsky gasket, 264, 278 Similarity transformations, 72-74, 167-168, 175, 176, 178 annihilation-fusion process, 206 density profile, 167-168 diagonal, 105 equivalence and enantiodromy, 72-74 factorized, 101, 189, 191 in free fermion models, 175, 176, 178 stochastic, 74, 185 typical results, 5-6 Single-site basis vectors, 239 Singularities, 430, 431,449 Site exclusion, 194 Six-vertex model, 5, 35, 52, 160 SLAC approach, 71 Smoluchowski theory, 4, 13, 196-197, 209-210, 212 Sociological behaviour, 3 Solid membranes see tethered membranes Solids, one-dimensional, 33 Solitons, 104 Space-time anisotropy, 68 Spanning tree, construction, 329-330 Spatial correlation length, 142 Spatial distribution function, normalized, 60 Specific heat exponent, 354, 431 Spectral properties, 62, 68, 72 Spectrin, in red blood cells, 257, 273, 274 Spin antiparallel, 36, 188 configuration, Ising model, 32, 33 decoupling, 427 flip events, 189 in free fermion systems, 186 Glauber dynamics, 48, 49-50, 76, 77, 213,214 Kawasaki dynamics, 43, 68, 108 and pair creation-annihilation, 188 rates, 212 symmetry, 76 term, ladder operators, 35-36 times, 49-50, 69

Index

transformation, 179 interaction, 36 matrices, 42 operators, 107 orientation, 32 parallel, 188 relaxation, 3, 108, 162-164, 212 -spin correlation, nearest neighbours, 206 systems classical and quantum, 30, 53 and particle systems, 24, 27, 30 two-state, 20, 21, 23 wave, 37-38, 81-82 Spin-(l/2) operators, 108, 175 Spring and bead model, 263, 279 Stationary algebra, 92 Stationary distribution ASEE 117 exponential, 66 grand canonical, 62 limiting, 54 Stationary states classification, 79-80 construction, 137-138 exclusion process, 95-97 free fermion systems, 19 l Heisenberg chain, 80 infinite time limit, 53-54 nonequilibrium systems, 53 obtaining, 5-6 perturbation, 59-60 and SU (2) symmetry, 79-80 Stationary vector, 53-54 Statistical mechanics, 3, 5 Statistical rotationally invariant force field, 390 Steady state selection, 125, 144-148 Step function, 107-108, ll7-118, 135, 158 Stochastic processes, 6, 72-78 Stochasticity conditions, 178-180 String expectation values, 164, 166-167, 183 String theory, 256, 257, 260, 262, 408 Strong coupling, 404 SU(2) symmetry breaking, 90 Heisenberg quantum spin system, 34, 35, 42 and particle injections and absorption, 88 and ,scaling behaviour, 101 and stationary states, 79-80 Subdivergences, 346-347 Subdominant operators, 259 Submanifold, 162, 164-165 Subtraction scheme, 409

Index

Surface configurations, 431 growth, 3 phase transitions, 230 Symmetric exclusion process, 43, 79-102, 217 see also asymmetric simple exclusion process (ASEP); symmetric simple exclusion process (SSEP) Symmetric Hamiltonian, Hermitian, 58 Symmetric simple exclusion process (SSEP), 10 Symmetry breaking phase transitions, 255 rotational, 431 spontaneous, 16, 32, 68--69, 375 non-Abelian, 80, 82 quantum algebra, 51 time reversal, 58-59 0 point, 363, 364, 369-370 Tableau formalism, 318, 324-327, 335-340 Tagged particle process, 44--45, 50, 52, 214 Taylor expansion, 321,322, 333,409 Temperature, increase, 389 Temperley-Lieb algebra, 50, 159, 188, 211, 213, 220 Tensor product notation, 239-240 Tensorial structure, 23-26, 294 Tethered membranes anisotropy, 375 critical behaviour, 259 dynamics, 377-388 experimental realizations, 262-265, 273-276 field theoretical treatment, 279-305 generalized O ( N ) model, 421-426 grand canonical ensemble, 457 isotropically polymerized, 374 long-range interaction, 304-305 renormalization, 318, 417 structure, 258, 263 Tethered polymerized surfaces, 256 Tetramethylammonium manganese trichloride (TMMC), 13, 176, 196, 223 see also polymers Thermodynamic limit, 67, 110, 119, 120, 131-134, 138-139, 212 Three-state systems, 363,365-366, 370-371 exact solution, 29 Time evolution, 76

