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
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Contributors G. M. SCHUTZ, Institut fiir Fesstk6perforschung, Forschungszentrum Jiilich, 52425 Jiilich, Germany K. J. WIESE, Fachbereich Physik, Universitiit GH Essen, 45117 Essen, Germany
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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
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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
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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.
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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.
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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
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
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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
1 Exactly solvable models for many-body systems
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.
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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
1 Exactly solvable models for many-body systems
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.
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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.
1 Exactly solvable models for many-body systems
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).
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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 Exactly solvable models for many-body systems
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
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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
1 Exactly solvable models for many-body systems
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
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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
1 Exactly solvable models for many-body systems
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)
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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.
1 Exactly solvable models for many-body systems
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.
1 Exactly solvable models for many-body systems
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.
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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)
1 Exactly solvable models for many-body systems
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
<--),
I
T I
Fig. 8 Two neighbouring down-spins (a) and two distant down-spins (b).
The scattering amplitude (3.17)satisfies S(pl, P2) = S-I (P2, pl) = S*(p2, Pl). It has a pole at (real) total momentum P = ( P l d- p2)/2 with complex individual momenta P l = P / 2 - iu, P2 = P / 2 + iu where u = In [cos (Pl + P2)]. This pole corresponds to a two-particle bound state where the wave function decays exponentially in the distance x2 - x l. Finally, the periodic boundary conditions imposed on the system have to be taken into account. Because of the ordering x l < x2 one has to satisfy the two equations qJ (Xl, x2) = qJ (x2, x I "+- L) = qJ (x I -~ L , x2 + L) which leads to the Bethe ansatz equations eiplL
=
S ( p l , P2)
e ip2L
=
S ( p 2 , Pl).
(3.18)
Because of translational invariance the total momentum P = 2zrn/L is quantized in a simple way. There is no closed expression for the quantized relative momentum. Because of translational invariance the solvability of the problem with two down-spins does not come as a surprise. In the centre-of-mass frame only the relative motion remains as single degree of freedom, i.e. essentially one deals with a one-dimensional single-particle problem. The nontrivial generalization is the extension of the Bethe ansatz to the case of three (and more) down-spins. In the framework of the coordinate Bethe ansatz we write qJ (Xl . . . . . Xn) as a superposition of plane waves with pseudomomenta Pi conjugate to the particle positions xl. Since all particles are identical this suu_perposition is a sum over permutations of the momenta in the plane waves e i 2...t pitt)xt. The magic of the ansatz (which originates in the underlying integrability of the system) consists in the fact that the amplitude of each permutation in the sum factorizes into a product
1 Exactly solvable models for many-body systems
39
of corresponding permutations of two-particle amplitudes. Following this strategy one constructs the N-body wave function ql = Z
Ail ..... iN e x p (ipilxl + . . . -k- ipiNXN)
(3.19)
perm.
with all N! possible permutations of the wave numbers pi and with amplitudes Ail ..... iN which are determined by the internal boundary conditions arising from the change of the eigenvalue equation when any two spins are on nearest-neighbour sites. One can satisfy all these extra equations (analogous to (3.14)) by Bethe's ansatz where the ratio of any two amplitudes for plane waves where two momenta Pi, Pj are interchanged to p j , Pi is the scattering amplitude Sji =-- S ( p j , Pi), in the same way as in the two-particle case (3.15). For instance, for three particles the wave function reads eiplxl +ip2x2+ip3x3 + $21 e ip2xl +iplx2+ip3x3
qlpl,p2,P3 (Xl, X2, X3)
+$21 $31 e ip2x~ +ip3x2+iplx3 +$21 $31 $32 eip3xl+ip2x2+iplx3
+ $31 $32 eip3xl +ipl x2+ip2x3 + S32eiplxl d-ip3x2+ip2x3 .
(3.20)
This construction takes care of the boundary condition (3.14) when (any) two particles are on nearest-neighbour sites. The crucial point is that for a higher number of particles there are no new constraints from the boundary condition when more than two particles are on adjacent sites. This can be seen by noting that satisfying the boundary condition for any given pair is independent of the coordinates of the remaining particles. So one constructs the Bethe wave function by starting from e iplxl+'''+ipuxu with amplitude AI2...N -- 1 and then performs all possible permutations of the momenta. For each permutation (i, j) ~ (j, i) one multiplies with a factor Sji a s in (3.15). The total 'energy' e corresponding to such a wave function is the sum of the single particle energies N
(Pl ..... PN = Z
(Pi"
(3.21)
i=1
The integrability is reflected in the quantum numbers Pi which characterize the eigenstates. Because of the indirect nature of the connection, the relationship to commuting transfer matrices was understood only more than 30 years after Bethe's original solution (Lieb, 1967a,b,c). The periodic boundary conditions q/(Xl,X2 . . . . .
XN; t)
=
q/(x2 . . . . .
XN,Xl + L; t)
40
G . M . SchQtz
X N , X l + L, x2 + L" t)
qJ(X3 . . . . . 9
=
.
o
qJ(xl+L,x2+L
..... xN+L't)
(3.22)
quantize the allowed values of the pseudomomenta pi. The structure of the Bethe wave function yields a coupled set of N Bethe ansatz equations (BAE) N I
eipjL = U Sjl
(3.23)
!=1
where the product over the scattering amplitudes (3.17) excludes j - I. This is a set of coupled transcendental equations for the pseudomomenta pi. They may take complex values, corresponding to bound states. The next step in analysing the physical properties of the quantum chain at low temperatures is the identification of the ground state quantum numbers and those of the excited states and then to calculate the partition function. The latter is not directly relevant in the context of stochastic dynamics. However, it is useful to get an intuitive picture of the structure of the low energy states. Consider the logarithm of the BAEs (3.23). The solutions of the BAE may be written in the form N
L p j -- 2zr lj - ~
|
P/)
(3.24)
1=1
with a set lj of L integers (if N is odd) or half-odd integers respectively (for N even) defined modulo L. The lj provide an alternative set of quantum numbers characterizing each eigenstate of H. Like in fermionic many-body systems all lj must be distinct, i.e. a quantum state labelled by an I is either occupied once or unoccupied. The ground state in a sector with N down-spins corresponds to the N consecutive quantum numbers lj taken from the interval (Bethe, 1931" Yang and Yang, 1966) N-I N-1 < lj < ~ . (3.25) 2 2 In field theoretic analogy this choice corresponds to a filled Dirac sea. For 0 < A < 1 and in the absence of a magnetic field the absolute ground state is in the sector with N = L / 2 down-spins, i.e. in the sector with vanishing total magnetization S z. The reference state 10) serves as the pseudovacuum state whereas the 'N-particle' state (3.25) is the true, physical vacuum state with the lowest energy. The low-lying particle-antiparticle excitations are built by 'minimal' modifications of the ground state filling, i.e. by occupying states close to the 'Fermi edge' + ( N - 1)/2 (Fig. 9). For A > 1 the ground state is twofold degenerate, corresponding to the two completely ordered states with 0 and L down-spins respectively.
1 Exactly solvable models for many-body systems 0
0
0
0 0
0
0
0 0
41
0
0
9
(a)
(b)
9
9
9
0
Fig. 9 Schematic representation of the filled quantum numbers of the ground state (a) and of the first excited state (b). Higher excited states correspond to other occupations of quantum states.
3.3 3.3.1
Quantum systems in disguise: some stochastic processes Hard-core lattice gases
After this brief excursion on quantum spin systems we focus now on the connection to classical interacting particle systems via the quantum Hamiltonian formalism. A classic example where this approach has turned out to be fruitful is the exclusion process. In its simplest form this is a one-species process where each particle hops between nearest-neighbour sites with constant rate D (Liggett, 1985; Spitzer, 1970). The particles have a hard-core exclusion interaction: hopping attempts which would lead to a double occupancy of a site are rejected (Fig. 10). This process, already discussed in the introduction in its one-dimensional version as a model for reptation dynamics, can be visualized by representing particles by the symbol A and vacancies by the symbol t3 and writing 0A ~ A0
with rate
D
for the elementary hopping events between sites k, I. According to the rules set out in the previous section, forward and backward hopping across a bond between two sites k, l is given by the matrix D(nkVl + Vknl -- s'~s I -- SkS~). In terms of Pauli matrices this matrix reads D(1 - a ~ a { aYa? ' - a ~ a [ ) / 2 . Hence the Hamiltonian for the full symmetric exclusion process is given by the zero-field Heisenberg quantum ferromagnet (3.4) HSEP _--
D 2 Z(a~a~ (k,/)
+ a ~ V + a~atz - 1 )
where D = 2J (Alexander and Holstein, 1978).
(3.26)
G. M. SchQtz
42
111,
2
l
; all
1
2
3
4
5
Fig. 10 An elementary move of the exclusion process defined on a square lattice. The particle at site (2,2) has moved to site (2,3). An attempted move of this particle to the occupied site (3,3) would have been rejected. In operator language the successful move is represented by the action of the matrix s(2,2)s(2.3 + ). -
Each two-site hopping matrix hk.l =_ I -- o'~ o'~ -- o'Y o ' / -
z cri
(3.27)
corresponds to the interaction matrices of the zero-field Heisenberg chain (3.4). Hence also H set' is symmetric under the action of the generators of S U (2). The commutativity with S z expresses particle number conservation since in terms of the number operator (2.20) S z = L / 2 - N . T h e nondiagonal conservation law [ H , S + ] = 0 will be discussed in detail in the next section. Indeed, any symmetric exclusion process which can be represented by some function of the hopping matrices hk.t is SU(2)-symmetric. The rates for hopping between sites k, l of the lattice can take arbitrary bond-dependent values. The lattice is also arbitrary and sites k, l need not be nearest neighbours. Moreover, it is not necessary to restrict oneselves to the spin-(l/2) representation. If one allows for a maximal occupancy of mk particles on site k and hopping from site k to l occurs with a rate w k , / n k ( m t - - n t ) (where wk,t = W l , k ) then this partial-exclusion process is also described by a Heisenberg Hamiltonian of the form (3.26) albeit with spin matrices acting on site k in the spin-(mk/2) representation of SU(2) (Schlitz and Sandow, 1994). Hence this system is capable of describing diffusion in rather inhomogeneous media. Random energy barriers between lattice sites (reflected in the bond-dependent hopping rates), structural inhomogeneities (site-dependent maximal occupation numbers mt,) and topological disorder (choice of underlying lattice) can all be taken into account.
1 Exactly solvable models for many-body systems
43
The one-dimensional symmetric exclusion process with nearest-neighbour hopping describes not only reptation, but in its most straightforward interpretation stochastic hopping of impenetrable particles in channels or on substrates so narrow that two particles can never pass each other. The partial-exclusion process describes wider channels with up to mk particles per site. Important examples of such systems, both of physical and commercial interest, are zeolites (Kiirger and Ruthven, 1992) whose crystalline structure admits long channels between the atoms. In this context it is of interest to understand the particle current through channels of finite length. But also other physical phenomena map to this process: fluctuations of one-dimensional interfaces (Krug and Spohn, 1991; Halpin-Healey and Zhang, 1995) (see also Appendix B) and Kawasaki spin-flip dynamics with conserved order parameter at infinite temperature (Kawasaki, 1966). A closely related process is the asymmetric exclusion process where hopping across bonds is biased. Physically, this may be the result of the action of an external field driving the particles in a preferred direction, or, as in the context of polymerization on RNA templates, the result of internal asymmetric forces between particles and the substrate, or be due to self-propelling of the moving entities as e.g. in traffic flow. The process with nearest-neighbour jumps (the asymmetric simple exclusion process, abbreviated ASEP) in one dimension with hopping rates DR to the fight and DL to the left gives the Hamiltonian H ASEP --- ~ [DR(nkVk+l -- s k+ Sk+ - 1) -k- DL(Vknk+l -k
sos;+
1
=
h,,
9
k
(3.28) The hopping matrices hk have been introduced here for later reference. This process is related to the (nonstochastic) anisotropic Heisenberg chain (3.6). To see the correspondence it is convenient to introduce the hopping asymmetry ~DR q=
(3.29)
DL
The diagonal similarity transformation H - /~HASEPB-I with B - e x~kk~'~ and imaginary 'phase' angle I. = (In q ) / 2 turns H ASEP into the zero-field Hamiltonian of the anisotropic Heisenberg quantum chain (3.6) H = - x/DRDL Z [4~ ~ k
+ 0.~'0"Y z _ l)] k+l + A(0.~ ai+l
9
(3.30)
The anisotropy parameter A = (q + q - l ) / 2 is related to the hopping asymmetry q and x/DRDt, = 2J.