497

density profile, 99, 117-118, 122 effect of driving, 203 operator, 18, 22, 56, 69-70, 83, 86, 109 spectral properties, 6, 72 shock distribution, 119, 121 TASEP, 136 reversibility, 58-59 scale separation, 69-70, 223 translation operator, 5 TMMC see tetramethylammonium manganese trichloride Totally asymmetric simple exclusion process (TASEP) as biopolymerization model, 221 collective velocity, 147 density profile, 139 exact results, 148, 150 maximal current principle, 144 with open boundaries, 125-126 phase transitions, 147 physics, 145, 146 Tracer diffusion, 44-45, 89, 101,220 Traffic models, 3, 12, 118, 122, 123-124, 148, 154-157, 160 Transfer matrices conserved charges, 35 energy gap, 69 equivalent, 72 formalism, 30, 31, 32, 225-226 statistical mechanics models, 5 in two-dimensional vertex model, 225-226, 227, 229 vertex models, 6 Transformation, 73-74, 179, 182, 189 Transition amplitudes, 166 matrices, 45--46 probabilities, 17, 199-200, 226, 255,408 Translation invariance, 128, 291 Transverse disorder see disorder, nonpotential Transverse projector, 391,454-456 Trapping, 389 Tricritical point, 259, 363-371,464 466 Tricritical state, 263, 273 Trilocal operator, 365 Truncation of series, 349, 350n Tubular phase, membrane, 374-377 Two-body interaction, 363, 364, 366, 370-371 Two-loop order, 259, 340-347, 349-350, 355-356

498

Two-membrane contact operator, 358 Two-particle transition probability, 166, 199-200 U (1) symmetry, 34, 35, 90 Ultraviolet (UV) convergence, 294, 322, 334, 381,423 cutoff, 272, 280 divergence, 280, 284, 285, 292, 299, 321, 340, 397, 414, 418,423, 428, 465 finite property, 281,301,438 Unbinding transition, 371-374 Uniqueness, 54-55, 56 Universality exclusion process, 199 field theoretical formulation of O(N) model, 414 interface fluctuations, 230--236 large order behaviour, perturbation, 437 nonequilibrium systems, 3 physical systems, 7, 255 U q [ S U ( 2 ) I symmetry, 108, 109, 119 UV see ultraviolet Vacancy excitation, 193 Vacuum state, 200, 202, 203, 441--442 Van der Waals forces, 363 Variational approximations, 352-354, 442--444, 445--449 Vector representation, particle (spin) configuration, 24 Velocity collective, 61-62, 123, 125, 147 pattern, 388

Index

Vertex models, 51,225-229 Viscosity see drift velocity Voter model, 49, 162-163 biased, 163-164, 185 Wave function, 37, 39, 84, 354, 439 Wetting, 371 Wick theorem, 214 Widom scaling theory, 230 Wilson-Fisher E-expansion, 446 World line, summing over, 255,408 Wulff shape, 230 X Y model, 6, 266 Z 2 symmetry, 181, 186 Z factors at one-loop order, 298, 301-303 at two-loop order, 342, 343 in double E-expansion, 366, 367 in E-expansion, 270 in perturbation expansion, 280, 286 in polymer modelling, 417, 424, 426 and R-operation, 318 and renormalizability, 384 and variational approximation, 351-352 Zel'dovich theory, 145 Zeolites, 43, 89 Zero-density phase, 171 Zero-field Hamiltonian, 43 Heisenberg quantum ferromagnet, 41-42 Zimm model, 259, 385-388 see also hydrodynamics

ISBN 0-12-220319-4

9

Phase Transitions and Critical Phenomena

Phase transitions and critical phenomena

Phase transitions and critical phenomena