44
G . M . SchLitz
Somewhat contrary to unreflected intuition it is crucial to specify the boundary conditions imposed on the system. This is easily understandable though when thinking of the system in terms of particle transport: particles have a bias and hence hop preferably in one direction. It is clear that the nature of the boundaries from which they flow away and towards which they move will determine the behaviour of the system also in the bulk. Still, the system holds surprises. Below, it will be shown that changing boundary conditions can actually induce phase transitions in the system (Krug, 1991). One distinguishes four main classes of boundary conditions: (i) periodic, (ii) periodic with a defect across a bond, (iii) open with injection and absorption of particles, and (iv) reflecting. Open boundary conditions are represented by single-site boundary matrices which in quantum mechanical language correspond to boundary fields. We denote injection (absorption) rates at the left boundary site 1 of a one-dimensional system by ct (y) and injection (absorption) rates at the fight boundary site L by 6 (/3). In matrix notation one obtains the boundary matrices bl = ff(Vl - s l ) -k- y(nl - s-~) , bL = 6(VL -- s L ) -k- ~ ( n L -- S +)
(3.31)
which replace the hopping terms across the bond L, 1 in the Hamiltonians (3.26) and (3.28) respectively. Reflecting boundary conditions correspond to ct = I3 = y = 6 = 0. Any hopping attempts off the boundary sites 1, L out of the system are rejected. In the case of defect boundaries, hopping across the bond (L, 1) occurs with a different rate than in the bulk. For the interpretation of the ASEP as a quantum system it is instructive to consider the boundary terms of the transformed Heisenberg chain (3.30). The boundary terms of (3.30) depend on the boundary conditions of the original process (3.28). For periodic boundary conditions (with or without defect) the off-diagonal part of the transformed boundary hopping matrix takes the form x / D R Dr. ( q L s + s f + q - L s - [ s ~ ) . This leads to a volume-dependent non-Hermitian boundary term. Also for open boundary conditions one obtains in the genetic case (3.31 ) volume-dependent non-Hermitian boundary fields. On the other hand, the system with reflecting boundary conditions gives rise to a quantum system in the usual sense with Hermitian boundary fields ( D R - Dt.)(cr~ - a ~ ) / 4 . In the symmetric exclusion process presented above we have assumed that the particles are indistinguishable. However, an important experimental problem is posed by tracer diffusion, i.e. diffusion of many exclusion particles which move identically, but where one or several particles are singled out and tagged with some additional identifiable label. Following the motion of these tagged particles (which may, e.g. have radioactive labelling) leads to an entirely different behaviour as the collective behaviour of untagged particles. Intuitively, this is clear for a single tagged particle in a one-dimensional system. Since hopping of
1 Exactly solvable models for many-body systems
45
this particle to an already occupied site is not possible, the actual space available for its diffusive motion is limited by the (fluctuating) position of its neighbouring particles. For a finite background density p of untagged particles the average distance to its neighbours is finite (proportional to 1/p) and hence one expects the diffusion coefficient of the tracer particle to vanish. Indeed, the variance in the position of the tagged particle grows at late times only proportional to 47 rather than linearly in t in normal diffusion (Arratia, 1983; van Beijeren et al., 1983). Given the importance of the system we derive the quantum spin chain representation. We consider the untagged particles as species A and the tagged particles as species B. Following the general rules for the construction of the quantum Hamiltonian detailed in Section 2.3.2 the process may be written as a SU(2)symmetric spin- 1 quantum chain of the form H = Y~k hk with the hopping matrix +
+
hk = vk(l--Vk+l)+(1--vk)vk+l--aka-~+ 1 - b k b k + l - a - ~ a k + l - b k b k + 1. (3.32) HerenkA - - E ~ l , n k8 =-- E33 and vk = 1--n kA _ n k8 =- Ek22 are projectionoperators on states with an A-particle, a B-particle and a vacancy respectively on site k. The operators ak = E 21, a~- = E22, bk = E k23 , and b k+ ~ E 3 2 are annihilation and creation operators for A- and B-particles. Boundary fields corresponding to particle injection and absorption are constructed analogously.
3.3.2
A single-species reaction-diffusion system
Following the rules described above one constructs the most general singlespecies exclusion process with two-site interaction. This process allows in addition to particle hopping also for 'chemical' particle reactions such as A + A ~ 0, where 0 represent inert reaction products which hinder neither the hopping process nor further reactions taking place on these sites. By not considering these objects one has a situation equivalent to vacant sites denoted below by 0. For the construction of the generator of the most general process of this kind we assume all rates to be space independent. This very general process includes as special cases many well-known processes discussed later. It defines a class of systems which we hope to be sufficiently wide to allow its use as a laboratory for the study of a large variety of dynamical nonequilibrium phenomena, but not so wide that one would get entirely lost when venturing to explore the consequences of the mechanisms which the model encodes. Any pair of sites k, l in a single-species exclusion process can take four different states which leads to 12 possible transitions between them. They are represented by the transition matrices s~:s~, sff(1 ~ 0-?)/2 and s~(1 -4- 0"3)/2 which
46
G.M. SchQtz
Table 1 Two-site reaction-diffusion processes on a pair of sites (x, y), their rates and representation by the off-diagonal matrices Process Rate Transition matrix Diffusion I ---> k Diffusion k --> l Pairannihilation Pair creation Fusion on k Fusion on l Branching to k Branching to l Death on k Death on l Birth on k Birth on l
0A --, A0 AO ~ OA A A ~ OO 00 ~ A A A A --. AO A A ~ OA OA ~ A A AO ~ A A AO ~ 0 0 (~A ~ 00 00 ~ A0 ~0 ---, 0A
W23
SkS ~S-~S1
1/314
s-~s-/-
l/)41
Sk S1
1/)34
W43
nkSf ~ s~nt s k nt nks l
1/113
S~-l;l
W l2
VkS/+
W32
W24
w42
m
W31
S k 1)1
w21
vks I
all contribute to the pair transition matrix Wll
w12
w13
w14
W21
W22
//323 W24
W31
W32
W33
W34
W41
1/)42 W43
W44
(3.33)
hk,l = --
k,l
acting nontrivially on sites k and l of an arbitrary lattice. This matrix is a sum of the tensor products of attempt matrices for the elementary moves given in Table 1. The diagonal elements tl)ii of hk,t satisfy 4
Wii :
-- ~ " Wi, i it=l i'~:i
(3.34)
which is imposed by conservation of probability and ensures (s Ihk,t = 0 for all k, l. In order to keep the interpretation of H as defining a stochastic process, (3.34) has to be supplemented by the condition ll)ii, > 0 on the rates. For a general process with space dependent rates the matrix elements are functions w i j ( k , I). We shall consider only processes where nearest-neighbour transition rates are nonvanishing and constant. In such a case one has h k,t = 0 if k, l are not nearest neighbours. The Hamiltonian of the process then reads H -hk,t where the sum runs over all distinct pairs of lattice sites. The processes
1 Exactly solvable models for many-body systems
47
described by H are reactions changing the configurations on at most two sites with rates lloii, given in Table 1. There is no generally accepted nomenclature for these processes. The fusion process is often called coagulation or coalescence. Branching, the back reaction to fusion is then called decoagulation. Many nearest-neighbour processes in one dimension where l = k + 1 are integrable and thus play a special role. It is worth introducing the special notation hk --= hk,k+ l and writing H = Z hk k
(3.35)
as already done in (3.28) for the asymmetric exclusion process. It is important to bear in mind that in one dimension with periodic boundary conditions tho matrices hk which are defined by the transition rates llOij may be redefined by adding an arbitrary divergence term of the form dk,k+l = Ak -- Ak+l without changing the process. This is so since H = Y]k hk = Y]k(hk + Ak -- Ak+l). The single-site matrix Ak has the general form Ak = dis~ + d2sk + d3nk and hence
d k . k + l = --
0
dl
-dl
0
d2
d3
0
-d]
-d2
0
-d3
dl
-d2
d2
0
0
(3.36) k,k + 1
Adding such a part to hk,k+l changes both the condition (3.34) of conservation of probability and the positivity constraints. For instance, instead of the four inequalities Wl2, Wl3, w24, 1/334 >__0 one has only the three independent inequalities (1/)12 + WI3), (W24 + W34), (WI2 q- 1/324) >__0.
3.3.3
Glauber dynamics
A phenomenon of wide interest in physics and chemistry is the growth of domains in nonequilibrium two-phase systems. The best-known example is perhaps the Ising model with domains of up- and down-spins, separated by domain walls. The energy of the zero-field Ising model is given by the nearest-neighbour sum E = - J y] sks/(3.2). Since the creation of a local domain wall costs an energy J the system tries to organize itself at low temperature into large domains of uniform magnetization. Starting from a high-temperature equilibrium state with many domain walls and quenching to low temperatures leads to a coarsening process: small domains of uniform magnetization merge to form larger domains since then the total length of the domain walls and thus the energy decreases (Fisher and Huse, 1988; Bray, 1994; Sire and Majumdar, 1995).
G.M. Sch6tz
48
The energy function (3.2) describes only the static equilibrium behaviour of the Ising model. In order to study dynamical effects Glauber (1963) introduced spin-flip dynamics for the one-dimensional Ising model which ensure that the system reaches the equilibrium distribution exp ( - t I E ) at temperature T = k / f t . In this model a spin within a domain of equal magnetization is flipped with a rate # = 1 - y, whereas a spin in a region of opposite magnetization is flipped with a rate ~. = 1 + t' with y = tanh (2fl J). At domain boundaries spins are flipped with unit rate, since no change in energy is involved (cf. Fig. 6). In one dimension, this process can be visualized in the following way: 'I" '1" ]' ~
]' ,1, 1' and ,1, ,1, ,1, ~
]',1, t ~
]' 1' 1' and ,I,l',l,--~ $4,,1,
]'1',1,~]',1,,1,
$ ~ ,1,
and $ $ ] ' ~ , 1 , ' 1 ' 1 "
with rate
/x
with rate with rate
1
To construct the corresponding quantum Hamiltonian we first note that the spin-flip event is represented by the Pauli matrix cr~ with corresponding diagonal part - 1 to conserve probability. The spin-flip rate w(r/) depends on the configuration of neighbouring spins and is given by Wk = 1 -- y s k ( s k - 1 + Sk+ 1)/2. Hence the generator of the process reads H
=
(1 - t y k ) ( 2 - yo"kz ( 4 - 1 + O.k+l) )z
~ k
-
2
(2
a/, - cr]+l)(1
~'a~cr~+ l)
(3.37)
Even though the interaction involves three neighbouring spins the translational rearrangement of elementary transition matrices terms performed here demonstrates that in one dimension the process can be interpreted in terms of two-site processes of the form (3.35). This process can then be mapped to a reaction-diffusion system in a straightforward way by identifying an up-spin with a vacancy and a down-spin with a particle. In particular, at zero temperature one obtains: A ~ or ~ A A O or O A
~ ~
AA
with rate
1
r
with rate
1
in which case one finds (Felderhof, 1970) Hv M
1 Z = .2
. (2
x . o-j~ .
x O'k+l)(1
z z crk ~ 1) 9
(3.38)
k
In the presence of a weak magnetic field one finds the same process, but with different rates for ending up in the states A A or ~30 respectively.
1 Exactly solvable models for many-body systems
49
The generalization of this nearest-neighbour process to higher dimensions is generally known as the voter model (Liggett, 1985, 1999). The flip rate for a given spin at some lattice site is equal to the number of nearest-neighbour spins of opposite value. Thus an atom (= human being) changes his magnetization (= opinion yes or no) at a rate proportional to the opinions of his neighbours!
3.3.4
Diffusion-limited p a i r annihilation
A different mapping to a one-dimensional reaction--diffusion system is obtained by identifying a domain wall (1' $ or ,[. 1") with a particle of type A on the dual lattice (defined by the links of the original chain) and no domain wall with a vacancy 13. The Glauber process turns into a reaction-diffusion system with pair annihilation and pair creation of exclusion particles (R~icz, 1985) t3 0 --+ A A
with rate
#
A A --~ 0t3
with rate
~.
13A ~
with rate
1.
A0
At zero temperature there is no creation (/z = 0, ~. -- 2). Thus the system evolves into the single absorbing state with no particles at all. In spin language this is the totally ferromagnetic state with all spins up or all spins down. This process of diffusion-limited pair annihilation (DLPA) is of interest not only for the study of spin relaxation and coarsening, but also for the understanding of the dynamics of laser-induced excitons on polymers as seen in experiments (see Section 9; for an extensive review, see Kopelman and Lin, 1997; Kroon and Sprik, 1997). The quantum Hamiltonian describing the stochastic time evolution of the domain walls is given by (Felderhof and Suzuki, 1971; Siggia, 1977) H DW -_ -y~
[r
sk+ , + Xs k+ sk+ + , + s k+ sk+ - , +
s~s-~+
1
+ (Ix-
1)cr~- 1].
k
(3.39) It is related to the quantum Hamiltonian of the anisotropic transverse X Y model in a magnetic field (Lieb et al., 1961) by a simple diagonal similarity transformation B = q s~ with q - v/-~--/~.. In the version (3.39) of the process the hopping time scale is determined by the spin-flip time scale of the Glauber process. In the general pair-annihilationcreation process it is an independent constant. In the limiting case without any hopping one obtains the quantum Hamiltonian for random sequential adsorption and evaporation (Stinchcombe et al., 1993) H R S A E -- -- Z
+ 1 -- nknk+l )]. [/Z(S/~ Sk+ 1 -- VkVk+l) + ~(S k+ Sk+ k
(3.40)
50
G.M. Sch6tz
In the absence of pair annihilation (,t. = 0) this process reduces to random sequential adsorption (Evans, 1993, 1997). There is no mapping of Glauber dynamics to DLPA via the domain wall picture in higher dimensions.
3.4 Algebraic properties of integrable models When studying a reaction--diffusion system it is generally not easy to check whether one has an integrable process or not. Some insights can be obtained from algebraic properties of the local interaction matrices. We recall the representation H = ~ k hk (3.35) of reaction--diffusion systems where the hk play the role of stochastic matrices for local (e.g. nearest neighbour) transitions. For the exclusion process, the pair annihilation process and some other one-dimensional reaction-diffusion processes Alcaraz and Rittenberg observed the remarkable property that the matrices hk satisfy the relations
hkhk+lhk-hk
=
[hk, htl = h2k =
hk+lhkhk+l--hk+l,
(3.41)
0 Ik-ll>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)
1 Exactly solvable models for many-body systems
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
1 Exactlysolvable models for many-body systems 4
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
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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* )
1 Exactly solvable models for many-body systems
=
(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)
1 Exactly solvable models for many-body systems
.
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
1 Exactly solvable models for many-body systems
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.
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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 << t << L the usual algebraic, diffusive behaviour of the conditional probability. On large scales the density profile is given (in good approximation) by the usual scale-invariant Gaussian distribution with diffusion constant D = Do(q + q - l ) / 2 = (De + DL)/2 (see Fig. 14). This is consistent with the vanishing energy gap of the periodic system even though, at first glance, it appears to be inconsistent with the asymptotically finite energy gap Emin. The point is simply that in this time range the probability distribution is exponentially small at the boundaries. The system did not have sufficient time to explore the full state space X -- 1. . . . . L and hence did not yet 'feel' the presence of the boundary. (ii) After a finite time of the order of the system size the particle has reached the neighbourhood of the right reflecting boundary with a probability of order 1. Now, for y 4- vt .~. L, the distribution cannot further spread diffusively around the centre of mass position. The temporal behaviour of the conditional probability crosses over to an exponential decay to its stationary value with relaxation time r = 1/Emin. One does not expect the details of the left boundary to play a role for the latetime behaviour of the system. The point we want to make here is that the system 'feels' the finite energy gap Emin only after the centre of mass of the density distribution has travelled in the vicinity of the boundary, where by necessity, it
68
G . M . Schfitz
cannot decay any longer algebraically. Only then the finite energy gap determines the asymptotic behaviour of the system. The same line of reasoning is applicable also to interacting particles. Below we show that for the asymmetric exclusion process at finite density the centre of mass of a local perturbation in a translational invariant background of finite density p travels with the velocity v = (DR - Dt.)(I - 2p). By going into a frame of reference with this velocity one expects the algebraic decay of the amplitude of the perturbation. On the other hand, the finite system with reflecting boundaries has a finite energy gap even in the thermodynamic limit. The reservations expressed above on the conclusions one is allowed to draw from the spectral properties of a given system somewhat temper the usefulness of the spectral classification of reaction-diffusion systems through generators of Hecke algebras. Some care needs to be taken when extrapolating spectral properties of a finite system to the relaxational behaviour in the thermodynamic limit. However, once one representative of a given quotient is sufficiently well understood, one may gain valuable information on the relaxational behaviour of other representatives of the same quotient.
Space--time anisotropy As discussed above, the time direction of the stochastic time evolution is one of the space directions of the associated (d + 1)-dimensional equilibrium system. By analogy to usual critical phenomena this might lead to the erroneous conclusion that algebraic decay of correlations in time would always imply algebraic spatial decay of correlations in the stationary state. This would be correct if the (d + 1)-dimensional system was invariant under rotations which turn the 'time'-axis into one of the other space directions. But in general this is not the case and spatial correlations in the stationary state may be characterized by length scales independent of the relaxation times. An example are Kawasaki dynamics (Kawasaki, 1966) for the Ising model where the relaxation is algebraic, but spatial correlations decay over the finite temperature-dependent length scale ~ characteristic for the Ising model (for a review, see Cornell, 1997). For the infinite-temperature limit where Kawasaki dynamics reduce to the symmetric exclusion process it will be shown below that the relaxation of local perturbations is algebraic in time, but the stationary distribution has no correlations in space (corresponding to vanishing correlation length).
Spontaneous symmetry breaking Since two-dimensional systems with shortrange interactions may show spontaneous symmetry breaking (an example is the Ising model), it is not surprising that this phenomenon may also occur in onedimensional stationary states of systems with local interaction (Evans et al., 1995, 1998; Godr~che et al., 1995; Arndt et al., 1998b; Helbing et al., 1999). Of course,
1 Exactly solvable models for many-body systems
69
in a finite ergodic system spontaneous symmetry breaking cannot occur, since the stationary distribution is unique. Hence the stationary expectation value of the order parameter does not reveal whether the symmetry is broken or not. The typical signature of spontaneous symmetry breaking in the finite system is an exponentially small energy gap Emin "~ e -aL in the transfer matrix (or quantum Hamiltonian respectively). Dynamically, the system flips between the state with broken symmetry, but the flipping times are exponentially large in system size. Correspondingly, a finite dynamical system reaches its true symmetric stationary distribution only after times which are exponentially large in system size. In the thermodynamic limit there are then two (or more) stationary distributions. The system becomes nonergodic and the symmetry is broken in the various stationary distributions. A simpler form of spontaneous symmetry breaking may occur in finite dynamical systems with two absorbing domains as shown in Fig. 12. In this case there are two stationary distributions already in the finite system. The 'flipping time' between these states is infinite.
4.3
Separation of time scales
In a complicated interacting particle system defined in discrete time some processes may occur with probability 1. The analogue of such a transition in the continuous-time description is a process occurring with infinite rate ~ --+ c~. Such transitions arise in the study of systems where some dynamical events are rare, whereas others occur, in comparison, frequently. Then taking such an infinite rate limit for the fast processes yields a zeroth order approximation of the slow process, perturbed by the fast process. In a number of cases this leads to a solvable problem while still retaining part of the essential physics of the system. Such a limit is a legitimate approximation for systems where different types of processes occur on strongly separated relaxation time scales rshort << Z'long. We stress that this limit is not equivalent to taking the limit of the rates of the slow process going to zero. In the infinite rate scenario one observes the process on time scales associated with the slow process (where the fast process is treated as a perturbation), whereas in the second case one would observe the process on time scales associated with the fast process (and where the slow process might be considered a perturbation). It is, of course, not possible to take the infinite-rate limit directly in a quantum Hamiltonian. Instead one has to study the time evolution operator for finite time intervals exp ( - H t ) and consider the limit l i m x ~ exp ( - H t ) where H is assumed to be a function of ~. One writes H = H0 + Z HI where H1 describes all fast processes. Using the quantum mechanical Schwinger-Dyson formula one
G. M. SchQtz
70
expands e-(Ho+~.H~)t
=
e -xnlt
=
[ /0' 1 --
drlHo(rl) +
drl
dr2
[fo t fOtfrtfrt 1-
--
dr2Ho(rl)Ho(r2)
fot frt drl
(4.37)
dr2Ho(-rl)Ho(-r2)
I
dr2
I
drl
dr3Ho(rl)Ho(r2)Ho(r3) + . . .
drl H o ( - r l ) +
drl
f0' f0
dr3Ho(-rl)Ho(-r2)Ho(-r3) 2
+...
]
(4.38)
• e -xnlt
where H0(r) = e ~'HIr Hoe -xH'r.
(4.39)
In the integral all factors H 0 ( r l ) . . . H0(rn) are time-ordered" in the first representation (4.37) 0 < rn _< . . . < rl < t and in the second representation 0 < rl < .-- < rn < t. In quantum mechanics this formula is normally used perturbatively for small k. In the same way it can be used in the context of stochastic dynamics to derive fluctuation-dissipation relations for equilibrium and nonequilibrium systems (H~inggi and Thomas, 1982). Here its value lies in taking the limit ~. ~ oo which gives e - ~ H ~ t H o ( r ] ) . . . H0(rn) ~ ( T ' H o T * ) n where T* is defined in (4.5). Hence lim e -(H~
= e -B~ T*
(4.40)
~.---, o o
where /)0 = T* H0 T*
(4.41)
is the generator of the limiting process. This result has a simple intuitive interpretation: first any initial state is mapped by T* into a stationary distribution by the infinitely fast process Hi. Then the process corresponding to H0 takes place, but immediately again the new state is mapped to the stationary distribution of Ht corresponding to the new state, and so on.
Comments
Section 4.2: (i) Many of the most interesting results on reaction--diffusion systems relate to critical phenomena which have a natural description in terms of the renormalization group approach. We refer for an introduction into critical phenomena in general to Yeomans (1992). An introduction to the renormalization group can be found in Cardy
1 Exactly solvable models for many-body systems
71
(1996), and specifically for dynamical critical phenomena in reaction-diffusion systems in the brief review (Cardy, 1997). Field-theoretical approaches to reaction-diffusion systems are extensively reviewed in Mattis and Glasser (1998). (ii) There are other renormalization techniques which directly use the quantum Hamiltonian representation of the generator. A promising real-space renormalization approach in the spirit of the so-called SLAC approach for quantum lattice models is capable of recovering exact results for some of the reaction--diffusion systems discussed in Section 9 (Hooyberghs and Vanderzande, 2000). Moreover, good estimates of critical parameters of the contact process (cf. (3.35) with w12 -- w13 -- w24 = w34 = 1, w42 = w43 = ~.) have been obtained. This process has a transition in the universality class of directed percolation from an active stationary state with finite density to an absorbing state with vanishing density (Liggett, 1985, 1999; Dickman, 1997). Despite its great theoretical importance few exact results are available about the contact process. No integrable model in this universality class is known. (iii) In another approach the density-matrix renormalization method (DMRG), also originally developed for quantum spin chains has been applied successfully to describe reaction-diffusion mechanisms (Kaulke and Peschel, 1998; Carlon et al., 1999). This is a numerical technique which uses a cut-off procedure in the underlying Hilbert space.
72 5
G.M. SchQtz Equivalences of stochastic processes
Many systems are far too complex to be amenable to analytical or even numerical investigation. However, particularly in the context of critical phenomena, simple toy models may suffice to determine universal properties correctly and to predict and explain observed power laws or universal amplitude ratios. Such universal behaviour is possible since many systems are known to be scale-invariant: irrespective of the microscopic details of the interaction they look identical on a wide range of length- and time scales (Yeomans, 1992). Hence it is of importance both to examine the behaviour of such models and to understand possible relationships between microscopically different processes and their characterization in terms of universality classes. A most appropriate approach to this problem is the renormalization group treatment (Cardy, 1996, 1997). In this way it can be understood why, e.g. the experimentally observed power law decay p ( t ) "~ t -1/2 of the density in onedimensional annihilation reactions of the type A + A ~ p r o d u c t s does not depend on whether the products still contain single A-particles (fusion reaction A + A --+ A) or not (A + A ~ 13) (Peliti, 1985). On the other hand, the results of the previous sections point to exact equivalences between stochastic processes and at the same time make clear that an understanding of these relations between processes beyond a purely algebraic level is required.
5.1
Similarity transformations revisited
There are two distinct types of relations which both involve similarity transformations between the generators of the process. We shall refer to these relations as equivalence and enantiodromy respectively, to be defined below. Similarity transformations between two stochastic processes (rather than between a stochastic and a nonstochastic Hamiltonian) have emerged as a tool of major importance in the study of reaction-diffusion mechanisms in the quantum-spin representation. A basic transformation of the stochastic variables of a given process may map this system to a microscopically quite different process. However, the spectral properties of the time evolution operators for such processes are the same. Moreover, if one manages to calculate expectation values for one system, one automatically obtains expectation values of the transformed systems as well. Two stochastic processes represented by Hamiltonians H , / q (or transfer matrices respectively) and related by 151--13H13 - l
(5.1)
for some invertible matrix/3 are called equivalent. For expectation values this implies that an expectation value of a function F for the process H with initial
1 Exactly solvable models for many-body systems
73
condition P can be expressed in terms of the expectation value of a transformed function F (which depends via the transformation B on F) for the process H with transformed initial condition P (which depends on/5). Inserting the definition of gives the relation between correlation functions ( nkl (tl ) " " " nkm ( t m ) ) Po - - ( fikl (tl ) " " " h k ~ ( t m ) ) Po
(5.2)
where the correlator on the r.h.s, is measured for the process H and the correlator on the l.h.s, is measured for the process/q. The transformed initial distribution is given by I/5o ) = BI P0 ). Note that the transformed observables (5.3)
tlki = ]~nki ]~ - 1
are in general nondiagonal. Since both H and H are stochastic, the summation vector ( s I remains invariant under such a transformation. In the theory of interacting particle systems the factorized transformations (5.4)
13 = Bl B2 . . . BL
where each matrix Bi acts nontrivially only on site i play a special role. Because of the tensor structure of the state space local observables for a given set of lattice sites are transformed into other local observables for the s a m e set of sites. Also local interactions of the process are transformed into interactions involving the same set of sites. Thus factorized transformations preserve the locality of the process and of the observables. In the simplest case all Bi are identical, i.e., B - B| Such a transformation is a homogeneous factorized transformation. With this technique the equivalence between the pair annihilation process with annihilation rate Wl4 - ~. and hopping rate w23 = w32 = D and the fusion process can be proved. The homogeneous factorized matrix (5.4) with B = e aCn-s+) and 0 _< ct < In2 yields the relation (Krebs et al., 1995; Simon, 1995) H DLFPA = B H DLPA/~-I.
(5.5)
The stochastic Hamiltonian H DLFPA describes the mixed process of diffusionlimited pair annihilation A A ~ 1313 with rate W14 ---- (2e -a - 1)X, fusion A A 13A, AI3 with rate tb24 = tb34 = (1 - e - a ) X and hopping rate tb23 = 5)32 = D. For c~ = In 2 one obtains the pure diffusion-limited fusion process H DLF. Correlation functions can be related by using the equivalence (5.5) inside the expression (2.25) for the expectation value. For equal-time correlators with uncorrelated random initial conditions (2.24) with density P0 the equivalence relations read nkm (t) }DLPA __ earn ( n k (t)
( nkl (t) 9
"
"
9 p0
~
nkm (t))DLFPA "
"
"
Po
(5.6)
with the transformed random initial state with density Pof = poe-~ This relation follows from the factorized form of the transformation which retains the
74
G.M. Sch~z
locality of the observable and from the explicit form of the action of the (nondiagonal) transformed observable on the summation vector, ( S Ihki = e ~ ( S Inki. Similar relations are obtained for the diffusion-limited pair-annihilation--creation process (3.39) and diffusion-limited branching-fusion (Krebs et al., 1995). Note that while the similarity transformation on the generator is defined in the full interval 0 < c~ < In 2, it is a stochastic similarity transformation (Henkel et al., 1995) only on the restricted interval defined by the requirement that the initial t t densities of both systems satisfy 0 < P0' ,o0 < 1. For initial densities/90 > 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.
1 Exactly solvable models for many-body systems
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) <131e x p ( - / - ) t ) ) and the first-passage-time probability is the (negative) time derivative of g (1" t). Using the infinite rate expansion of the last section one may write 131exp (-/-) t) = limx_,~ exp [ - ( H + kn I )t ]. More generally, one may consider the persistence probability p ( Q ; t) <7~t )P0 that the system has always remained in the subset of states Q up to time t under the initial condition P0. This problem plays an important part in the
G.M. SchQtz
76
study of interface dynamics where one investigates the probability that a growing interface remains above (or below) its mean height (Kallabis and Krug, 1999) or the first passage probability for returning to its initial height (Kruget al., 1997). Such a generalized persistence probability is generated by T = Q T Q where Q is the projector on this set of states. In continuous time one projects with Q on the initial distribution I P0 ). The time evolution is then given by exp ( - H t ) with /-) = Q H Q. This makes sure that the condition defined by Q is never violated in the continuous time evolution, i.e. all paths the stochastic time evolution may take and which leave Q are projected out. Finally one takes the sum over all resulting final probabilities, i.e. p(Q" t) = ( S IQ exp ( - Ht)l P0 ). Notice the commutation relation [ Q, exp ( - / - ) t ) ] = 0 . An interesting question may serve to illustrate the importance of the concepts of infinite rates, first-passage-time problems and equivalence. In the language of kinetic Ising models g(l; t) is the probability that the spin at site 1 has always been an up-spin until time t. The probability that the spin has always been down is given by a similar function ~(1" t) where Vl is replaced by nl. Consider now the probability p(l" t) that in zero-temperature zero-field Glauber dynamics a spin at site k - 1 has never flipped up to time t. For definiteness consider a random initial distribution (2.24) in which each spin configuration has equal probability. This distribution is given by the vector I P0) = Is )/2 L. Because of spin-flip symmetry p(l" t) is twice the probability that the spin has always been down. Thus p(1; t) = 2 lim ( S l e x p [ - ( H + ~.vl)/]l P0). (5.10) ~.---~oo
The point we want to make is that this quantity which is defined by a continuous measurement over the whole interval of time [0, t] (and hence is nonlocal in time) can be expressed in terms of a quantity which is measured only at the end of this interval (and hence is local in time). It is remarkable that in some cases the non-stochastic generator H + ~.Vl for this quantity is equivalent to a stochastic Hamiltonian. Then one can relate the persistence probability to an ordinary expectation value at time t. To show this for the problem at hand we use the tensor property of the summation vector and the decomposition of the (unnormalized) product state L
e=S-10 ) - ~ N=0
(ots-)N
N~I0)--
L
ZotNIN )
(5.11)
N=0
with density p = c~/(l + ct) in random states with fixed particle number. IN ) is the unnormalized distribution where all configurations with N particles (downspins) have equal weight 1. Analogously one defines the transpose ( N I = I N )T by replacing S - by S + in (5.11 ) and acting with the exponential to the left on { 0 I. The row vector ( N I is the summation vector restricted to states with N particles
1 Exactly solvable models for many-body systems
77
(down-spins). Using the factorization of the tensor space into two-dimensional spaces C 2 and the factorization of e s§ one can now easily verify L
eS+l po)=
Z2-NIN).
(5.12)
N=0 Finally define the matrix H,,~ = e s + ( H + ~.vl)e- s + 9 Then, by transposition, (5.10) can be rewritten L p(l" t) -- 2 lim ---- ~ 2 - N ( N ~.---~oo N=0
lexp-/Qxtl 0).
(5.13)
It remains only to calculate/4z. The transformation e s+ applied to the Glauber s + . The e s§
pair transition matrix (3.38) yields two-site matrices /~" =
transposed/~k is stochastic with nonvanishing rates w23 = w3e -- w42 =//)43 : 1. This is the fusion process with symmetric hopping (Table 1). The transpose of the transformed boundary matrix ZVl is a stochastic particle injection matrix bl = ~.(Vl - s~-). Hence p ( l ; t) is given by the fusion process with hopping and injection at site 1 with infinite rate. We conclude that the quantities entering the sum in (5.13) are the probabilities of finding exactly N particles at time t if initially the lattice was empty. This is an expectation value which is local in time and both simpler to measure and to treat analytically. Using the same formalism one can calculate p ( l ; t) for other initial conditions.
Comments Section 5.1" The equivalence between DLPA and DLFPA can also be formulated for non-interacting particles with finite annihilation rate (Schfitz, 1997c). Because of the factorized form of the transformation it holds even if the process takes place in a fractal or a disordered environment. Thus the usefulness of similarity is not restricted to integrable systems. Section 5.3: The relation (5.13) and its generalization to Glauber dynamics for the q-state Potts model was first found by Derrida (1995) by tracing back the history of the value of a Glauber spin. This corresponds to considering coalescing random walks as described by the fusion process, but going backward in time. This argument expresses the physical interpretation of the enantiodromy relation under the transformation e S§ between Glauber dynamics and the diffusion-limited fusion process. Taking an infinite injection rate ensures that at the injection point there is always a particle. This corresponds to selecting only such walks which correspond to a spin at the injection point that has never flipped. Assuming random initial conditions and going back in time to t = 0 these walks lead to
78
G.M. SchQtz
N different ancestors with probability P L ( N ) = ( N IP*) and therefore to (5.13). Using free-fermion techniques based on those described in Section 9 the persistence probability p(l" t) could then be calculated (Derrida et al., 1996).
1 Exactlysolvable models for many-body systems 6
79
The s y m m e t r i c exclusion process
In Section 2.3.1 we defined the symmetric partial-exclusion process on an arbitrary finite lattice with L sites and with arbitrary space-dependent hopping rates wk,t -- wt,k. In this general form the process is not integrable, but the SU(2)-symmetry has some surprisingly strong consequences. We present an improved discussion of the rederivation and extension (Schlitz and Sandow, 1994) of Spitzer's duality relations (Spitzer, 1970) for total site exclusion using the S U (2)-symmetry of the process. These relations are valid in all dimensions. The generalization to partial exclusion (SchLitz and Sandow, 1994) proceeds along analogous lines and is therefore not treated here. In a further step we exploit the integrability of the one-dimensional model by using the Bathe ansatz (Bethe, 1931). We show that all m-point correlation functions reduce to conditional probabilities of noninteracting particles plus subleading correction terms due to the hard-core interaction. In Appendix B we use this result to prove universality of interface fluctuations in a class of two-dimensional two-phase Ising systems and to calculate finite-size scaling functions for interface fluctuations. A derivation of expressions for first-passage-time distributions leads us to consider the exclusion process with particle injection and absorption. The one-dimensional process with open boundaries is reformulated in a purely algebraic manner (Stinchcombe and Schlitz, 1995a,b). The representation-free treatment of the algebra (Schiitz, 1998) yields explicit expressions for both stationary and time-dependent correlation functions (Santos, 1997b) and leads us to conclude that for random initial states the open dynamical system shows quasi-stationary behaviour.
6.1
SU(2)-symmetry and stationary states
As pointed out above, any symmetric exclusion process generated by some function of the hopping matrices (3.27) is S U (2)-symmetric and therefore all results derived in this and in the next two sections hold for all such processes. It is natural to begin with a classification of the stationary states of the system. Since H is symmetric and conserves particle number it follows immediately that any constant distribution IN* ) of N particles is stationary. This state is nothing but the normalized transposed vector of the constant summation vector ( N I restricted to the N-particle sector. It is instructive to see how this property follows from the S U (2) symmetry (3.5). Since the empty lattice 10 ) is a stationary state, and since H commutes with S-, each unnormalized N-particle state (S-) N
IN) - ~10) N!
(6.1)
has eigenvalue zero and hence defines also a stationary distribution IN*) =
G . M . Sch(itz
80
IN )/ZL,N, normalized by the trivial partition function ZL,N = ( S I N ) = L ! / ( N ! ( L - N)!) which gives the number of possibilities to place N hard-core particles on a lattice L sites. Algebraically speaking, these states are the highest weight states of S U (2) with total angular momentum S = L / 2 and z-component S z = ( L - N ) / 2 . T h e point of this derivation is its straightforward generalizability to stochastic processes with other global non- Abel i an symmetries. Examples include the partial exclusion process introduced by Schlitz and Sandow (1994) or the S U ( N ) - s y m m e t r i c pair-exchange process where each lattice site can take N different states and in each elementary move the states of sites x and y are interchanged (Alcaraz and Rittenberg, 1993; Albeverio and Fei, 1995) and, in one dimension, to the asymmetric exclusion process with the q-deformed analogue of S U (2) (see Section 7). The stationary distribution defined by (6.1) is a canonical distribution with a fixed number of particles. A special linear combination of these stationary states is the grand canonical factorized state (2.24)
Ip ) = ~
L
(] -
p)L-NpNI N )
(6.2)
N=O
with a binomial distribution of the total particle number N and vanishing correlations, i.e. ( S Ink, . . . nkm I P ) = pro. On the other hand, ( S Ink, . . . nk m IN* ) = pm [ - l , ~ l (1 - I / N ) / ( 1 - l / L ) . This shows that the N-particle distribution IN ) converges to I,o ) in the limit L, N ~ oo (with ,o = N / L fixed) in the sense that all correlation functions of finite order m converge to the uncorrelated value pro. Any finite subsegment of a large canonical ensemble behaves essentially like the grand canonical ensemble defined by the product state (6.2). Intuitively speaking this means that on a finite interval the stationary system does not 'know' that the system contains a specific number of particles. Finite-size corrections of the m-point correlator are of order 1/ L.
6.2
Nonequilibrium behaviour
In connection with the stationary states of the Heisenberg chain S U (2)-symmetry shows little more than what is obvious. Its real value is the resulting selfenantiodromy of the process with respect to any homogeneous factorized similarity transformation of the form 13 - B | i.e. H r = BHB -l.
(6.3)
In the quantum Hamiitonian formulation of the process the proof of this relation is a trivial consequence of the SU(2)-symmetry of H" since any homogeneous
1 Exactly solvable models for many-body systems
81
factorized transformation 13 can be written/3 = eaS+el3S-e• and since H is symmetric, the statement (6.3) follows. This property has a number of striking consequences, some of which are the famous duality relations originally found by Spitzer (1970) using a property of the process known as self-duality.* For the m-point equal-time correlator with an arbitrary initial distribution P0 one has ( nkl (t) . . . nkm (t) ) eo
y~
( nil ( 0 ) . . . nlm ( 0 ) ) P o x
ll ..... linES
P(l]
.....
In; tlkl
.....
km; O)
(6.4)
where P ( l l . . . . . lm; tlk] . . . . . km; 0) is the m-particle conditional probability which gives the probability of finding particles on sites l] . . . . . lm at time t given that they were on sites kl . . . . . km at time 0. In other words, relation (6.4) states that the m-point equal-time correlator with initial distribution P0 (i.e. the m-point joint probability of finding particles on sites k l . . . . . km at time t in an arbitrary N-particle state) is given by its initial value and the m-particle conditional probabilities P(ll . . . . . lm; tJkl . . . . . km; 0). Thus self-enantiodromy reduces the calculation of the m-point correlator for the many-particle system to the solution of the symmetric exclusion process involving only m-particles. We do not review the derivation by Spitzer, but derive these relations from the S U (2)-symmetry. Notice that s~- creates a particle when acting to the left on the vacuum, i.e., ( 0 I s ~ . . . s km + = ( kl . . . km J. With relation (2.23) one then finds ( S Ink, . . . nkm -- (kl . . . . . km le s+
(6.5)
The central idea is to use now the SU(2)-symmetry to commute e x p ( S +) with exp ( - H t ) . In the final step of the calculation one uses conservation of the zcomponent of the angular momentum of the Heisenberg chain to insert a unit operator I = Y~4~..... /,,~s J I I . . . . . lm )( l] . . . . . In I restricted to the m-particle sector and applies again (6.5). Here we use the expression Y~ll linES as a sum over all distinct sets of m lattice sites on the lattice S. This proves (6.4). Adapting a spin-wave argument by Alexander and Holstein (1978) the enantiodromy relations may be understood algebraically as a consequence of the fact that the projector nj, is a spin-(I/2) operator and of the corresponding selection rules of SU(2)" as observed above, the state ( N I in the chain with L sites has total angular momentum S = L / 2 and z-component S z = L / 2 - N. Therefore ( N In/,, -.-nk,, may be decomposed into states with L / 2 > S' > L / 2 - m and .....
'This duality is not to be confused with the domain wall duality in spin systems, but is a special case of enantiodromy. In order to avoid possible misunderstandings and for consistency of language we shall use the notion of (self-)enantiodromy, even where only (self-)duality in the probabilistic sense is meant.
82
G.M. SchOtz
S z' = L/2 - N which are obtained from m-particle states using the lowering operator S - . Thus only m-particle amplitudes enter into the r.h.s, of (6.4). Seeing the enantiodromy relations in this way provides an algebraic interpretation of the relations which allows for generalizations to other systems with non-Abelian continuous symmetries. For genetic N-particle initial distributions the self-enantiodromy relations (6.4) imply that the dynamics of the density profile (nk(t) ) are determined by the conditional probability P(k; tll; O) = ( k le-Ht l l ). This is the probability for a single random walker on an otherwise empty lattice to be found at point k at time t given that it started at site l at time t = 0. This relation is valid on any lattice. On an infinite one-dimensional lattice with nearest neighbour hopping rate D the conditional probability satisfies the differential-difference equation (2.3) with DR = DL = D and with initial value P(k" 01l; 0) = ~k,l. This equation is readily solved by Fourier transformation (see Section 4.2.4) and one finds P(k" tll" O) = e -2Dt Ik_t(2Dt) (6.6) The enantiodromy relations have a straightforward extension to correlators at unequal times (Schiitz and Sandow, 1994). We consider time-delayed correlation functions in the stationary state which describe the density fluctuations in equilibrium. Using the algebraic property (6.2) of the stationary distribution yields I P ) = (1 - p)L/2+SZpL/2-SZls ). With the stationarity of the one-particle state I 1" ) = ~ y l Y )/L one finds the simple expression for the structure function
Cp(ki,t" k2, 0) = ( S I(nk~ - p)e-ttt(nk2 - P)I P ) = p(l - p)P(kl; tlk2" O) (6.7) and for the corresponding two-time, three-point correlator Cp(kl, tl" k2, t2; k3" 0) = p(l - p)(l - 2p)P(kl" tl - t21k2" 0)P(k3; t21k2" 0). (6.8) Because of particle-hole symmetry this quantity vanishes in a half-filled system where p = 1/2. In a similar way one finds that the four-point correlator is given by one- and two-particle conditional probabilities rather than by four-particle conditional probabilities as the genetic four-point correlator. From (6.6) one finds the simple expression
S(p, t) = p(l - p)e -2D~l-c~ p)t
(6.9)
for the dynamical structure function S(p, t) = Y~r eipr Cp*(r, t; 0, 0). The largescale behaviour of the structure function is determined by small p. In the scaling limit p -+ 0, t ~ e~ the structure function becomes a function of the scaling variable u = pZt with the dynamical exponent z = 2 already encountered in single-particle diffusion. Thus the collective diffusion coefficient Dr (4.16) of the
1 Exactlysolvable models for many-body systems
83
symmetric exclusion process on an infinite one-dimensional lattice is given by Dc = D. One realizes that the collective diffusion coefficient of the many-particle system is identical to the single-particle diffusion coefficient (Kawasaki, 1966; Dieterich et al., 1980; Kutner, 1981). This holds also for higher-dimensional lattices, since for any lattice the normalization factor Y]k Cp (k, t; l, 0) = p ( 1 - p ) cancels the p-dependence in the amplitude of the correlator.
6.3
First-passage-time distributions
The preceding subsection has elucidated the role of the SU(2)-symmetry for the symmetric exclusion process. This symmetry is broken in the presence of particle creation and annihilation terms. Nevertheless, the invariance of the pure hopping process under homogeneous factorized transformations can be employed for the derivation of expressions for persistence- and first-passage-time distributions (FPTDs) in terms of usual expectation values. We define Vpo (k, t) as the probability that site k in the lattice has always been empty up to time t, assuming an initial state or initial distribution P0. In addition to the symmetric hopping process we allow for spontaneous particle decay at site k with rate ~.k- The quantum Hamiltonian for this process reads H = H0 + HI where H0 is the Heisenberg Hamiltonian (3.26) for hopping only and Hi = Y~'~kes~.k(nk -- s+). Without loss of generality we take k = 0. Then one finds following the steps detailed in Section 5.3: Ve0(0; t) = { S le-Jqtv0l P0 )
(6.10)
where vk = 1- n k and/-) - voHvo. For definiteness we take an initial state which has N particles placed on sites k l . . . . . kN. In order to calculate the persistence probability (6.10) one takes the transpose of the matrix element and writes I S ) = 131 L) with /3 = e s§ and I L) representing the completely filled lattice. One obtains VPo(0; t) = ( Slnk~ ...nkuQoe-Jqtl L ) where Q0 = ~ - l u 0 ~ and H -B-1/_) r/3. The transformed operator Q0 may be written as the infinite-rate limit of the annihilation process at site k = 0, lim ea(S~ -n~
Qo -
(6.11)
ff---~ o o
After computing/4 one realizes that
Qo e-l:lt = lim e -H't
(6.12)
Ol----~(X)
is the time evolution operator of a process generated by
H' = HO + or(no - s~) + Z kk(vk -- s f ). k6S
(6.13)
84
G . M . SchQtz
This process is the infinite-rate limit of the symmetric exclusion process with particle creation with rates ~.k and particle annihilation with rate a at site k ---- 0. Thus -- l-tit VP0(0; t) = lira ( S Ink~ . . . n ~ e I L). (6.14) o~----~ O O
To obtain an expression for the persistence distribution for a general initial distribution I P0 ) one replaces in (6.14) ( S Ink1 ... nku by ( Po I/3. By taking the negative time derivative of (6.14) one obtains the normalized first-passage-time distribution (FPTD) fPo (0; t) = - d / ( d t ) Vp o (0; t ) / ( V p o (0; 0 ) VPo(0; c~)). To calculate the mean first-passage time r Po MFPT(0) = f dt t f e o (O; t ) one integrates by parts and gets MFPT
rP0
(0) =
fo dt
(( nk (t)
~
1
"'" --
n k N ( t ) ) -- ( nk, . . .nkN )*)
(nkl...nkN)*
(6.15)
where (nkl ( t ) . . . n k N ( t ) ) is the time-dependent correlation function of the process H' with infinite rate tr and (nk~ ... nkN )* is the stationary correlator. In the absence of particle annihilation in the original process H there is no particle creation in the related process H'. In this case the stationary correlator vanishes.
6.4
Bethe ansatz solution
We have seen above that the relevant quantities that characterize the dynamical behaviour of the process, i.e., the time-dependent density correlators, can be expressed in terms of conditional probabilities. Since the particles have no longrange interaction, but only on-site repulsion one might wonder to what extent the conditional probabilities deviate from those obtained for completely noninteracting particles. We address this problem for the integrable one-dimensional system with nearest neighbour hopping where one expects the hard-core constraint to be most relevant for the dynamics of the system. These quantities can be calculated using the Bethe ansatz adapted to the problem at hand. We study the system directly in the thermodynamic limit L ~ o0. In order to avoid notational difficulties we label in this context lattice sites by xi and yi and always assume these to be ordered, Xl < x2 < ... < Xn, Yl < Y2 < "'" < Yn. For brevity we shall drop the initial values yl . . . . . Yn in the expression for the conditional probability. A straightforward approach to a Bethe solution for time-dependent conditional probabilities would be the calculation of the energy eigenstates of the Heisenberg chain with the original coordinate Bethe ansatz and then insertion of a complete set of energy eigenstates in the expression P ( x l . . . . . Xn; t) = ( x l . . . . . x . [e-Ht[ Yl . . . . . Yn ). However, this requires proper normalization of the wave function, which is a difficult problem. Therefore we use a slightly different approach by turning the master equation for
1 Exactly solvable models for many-body systems
85
the n-particle conditional probability directly into an eigenvalue equation by the ansatz P (x l . . . . . Xn ; t) = e -el P~ (x l . . . . . Xn ). The Bethe ansatz for the single-particle conditional probability is nothing but the Fourier ansatz P~ (x) = e ipx that led to (4.27) and has been solved above (6.6). For symmetric hopping with DR = Dr. -- D the ansatz yields the integral representation (4.29) of the Bessel function with 'energy' = 2D(1 - cos p).
(6.16)
For two particles the solution of the master equation for the conditional probability becomes nontrivial. For particles at a distance of more than one lattice unit one has to solve the two-particle master equation
EPe(xI,x2)
1 , x 2 ) + PE(Xl,X2- 1 ) - 2PE(xl,x2)
-D[P~(xl-
-t- PE(Xl -+- 1,x2) d- P~(Xl,X2 + 1) - 2 P e ( x l , x 2 ) ] .(6.17)
In the expression on the r.h.s, one recognizes the gain terms (positive sign) resulting from nearest-neighbour hops into the configurations with particles at sites (x l, x2) and the loss terms resulting from all possible moves out of this configuration (Fig. 15). In a more formal way (3.12) can be obtained by calculating the action of H to the left in time derivative - ( X l , X2 I H e - H t l Yl, Y2) of the conditional probability. loss
loss
k.L/..._.~ I P ' ~__ kJJ
:
kd.J ..._.~
gain
gain loss
<---
(b)
~
+__.
~k.L/.._). IAF
0(__.
0
J
J
I
gain
Fig. 15 Two exclusion particles on the integer lattice with the allowed elementary moves for distant particles (a) and neighbouring particles (b). The arrows indicate the origin of the loss terms (outgoing arrows) and of the gain terms (ingoing arrows from possible configurations which lead to the configuration shown here). For neighbouring particles E P ~ ( x , x + l) - - - D [ P ~ ( x -
X2 =
Xl
-+- 1 -- X one has to satisfy
l , x + l ) + P ~ ( x , x + 2 ) - 2 P ~ ( x , x + I)] (6.18)
86
G . M . SchQtz
with the same E. This 'boundary condition' in the two space coordinates expresses the exclusion interaction. There is no hopping from sites (x, x) or (x + 1, x + 1) to sites x, x + 1 or back (Fig. 15). One recognizes in (6.17) and (6.18) the eigenvalue equations (3.12) and (3.13) with A = 1 for the Bethe wave function ~ . This leads us immediately to the integral representation
'ff
P ( x l , x 2 ; tlyl, Y2" 0) = (2yr) 2
dp!
dp2e-(~l+~2)t-iplyl-ip2y2qlpl.p2(Xl,X2) (6.19)
with the single-particle 'energies' 6i =
2D(I - cos Pi).
(6.20)
The energy expression arises from the diffusive motion of the particles: the time evolution operator acts on the conditional probability like a lattice Laplacian if the difference between the coordinates is larger than 1, i.e. if the two particles do not 'feel' the presence of each other (3.12). The exclusion interaction determines the 'scattering amplitude' S (3.17) appearing in the Bethe wave function. We note that if the particles were identical, but completely noninteracting, one could have two particle on the same site. In this case there would be a symmetry constraint on the conditional probability under exchange of particle coordinates: P~(xl, x 2 ) = P~(x2, X l ) . This boundary condition is the noninteracting analogue of (6.18) for exclusion particles. The Bethe ansatz would then give S = 1, independent of the pseudomomenta pl, p2. The expression (6.19) is, as it stands, ill-defined because of the pole in the scattering amplitude (3.17). The pole needs to be taken into account in the definition of the contour of integration which is determined by the initial condition
e ( x l , x2" 01yl, Y2; 0)
=
6xl,yl~x2,y2.
(6.21)
The bound state contribution to the conditional probability resulting from the pole is discussed in Dieterich and Peschel (1983). In order to obtain an explicit expression for the conditional probability we choose a different approach. We prescribe the appropriate contour of integration by isolating in S the constant part 1 which corresponds to noninteracting particles. One writes S2I =
1 + 2(1 - eiPl)(l - e -ip2)
due -u(eipl+e-ip2-2eipl-ip2)
(6.22)
and integrates both pl and P2 from 0 to 2rr along the real axis before integrating over u. Both this definition of the integration and the choice e -iplyj -ip2y2 for the amplitude ensure that the initial condition (6.21) is indeed satisfied. Thus the conditional probability has the form P(x, y; tlxo, Y0; O) = P(x, y; tlxo, Y0" 0) + 2Vr+VZ Q(r, s" t)lr=y-xo;s=x-yo (6.23)
87
1 Exactly solvable models for many-body systems
where /5(X, y; tlxo, Y0; 0)
=
P ( x ; tlx0; 0)P(y; tlY0; 0)
+ P ( y ; tlx0; 0)P(x; tJy0; 0)
(6.24)
is the conditional probability for identical, noninteracting particles, Vm ~ is the forward (backward) lattice derivative with respect to the coordinate m, i.e., Vm~A(m) = + ( A ( m 4- 1) - A ( m ) ) . The term Q ( r , s ; t ) arising from the interaction is given by (6.19), (6.22). Higher-order conditional probabilities are obtained by generalizing the strategy. The initial condition determines the overall normalization of the wave function and the position of the poles arising from the integration over the various Sij appearing in the wave function. It is satisfied by the choice f ( P l . . . . . PN) = e -(iplyI+'''iPNyN) for the amplitude and by placing the poles analogously to the two-particle case. This construction provides an integral representation of the conditional probabilities. Therefore the general solution of the master equation may be written P ( x ] . . . . . XN; tlyl . . . . . YN; O)
N " ~1 I-I
f02~rdpje-EPJ
t-ipjyj X
j=l q~pl ..... PN (X] . . . . . XN)
(6.25)
with the Bethe wave function , and contour of integration as defined above. It is important to note that as a function of its arguments xi the function P is well-defined in Z N, i.e. also for the ghost coordinates xi - Xi+l or Xi > Xi+l. However, in this domain P is not a probability. In other words, in the domain f2N = Xl < x2 < . . . < XN C Z N, the function P is the probability defined above, whereas in Z N \ f2N it is defined by the master equation, but is not a probability. To analyse the late-time behaviour consider first the two-particle case (6.19). At late times the main contribution to the integral arises as in the single-particle case only from small values of pl, P2. So we make a substitution of variables Pi ~ fii = Pi V/~, Xi, Yi ~ 3~i, Yi = Xi/~/t, Y i / x / ~ and expand the cosine in the energy term to first nonvanishing order. This gives an expression of the conditional probability in terms of the scaling variables s )7i. Expanding S (6.22) for small arguments/~i/x/t leads to S = 1 + O(t-l/2).
(6.26)
Thus we arrive at the conclusion that the leading contribution to the conditional probability comes from S = l, corresponding to undistinguishable, noninteracting particles. Simple power counting shows that for large times the correction to
88
G.M. SchQtz
noninteracting particles vanishes proportional to t -3/2, i.e. faster than the leading part which decays proportional to t -l. With the linear combinations of scaling variables w = ( x l + x 2 - Yl - Y 2 ) / ~ / 8 D t , u - - (x2 - Xl - Yl + y 2 ) / x / ~ D t and v - (x2 - x l + Yl - y z ) / x / ' - 8 D t one obtains the scaling form and leading correction term for the two-particle conditional probability e-W 2 P(x1, X2" tlyz, Y2; O)
=
[e -u2 + e -v2
2rrt + ~ 1 [ve_V 2 + (1/2 - w2)~/-n-effc(v)] (6.27)
-t-O(l/t)}
in terms of the complementary error function erfc(v) = 2/4%-
fo ~
dxe -x2.
(6.28)
We note that the large-scale behaviour of correlation functions shows dynamical scaling with dynamical exponent z - 2. As seen above, the integrability implies a factorization of the plane wave amplitudes A for m-particle conditional probabilities into two-particle amplitudes. Hence ail iN = 1 + O(I/x/7) which implies that the leading part of the conditional probabilities for m-particle conditional probabilities decays like in the case of noninteracting particles proportional to t - m / 2 . T h e correction term resulting from the excluded-volume interaction decays faster, cx t -(re+l)~2. T h u s we arrive at the main conclusion that all m-point correlation functions of the symmetric exlusion process are, to leading order in time, identical to the same m-point correlators of identical, noninteracting particles. .....
6.5 Algebraic formulation and solution While one expects bulk properties of the symmetric exclusion process to be unaffected by the choice of boundary conditions, this cannot be expected for exact scaling functions, and, of course, not for boundary phenomena. To gain some insight we consider now open systems with particle injection and absorption at the boundary sites. Any such term breaks the S U (2)-symmetry of the system, but, as already encountered in the consideration of the first-passage-time problem, one can still obtain useful information from certain similarity transformations. In fact, the results of Section 6.3 provide additional motivation for tackling the system with open boundaries: the FPTD for the one-dimensional system with particle absorption at a single boundary site L is given by the correlation functions of the
1 Exactly solvable models for many-body systems
89
system which is open at both ends x0 = 0 and x = L with injection rate 6 = ~.t~ at site L and infinite absorption rate at site 0. Yet another aspect to consider is the possibility of maintaining a genuine nonequilibrium situation (even for very large times) by coupling the system at its boundaries to particle reservoirs of different constant densities pL ~ PR. This is again equivalent to particle injection and absorption at the boundaries of the system and leads to a stationary particle current j cx I/L (Spohn, 1983). Materials such as zeolites consist of interconnected narrow tubes (Kfirger and Ruthven, 1992) and the exclusion process appears to be a good candidate for a model of the penetration into and diffusion within such substances. This is confirmed by experiments on tracer diffusion in single-file systems (Kukla et al., 1996). It is therefore of interest to study not only the stationary behaviour, but also the dynamics of the one-dimensional process with open boundaries.
6.5.1
The dynamic matrix ansatz
Consider the simple symmetric exclusion process with bulk hopping rate D = 1/2, but injection (absorption) of particles at the left boundary site k = 1 with rate c~ (y respectively) and at the left boundary site L with rate 3 (/3 respectively) (Fig. 16). The dynamics are given by the ferromagnetic Heisenberg chain 1 L-I
L-I
H = ~ )---~(1- 6k"
~k+l)"q'-bl-k-bL =-- y ~
k=l
hk + bl h-bL
(6.29)
k=l
with nondiagonal boundary fields (3.31).
/
I
A IP'
t
t
A IP'
A IP'
./
Fig. 16 Symmetric exclusion process with open boundary conditions. Injection and absorption of particles at the boundary sites is indicated by the arrows together with the respective rates. The SU (2)-symmetry is manifestly broken, but the transformation 13 = e s+ leaves the bulk part invariant and transforms the boundary matrices bl,L
90
G.M. Sch~J~Z
simultaneously into triangular form: 13H/3- l = Hbulk + (t~ + y)nl -~(~ + 8)nL -- cts~ -- 8s-[. T h e bulk Hamiltonian together with the diagonal part of the boundary fields has a residual U(1)-symmetry generated by S z, expressing conservation of the z-component of the total angular momentum. Thus this part of the Hamiltonian splits into blocks with fixed S z. T h e off-diagonal boundary fields (which break even this symmetry) connect the block corresponding to some value S z with that of lower spin SZ _ l, but not with blocks of larger S z. Hence the off-diagonal boundary fields determine the form of the eigenvectors, but do not change the spectrum of the chain.* As a result, one may use the coordinate Bethe ansatz to determine the spectrum of (6.29) (Alcaraz et al., 1987) by just omitting the off-diagonal part of the boundary fields. However, a problem occurs if one wants to calculate correlation functions which require knowledge also of the eigenvectors. Even though the system was shown to be integrable (Inami and Konno, 1994; de Vega and Gonzalez-Ruiz, 1994), this alone is not sufficient actually to construct the eigenstates in the presence of the off-diagonal boundary fields. In a very different, algebraic approach the ground states of one-dimensional spin Hamiltonians are formulated in terms of matrix p r o d u c t states (Affleck et al., 1988; Fannes et al., 1989; Kliimper et al., 1991, 1993) where the ground state wave function is expressed in terms of a trace over a product of matrices. They may be seen as representations of an operator algebra which is determined by the requirement that by acting with the Hamiltonian on this state one obtains an eigenstate (the ground state) of H. The correspondence between quantum spin chains and stochastic Hamiltonians suggests extending this approach to the calculation of stationary distributions (Derrida et al., 1993a). Here we proceed along similar lines, but go one step further to employ a matrix product ansatz for the calculation of t i m e - d e p e n d e n t probability distributions (Stinchcombe and Schiitz, 1995a,b). This leads to an algebraic formulation of the stochastic dynamics of a given process which we shall call the dynamic matrix ansatz. The dynamic matrix ansatz is an ansatz for the time-dependent probability distribution which has the form of a product measure (2.24) where the local weights are not c-numbers, but matrices. The actual probabilities for a given particle configuration are recovered by taking a scalar product with suitably chosen
*A similar observation was made in the context of certain reaction-diffusion processes (Alcaraz and Rittenberg, 1993).
1 Exactly solvable models for many-body systems
91
vectors (( W I, IV )).+ We write IP(t))
=
((WI
l-I [E(t) + D(t)trk- ] 10) I V))/ZL
( )o, k=l
=
((WI
D(t
I V))/Zt.
(6.30)
where 10) is the vacuum state with all spins up and D, E are time-dependent matrices satisfying an algebra determined by the master equation (2.11) with the Hamiltonian (6.29). The vectors ((WI and IV)) on which D and E act are determined from the boundary terms in the master equation and Z t~ = ((wICLIV)) where C = D + E is a normalization. In this framework the m-point density correlation function is given by
(nk~(t)...nkm(t))Po = ( ( w l c k ~ - l o c k 2 - k ~ - l o . . . c L - k m l v ) ) / Z L .
(6.31)
Therefore, given a matrix representation of the algebra satisfied by D, E, the computation of time-dependent correlation functions is reduced to the calculation of matrix elements of a product of L matrices. Alternatively, as shown below, the algebraic properties of the matrices D, E may be used for the derivation of explicit expressions for correlation functions. This eliminates the need to find representations. The key idea for the derivation of the algebra is to insert the ansatz (6.30) in the master equation (2.11) with H given by (6.29) and to try to satisfy the equation for each individual hopping matrix hk and boundary field bl,L. This can be achieved by adding a divergence term to each individual equation involving auxiliary operators S, T. This leads to the set of four bulk matrix equations d
E
E
T
E
S
(6.32) (We have written only the part of the L-fold tensor product (6.30) on which hk acts nontrivially.) In a periodic system the summation over k leads to a cancellation of all terms involving the auxiliary matrices and thus one satisfies the correct master equation. In the case of an open boundary this cancellation is achieved by imposing conditions involving the boundary fields and the vectors (( W I, I V )) on which the time-dependent matrices D, E, S, T act: ((W.[(I
d
_+bl)(
E D )+(
S T )]
=
0
(6.33)
+We use double brackets for vectors in the space .A4 where the dynamical matrices act. This is to avoid confusion with vectors in the tensor space X = C L. Hence a vector of the form I. )1" )) is understood as a vector in the tensor space X | .A4.
92
G.M. Schfstz ld
E
S
=
0.
(6.34)
The four terms in (6.32) yield four quadratic relations for the operators D, E, S, T. Equations (6.33) and (6.34) give two pairs of equations which define ((WI and IV)). Since the ansatz may be applied in the same way to the general reactiondiffusion system (3.35) with open boundaries we write down the general form of the matrix algebra (Schiitz, 1996a). Introducing A (1)
=
--(1/)21 + to31 + to41)E 2 + w l 2 E D + wI3DE + to14D2
B (1)
=
w21E 2 - (w12 + 1/)32 + w42)ED + w23DE + 1/)24D2
B (2)
=
W31E2 + w32ED - (w13 + w23 + w43) D E + w34 D2
A (2)
=
1/341E2 + w42ED + w43DE
-
-
(1/314 + 1/324 + w34)D 2
(6.35)
one finds from (6.32)
1 dE2_[S,E] 2 dt
A (1)
ld - - - E D - SD + E T 2 dt ld ---DET E + DS 2 dt
=
B (l)
=
B (2) A (2)
1 dD2_[T,D]
(6.36)
2 dt and 1 d E_otE + yD + S ]
((Wl
~ dt
((WI
-~--~D + a E - y D + T
1' {laot
'1 ]
= - : - D + ~ e - t~D - r
IV))
-
0
--
0
-
0
=
0.
(6.37)
By requiring the time derivatives to vanish one obtains the stationary algebra of Hinrichsen et al. (1996a). It is difficult to analyse this algebra without any further input. One may reduce this algebra by assuming that C is time-independent. This is motivated by conservation of probability, i.e. time-independence of the unnormalized sum
1 Exactly solvable models for many-body systems
93
of probabilities Z L -" Y'~rl Po (t) = (( W IcLI V)). Adding all four of equations (6.36) then gives [C, S + T] -- 0 and equations (6.37) imply ((WI(S + T) -0 = (S + T)IV)). Without loss of generality (Santos, 1997b) these relations are solved simultaneously by S + T -- 0. Assuming further that C is invertible one can express S in terms of C and D and is left with only two further relations to be satisfied by D(t) and C, and with two relations defining ((WI and IV)). For the exclusion process with w23 -w32 = 1/2 one obtains the auxiliary matrix S = ( C D C -l - C -l D C ) / 2 and the matrices C and D satisfy d D dt
=
1 (CDC_ l +C- IDC_2D] ! 2 \
(6.38)
D2
=
_1 ( C D C - I D + D C - I D C ] ] 2 \
(6.39)
0
=
( ( W l [ ( 2 a + 27 - 1) D - 2otC + C - l O C I
(6.40)
0
=
[(2/3 + 26 - 1) D - 26C + C D C - I ] I V
)).
(6.41)
These relations provide an alternative, purely algebraic definition of the symmetric exclusion process. Equation (6.38) expresses the diffusive motion of particles and in (6.39)one recognizes the two-particle boundary condition (3.14)encountered earlier in the Bethe ansatz solution. Defining
Dk =-- C k- 1DC-k
(6.42)
the m-point density correlation function can be written
(nk,(t)...nk~(t))--((WIDk,...DkmCLlV))/Zt.
(6.43)
where ki + l > ki . Equation (6.38) is linear in D and we procede by constructing the Fourier transforms 79p -- Y~k eipk Dk to reformulate the algebra in terms of the Fourier components. Since C ~ ) p C -1 -- e - i p ~ ) p , (6.44) the time-dependence of Dp is now simply obtained from (6.38), 7)p(t) Ep
--
e-~ptT)p(O)
(6.45)
~
1-
(6.46)
cos
p
in terms of the initial matrix Dp(O) and the 'energy' ep. The inverse relationship yields Dk in terms of Dp. It is useful to separate the static and dynamical parts of Dk and to write Dk = (1 -- k)Do + Z +
f
' dp -ipk ~-~Dpe
(6.47)
94
G . M . SchQtz
where the primed integral excludes the point p = 0 and is defined by
I'"/"[ 2= =
~
1 - 2zr6(p) - 2n'i dp (p)
]
"
(6.48)
This splits Dk into a purely static part D~ = (1 - k ) D o + Z with Do = Y~n Cn-] D c - n and Z = Y~n n C n - l D c - n (which yields the stationary correlations) and into an initial-value dependent part with nonvanishing p which therefore decays to zero in the limit t ~ 0o. This is not surprising as the stationary state of the system is unique and hence contains no information about the initial state. In terms of the Fourier components, (6.39) can be written as 0
=
dpl 71"
=
d279p179p2(l -~- e i p l + i p 2
dpl 71"
+
--
2eip2)
7r
dp2I)pj'Dp2(l +
e ipl+ip2 --
2e ip2)
I
ff f= dpl
7r
dp2Dp2Dpl(l
+e ipl+ip2 -
2eipl).
(6.49)
I
Since this holds for all times the integrand must vanish. If not both pt and P2 are zero leading to
~)PI 'DP2 = S(p2, pl )~)pz ~)pl
(6.50)
with, quite remarkably, the two-body scattering amplitude (3.17) of the exclusion process. The relations for D p derived so far apply to the dynamic components with p # 0. The static parts need separate treatment (Santos, 1997b). It is easy to verify that the time derivatives satisfy D0 = 2- = 0. Setting pl = 0 in (6.50) yields [Do, Dp ] = 0 (6.51) and furthermore, by differentiating (6.50) with respect to pl and then taking pl = 0 yields [Dr,, Z] = 2DoD t, (6.52) for p ://: 0. The constraint equation (6.39) together with the infinite-time limit of (6.47) finally gives [Do, 2 ] = D~. (6.53) for the relation between the static components alone. It remains to reformulate the vector relations (6.40), (6.41) resulting from the boundary conditions in terms of C and 79p. With f p(a, b) =- 2a + 2b - 1 + e ip
1 Exactly solvable models for many-body systems
95
one obtains two sets of equations, one each for the static and the dynamical part respectively: (6.54)
0
--
(W] {Do + 2(ct + y ) Z -
0
--"
(wlC{fp(Ol,),)~)p
0
=
{(2/3 + 28 - 1)D0 + (2/3 + 2 8 ) 2 - 28} IV)
(6.56)
0
=
{f_p(~,8)Dp + fp(~,8)D_p}lg)
(6.57)
2c~}
-Jr f_p(Ol, y)D-p)}
(p # O)
(p # 0).
(6.55)
The momentum space formulation of the algebra provides an equivalent formulation of the process. In terms of the Fourier transforms the correlator (6.43) reads
( nkl (t)... nkm(t)) =
((-If
dpie-piki-~pit)T({pi}) 2Jr
(6.58)
i=1
where the so far undetermined matrix element
T({pi}) = (( W IZ)p, (0)...Z)pm(O)CLI V ))/ZL
(6.59)
depends on the initial distribution. In order to calculate correlation functions explicitly one can either proceed by trying to find a representation for the timedependent algebra (Schiitz, 1996a) or by trying to exploit directly the algebraic relations of the Fourier components (Schiitz, 1998). We choose the second route.
6.5.2
Stationary state
In the infinite-time limit the dynamical contributions Z)p(t) vanish and one is left with the algebraic relations w h e r e / ) = E = 0. Therefore only the static parts Do and 2" have to be considered. We first calculate the stationary density profile
p; = (nk(oo) ) -- (( D~C L )). Equations (6.54) and (6.47) determine Pk in terms of the matrix element gl =
(( w 17901 v ))/ZL: a
(2(c~ + y ) -
1 _ k')gl.
/
(6.60)
A similar equation can be obtained from the right boundary equation (6.56). Consistency with (6.60) determines
gl
=
2(0q5- yS) 2(ct + ),)(/3 + 8)(L - 1) + a + y +/3 + 8'
(6.61)
thus generalizing an old result by Spohn (1983). We also find the stationary current j = ( nk+l )* -- ( nk )* = gl. For a large but finite systems the stationary
96
G. M. SchOtz
density profile takes the asymptotic form p k ( O 0 ) - - PL -t- (PR -- p L ) k
(6.62)
where PL = ot/(ot -t- F) and Pn = 8/(fl + 8) are the left and right stationary boundary densities and fr = k / L is the rescaled bulk coordinate. Higher-order correlation functions are computed in a similar manner. For the purely static part one uses (6.53) to commute all 27 to the boundary vectors and then the static boundary relations (6.54), (6.56) to express the matrix element in terms of matrix elements gn : (( W I(Do)"CLI V ))/(( W IcLI V )). Consistency gives a set of equations which determine gn -- I 7 A/,
(6.63)
k=l
with Ak -- 2(a~ + y ) ( f l + 8)(L - k) + (a + F + fl + 3)"
(6.64)
Specifically for the stationary two-point correlator CL,(k, I)* - ( n k n l ) -( nk ) ( nt ) one finds in terms of the shifted coordinates k' = k - 1 + 1/[2(a + F)], l' -- l - 1 + 1/[2(a + F)] the exact result
C L ( k , l)* -- - A I k' [A2 + (A1 - A2)I'].
(6.65)
There is a weak anticorrelation due to the hard-core interaction. For a large system size L this expression reduces to the asymptotic scaling form
C L ( k , I ) * - - ~1 (PL -- PR) 2h , (~:, D
(6.66)
with the scaling function h(k, l-) - / r - D for the scaled bulk coordinates ~: -k'/L,l=Z'/L. An alternative, but equivalent treatment of the stationary matrix algebra is obtained by considering the algebra of C and D* = Z C . The commutator yields the static component S - D o C / 2 which commutes both with C and D*. Hence one may represent S by the unit matrix. In this way one obtains the stationary matrix algebra first obtained by Den'ida et al. (1993a) for the exclusion process, both symmetric and asymmetric. Generally, the stationary algebra for a given process may be defined by imposing [ S, D ] -- [ S, E ] - 0 instead of setting S + T -- 0, i.e., by treating S as a c-number. For the partially asymmetric exclusion process (Section 7) this algebra is a q-deformed harmonic oscillator algebra (Sandow, 1994) for which both finiteand infinite-dimensional representations have been constructed (Derrida et al.,
1 Exactly solvable models for many-body systems
97
1993a; Sandow, 1994; Essler and Rittenberg, 1996; Mallick and Sandow, 1997). This approach has been extended to lattice gases with more than one species of particles (Derrida et al., 1993b; Mallick, 1996; Kolomeisky, 1997; Arndt et al., 1998a; Alcaraz et al., 1998; Karimipour, 1999a,b,c; Mallick et al., 1999) and to diffusion processes with discrete-time dynamics (Honecker and Peschel, 1997; Rajewsky and Schreckenberg, 1997; Hinrichsen, 1996; Rajewsky et al., 1996, 1998; Hinrichsen and Sandow, 1997; Fouladvand and Jafarpour, 1999). The algebraic construction has, in similar form, been applied to a matrix description of the ground states of quantum spin chains (Affleck et al., 1988; Fannes et al., 1989; Kliimper et al., 1991, 1993). The purely stationary matrix ansatz does not require integrability of the quantum spin system. Hence one can obtain with this method exact results for the stationary states of non-integrable processes.
6.5.3
Density relaxation
We are now equipped to calculate the local dynamics of the process. The strategy is best explained in terms of the density profile pk(t) = ( n k ( t ) ). The generalization to higher-order correlators requires only a little further discussion. For the dynamical part (6.55) implies T ( p ) = -e2ip f_p(Ot, y ) / f p ( ~ , y ) T ( - p ) .
(6.67)
It is easy to see that this functional equation has the solution
T(p) = E al (eipl -- e2ip f-P(~ Y)e-ipl) 1>0 f p(Ot, y)
(6.68)
with arbitrary constants at which have to be determined from the initial density at site l. In this expression one recognizes the Bethe ansatz wave function for the single-particle sector of the Heisenberg chain with diagonal boundary fields (AIcaraz et al., 1987). The second equation (6.57) coming from the fight boundary yields a similar functional equation for T (p). Comparing these equations shows that T (p) can be nonzero only if
e2ip(L-1) =
B(p)
(6.69)
where
O(p) -- f _p(Ot, y) f _p(~, ~) . f p(a, y) f p(~, ~)
(6.70)
This equation quantizes the allowed values for p in a finite system and reduces the integration in (6.58) to a summation over the solutions of (6.69). As in the solution of the eigenvalue problem of an ordinary second-order differential equation for a
98
G.M. SchQtz
quantum particle in some potential, the boundary terms both determine the form of the general solution and quantize the spectrum. In a semi-infinite system (L ~ oo) the constraint (6.69) disappears and the calculation of pk(t) reduces to the evaluation of the integral (6.58). The solution is now determined by the requirement that the expectation value should remain finite for all k > 1. Inspection of the integral determines the contour of integration which is nontrivial because of the pole in the boundary phase shift e2ipf-p(a, F)/fp(a, F) for a + y g: 0, 1/2, 1. Rather than explicitly defining the contour (which depends on whether a + y < 1 or a + y > 1) we proceed in a way similar to the treatment of the conditional two-particle probability in the preceding section. A convenient prescription which yields the constants of integration at in the simple form at = pl(O) - p(oo) is captured in writing
e2ip f_p(Ot, ~') = 1 + 2 i ( 2 a + 2 y f p(ot, y)
l)sin p
due -u(e-~p(2a+2•
(6.71)
f 0 ~176
and defining the integral by performing first the integration over p along the real axis [-rr, n']. For 0 < a + y < 1 one finds oo
pk(t) =
p(oo) + Z
ate-t(Ik-t(t)
-
-
Ik+l(t))
1=1 oo
+Z
oo
Z
at(2u + 2F - l)ne -t (lk+t+n(t) - lk+t+n-2(t))(6.72)
l=l n=l
with p ( o o ) = a/(a + y). With (6.72) we can study the effect of the open boundary on the relaxational behaviour of the system. Connecting the system to a reservoir of constant density PL at its left edge (at site k = 0) corresponds to the choice t~ = pL/2, y = (1 - - P L ) / 2 . ' For the time-dependent equilibrium autocorrelation function C * ( l , 1 ; t ) -- l i m r ~ o o ( n l ( r + t ) n l ( r ) ) - (nl(r+t))(nl(r)) = (S Inle-Htnll P*) - p2(oo) which measures the relaxation of the boundary density in equilibrium one reads off 2 C*(I, 1; t) = - p L ( I -- p L ) e - t l l ( t ) t
(6.73)
From the asymptotic behaviour of the modified Bessel function one finds an algebraic decay cx t -3/2 for large times. In an infinite system (i.e. far from any boundary) with the same density PL one finds from (6.7), (6.6) the much slower tThis can be shown formally with the infinite rate formalism of Section 4.3 by both creating and annihilating particles at the reservoir site k = 0 with infinite rates &, ~ such that the ratio &/(& + ~) takes value PL.
1 Exactly solvable models for many-body systems
99
decay of the autocorrelation function C * ( k , k ; t) = PL(I -- p L ) e - t lo(t)
(6.74)
which vanishes only cx t -1/2. Intuitively this is not surprising. In the main one has particle conservation which severely constrains the dynamics of the system. Correlations induced by perturbations of the equilibrium state can relax only much slower than on the boundary, where particles can be created and annihilated and the number of possible stochastic paths that lead to the equilibrium state is much higher. Indeed, slow relaxation of the order parameter (here the density) is a genetic feature of systems with conservation laws (Hohenberg and Halperin, 1977). From this point of view the algebraic decay of the autocorrelation function close to the open boundary is actually surprising, since in the absence of conservation laws one usually has exponential relaxation. Finally we investigate the time evolution of the density profile for an initial state with constant, but nonstationary density O0. We assume again the semiinfinite system to be connected to a reservoir of density PL. This initial state yields at = Po - PL and (6.72) gives the exact result k-I pk(t) = PL + (190 -- PL) Z e-t l l ( t ) l=-k
(6.75)
shown in Fig. 17. In a region much smaller than the diffusive length scale ~ ~ 4r/ the expression for the density reduces to Pk (t) = PL -F (t90 - PL)
2k 2/~_~..
(6.76)
This linear behaviour is reminiscent of the stationary density profile (6.62) in a finite system of length L cx x/7 and fight boundary density PR = p0. It is remarkable that the profile is determined alone by initial profile, but not by the initial correlations.
6.5.4
Quasi-stationarity
Also, equal-time m-point correlations can be studied without explicit construction of representations of the dynamic matrix algebra. The strategy in determining the matrix element T ( { q i } ) is the following. From the action of the Dpi on the left boundary vector (( W I and from their mutual commutation relations (6.50) one finds by permuting the order of the D pi a functional equation for the dynamical part of T ({pi }). This functional equation is solved by the m-particle Bethe ansatz
G. M. Schi~12
100
PL
oooeeoeeeeeoeeeeeeoeoeooeoeeeeeoeeeeoooeeoeoooeeoooeoeeeeooeeeooooeeooo "~~ ~
pk(t)
ee
9
~
9 9
go o
o o
~176176176176
9
"" . . . . . . . . . . . . . . .
""'""~
.................................
PR
t
F
t
t
t
t
:
10
20
30
40
50
60
70
Fig. 17 Exact density profiles of the semi-infinite symmetric exclusion process at time t = 40 (lower curve), t = 250 (intermediate curve) and t = oo (straight line), with fixed left boundary density PL and homogeneous initial density P0 = ,OR. At time t = 250 the profile is linear to good approximation up to site k = 10.
wave function found in Alcaraz et al. (1987) for the system with diagonal boundary fields. Acting on the right boundary vector gives consistency conditions for T ({ Pi }) not to vanish: unless
m' S(pj,_ p~_) e2ipi(L-l) = B(pi)I-I --~j, Pi) Vi
(6.77)
j=l this matrix element is identically zero 9 (The primed product excludes the term with i = j.) Equations (6.77) are in fact nothing but the Bethe ansatz equations of Alcaraz et al. (1987) that one would have obtained from the coordinate Bethe ansatz for the transformed Hamiltonian/3H/~ - l . In a final step one determines the free constants (the contours of integration in a semi-infinite system) such that the initial value problem for the m-point correlator is solved. In the same way one determines dynamical matrix elements involving static contributions. One obtains functional equations solved by Bethe wave functions with m' < m particles. The calculation of the dynamical part in a finite system is not easy as it requires careful analysis of the Bethe ansatz equations (6.77). For a semi-infinite system coupled to reservoir of density pt. the time-dependent two-point correlator C(k, l" t) = (nk(t)nl(t) ) -- (nk(t) )(nl(t) ) has been calculated for a random initial state of density P0 by Santos (1997b). We remark that the asymptotic behaviour of the semi-infinite system for large times can be treated as in the previous section by expanding S(pi, p j) around l and the boundary phase shift B(pi) around - 1 , unless a = y = 0 in which case one expands also B(pi) around + 1. The correlator turns out to have the scaling form C(k, l; t) = AO(PL-po)2/~/Tf(k/~/'i, l/v/-i) with some numerical constant A0. An interesting result is the observation of quasi-stationarity of the system close to the boundary in a semiinfinite system. Santos has shown that within a distance from the boundary which
1 Exactly solvable models for many-body systems
101
is small compared to the diffusion length scale x/7 one finds f ( x , y) = h*(bx, by) where h is the scaling function of the stationary correlator (6.66). This shows that close to the boundary the system behaves essentially like a finite stationary system with boundary densities PL, PO which slowly changes its effective length
L~ff cx ~/7.
Comments Section 6:
By using factorized similarity transformations the equivalence between the symmetric exclusion process on a bipartite lattice and random sequential adsorption and evaporation (3.40) with equal rates L = /~ can be proved (Stinchcombe et al., 1993). On a one-dimensional lattice with an even number of sites L = 2N the equivalence is established by a transformation with the factorized matrix /3 = (1 | trx) | The generalization to higher-dimensional bipartite lattices is analogous.
Section 6.4: (i) The integrability and the simple asymptotic behaviour of the scattering amplitude is the mathematical origin of the reflection principle underlying the continuum treatment of tracer diffusion of a single tracer particle in R6denbeck et al. (1997). (ii) We have shown that the behaviour of all finite-order correlation functions is determined by the sectors with finitely many particles. The dynamical exponent is z = 2, which suggests a finite-size scaling behaviour of the low-lying energy gaps of the form E cx 1/ L2. One may wonder whether the energy gaps in the sectors with finite density (i.e. infinite particle number in the thermodynamic limit) show the same finite-size scaling behaviour. This question has been answered using the SU (2)-symmetry of the process (Quastel, 1992; Gwa and Spohn, 1992b). In all sectors the lowest energy Emi n of a periodic system is bounded by 2/L2 _< Emin _< 2rr 2/L2. Similar 1/L2-bounds have been obtained in more general, nontoroidal geometries using probabilistic comparison techniques (Diaconis and Saloff-Coste, 1993). (iii) An interesting generalization of the symmetric exclusion process is a model where particles hop two lattice units, provided the intermediate lattice site is occupied. This may be interpreted as diffusion of reconstituting dimers. The model has a large number of conserved quantities, diverging exponentially in system size. The various sectors can be shown to be equivalent to the integrable Heisenberg chain (Menon et al., 1997). Section 6.5: (i) A major puzzle is the relationship between integrability and the properties of the corresponding matrix algebra. From the dynamic matrix ansatz for the symmetric exclusion process, for the asymmetric exclusion process with periodic boundary conditions (Sasamoto and Wadati, 1997a,b) and also for some reaction-diffusion systems (Schiitz, 1998) one can construct the Bethe ansatz equations and obtains functional equations which solve the master equation in terms of Bethe wave functions. On the other hand, not all lattice gas models for which similar (stationary) algebras have been considered appear to be integrable (Alcaraz et al., 1998). It seems that matrix algebras of
102
G.M. SchQtz
the form (6.36) which correspond to integrable systems have additional, so far unidentified properties. It is intriguing that (6.50) is the algebra satisfied by the reflection matrices arising in the treatment of the nonlinear Schr'6dinger equation and the Heisenberg chain (Thacker, 1981). The associativity of the algebra (which guarantees that there are no separate equations for D n, n > 2) is the analogue of the factorization of the many-body S-matrix into two-particle amplitudes. (ii) Krebs and Sandow (1997) have proved that in the stationary limit/~ = /~ = 0 of the algebra (6.36) one can always find a representation of the full algebra involving D, E, S, T. The proof is nonconstructive and hence not useful for practical calculation of a stationary distribution. It is interesting, however, that the stationary master equation for the general reaction--diffusion process (3.34) can be formulated in a purely algebraic way, without any constraints on the reaction rates. To date, there is no equivalent theorem for the general time-dependent algebra.
1 Exactly solvable models for many-body systems 7
103
Driven lattice gases
Driven lattice gases are interacting particle systems where particles move under the action of some external force field. In the simplest case one assumes the hopping rates in the absence of the field to satisfy global detailed balance and the driving force imposes a bias on the hopping rates such that detailed balance remains satisfied locally (Katz et al., 1984): if a hopping in the field direction leads to an energy loss ~ E, the equilibrium forward hopping rate gains a factor q = e ~E/~kr) whereas the backward hopping rate (corresponding to energy gain) is reduced by a factor q - I (Fig. 18).
1
t
1
9
I
w
: q 1: 9
9
_
q
1
t
o
o
.
:
."
t
1
9 :
: :
.
9
:
o
9
:
I
9 I
k-I
k
k+l
I
Fig. 18 Energy landscape with local detailed balance in a driving field 9The particle hops from site k preferably to the right (where it gains an energy ~E) with an enhanced rate q DO and to the left with reduced rate q - 1 D O . The equilibrium rates DOare determined by additional energy barriers of height E (not shown) between sites k, k + 1. The exclusion process is one of the conceptually simplest lattice gas models, describing diffusing particles with nothing more than a hard-core on-site interaction. Yet, in the presence of an external driving field, this system develops a rather complex behaviour. In this nonequilibrium situation even in the stationary state a finite particle current can be maintained, and, as a result, the boundary conditions imposed on the system start playing a decisive part in the behaviour of the system 9 Unlike in equilibrium systems, details of the boundary conditions do not get washed out by fluctuations as one probes the system deeper and deeper into the bulk, but get 'transported' by the current into the interior of the system. While this intuitive insight may predict the general importance of boundary conditions, quantitative results are necessary to understand some of the specific consequences of the boundary dynamics. In particular, it is not obvious why coupling of the one-dimensional driven system to external reservoirs of constant density leads to various kinds of phase transitions with divergent spatial (Krug, 1991; Schlitz and Domany, 1993; Derrida et al., 1993a) and temporal (Schiitz, 1993b) length scales 9 For the periodic or infinite system it is also of great theoretical interest to get quantitative insight into the role of the hard-core interaction in
104
G.M. SchQtz
the presence of a drift. The reasons for the interest particularly in the asymmetric exclusion process are not only its rich physical behaviour and its many mappings to both experimentally relevant systems and to problems of wider theoretical interest, but also the mathematical properties of Heisenberg quantum chains. Sections 7.1 and 7.2 discuss exact results on the asymmetric exclusion process (ASEP) which was originally introduced by MacDonald et al. (1968); MacDonald and Gibbs (1969) as a model for the kinetics of protein synthesis on RNA templates (see Section 10). This process is a lattice realization of the noisy nonlinear diffusion equation (Burger's (1974) equation) which has become relevant, e.g. for the description of interface growth (Krug and Spohn, 1991; Halpin-Healey and Zhang, 1995). The nonlinearity-allows for the formation of shocks which may be seen as a gas of solitons (Fogedby et al., 1995). Its many applications, rich dynamical behaviour and an abundance of interesting exact results make the ASEP one of the standard models in the field of nonequilibrium statistical mechanics. The exact solution renders the model a testing ground for more widely applicable, but usually nonrigorous macroscopic hydrodynamic theories and approaches such as the renormalization group treatment (Janssen and Schmittmann, 1986; Krug and Tang, 1994; Janssen and Oerding, 1996; Oerding and Janssen, 1998). In particular, we shall develop a theory of boundary-induced phase transitions for the steady-state selection of driven diffusive systems with open boundaries (7.4). Some comments on nonintegrable traffic flow models conclude this section.
7.1 7.1.1
The asymmetric exclusion process ASEP in a finite box: stationary states and self-duality
In this subsection we shall assume that exclusion particles hop in a closed nonperiodic system of L sites with rates DR,L to the right and left respectively. We interpret the hopping asymmetry (3.29) q = ~/DR/Dt~ - e 13~E as resulting from a linearly decreasing external potential E(O) = ~ E Y~k ko(k). The simplest realization of such a potential is the gravitational field, but other experimental realizations involving, e.g. a constant electric field, are conceivable. The local energy difference 3E between neighbouring sites provides the hopping bias at temperature T = 1/(kfl). The stochastic Hamiltonian is given by L-I
H-Zhk
(7.1)
k=l
with the two-site hopping matrix defined by (3.28). Because of the reflecting boundaries there can be no stationary current and after long times the system enters an equilibrium state satisfying global detailed
105
1 Exactly solvable models for many-body systems
balance. By using the explicit representation of the hopping matrices hk one verifies the detailed balance condition
H T = V-2HV 2
(7.2)
with the diagonal similarity transformation V = qZ~L, knj,
(7.3)
which gives the equilibrium weights of the system defined by the external potential. Combined with particle number conservation this implies that a oneparameter family of stationary product measures is given by P~ cx q2 S.~,(k+U)nk. Here the chemical potential/z fixes the overall particle density. The stationary density profile Pk takes the form l
Pk = (S Inkl P*) = ~ (1 + tanh [/~(k - a)3E])
(7.4)
with a constant ct determined by lz. This density profile has the shape of a step extending over a finite region of length cx 1/(3E) (Fig. 19). There are no correlations in this stationary grand-canonical distribution which is also defined for the infinite system (Liggett, 1985). In analogy to phase separation in usual equilibrium systems one may regard the region of the step as a domain wall separating an empty and a completely filled region of the system.
l
oooooooooeoooeoooooooooooeoooeoooooooooooooooooooooeooooQooooeoooooo
,Ok
~ o~ e
I
t
I
t
I
t
t
t
t
t
I
Fig. 19 Exact stationary density profile of the ASEP with reflecting boundaries with 100 sites. The position of the step is determined by the particle number, its width depends on the driving field. Here we have chosen ,83 E = 1/2, corresponding to q = x/e. It is instructive to investigate the canonical equilibrium distribution with fixed number of particles N. Up to normalization the stationary N-particle state is given by the vector V21 N ) with the constant N-particle vector IN ) which represents the (trivial) stationary distribution of the symmetric exclusion process. The difficulty in analysing the properties of this distribution lies firstly in the
106
G.M. Schl~z
need to compute the 'partition function' ZL,N = ( S IV21N ) for a system of L sites and, more importantly, in the fact that the correlations in the state I N ) are nonvanishing. It is not obvious how this translates into equilibrium correlations of the driven system. In the symmetric exclusion process we showed how the N-particle equilibrium states could be constructed using the SU(2)-symmetry. The quantum spin chain defined by H (7.1) is not SU (2)-symmetric, but is symmetric under the action of the quantum deformation Uq[SU(2)] of SU(2) (Kirrilov and Reshekikhin, 1988; Pasquier and Saleur, 1990). The generators S +'z of the quantum algebra Uq[SU(2)] satisfy, by definition, the following relations: [S +, S-] = [2SZ]q,
[S z, S +] = dzS +
(7.5)
where the expression [X]q is defined by qX __ q--X
[Xlq
q - - q -I "
=
(7.6)
In the limit q -+ l (symmetric hopping) one has [X]q = X and the quantum algebra relations (7.5) reduce to the usual commutation relations for SU (2). A representation of the generators of the quantum algebra in terms of Pauli matrices is given by + L
L
S + = y ~ s-~ (q),
S- = Z
k=l
L
s ; (q),
k=l
S z = y ~ ' ( l / 2 - nk)
(7.7)
k=l
with sk (q)
=
(qZ~2~vj ) s~- (q
s-~(q,
:
(q-~2-"nJ)s-~(q~='+'ni).
ZJt--k+'~
),
(7.8) (7.9,
To prove that this is a representation of (7.5) one uses the elementary relations qnks-~ : S-~, s-~q nk = q s-~ and qnks~ -- q s ; , s~qnk _ s~ for the usual Pauli matrices to verify that [s~ (q), s~(q)] = 0 for k ~: 1. For k = l one observes that crz = [crZ]q = [2sZ]q for the spin-(l/2) Pauli matrix and thus obtains (7.5). +Our representation is related to the representation given in Pasquier and Saleur (1990) by the similarity transformation V (7.3). In Kirrilov and Reshekikhin (1988) the same representation as in Pasquier and Saleur (1990) is used, but with the replacement q -+ ql/2. This quantum algebra symmetry, where the deformation parameter q is the hopping asymmetry (3.29), arises from the algebraic structure of H, which is discussed in Section 3.4. It would lead much too far off the main route of investigation to discuss quantum algebras as such. The interested reader is referred to Fuchs (1992). Here we introduce the quantum algebra in an elementary way in terms of its defining relations. This is all that we actually need.
1 Exactly solvable models for many-body systems
107
Notice that unlike the usual local spin operators s~: the q-spin operators do not commute among themselves at different sites: s~: (q)s~: (q) = q :F2s/: (q)s~: (q) for l > k.
(7.10)
Each term hk in the Hamiltonian H commutes with S + and S z = L / 2 - N. Hence [n, S • = [n, S z] = 0. (7.11) This can be derived on a purely algebraic level by expressing hk in terms of quantities related to the generators of the algebra (Kirrilov and Reshekikhin, 1988). However, these commutation relations are in fact straightforward to verify by using the explicit representation (7.8), (7.9) in terms of Pauli matrices. For [q[ ~ 1 the representations of the q-deformed algebra are isomorphic to those for q = 1. Hence the canonical N-particle stationary distributions with fixed N can now be obtained as for the symmetric exclusion process by using the symmetries of the system. One obtains essentially the same formulae, except that the integers appearing in the factorials of Section 6 have to be replaced by the q-integers (7.6). The unnormalized zero-energy eigenstates and N-particle summation vectors are constructed by applying the q-deformed ladder operators on the vacuum state 1
1
I/V) = [N]q------~.(s-)N[o)and (NI = [N]q! (0[(s+)N"
(7.12)
Here the q-factorial is defined by [m]q! = [l]q[2]q ... [m]q. The partition function ZL,N = (s IN) is given by the q-binomial coefficient Zt,,N =
[L]q! . [ L - N]q![N]q!
(7.13)
Unlike in the symmetric case, here we get an interesting result. In order to derive the density profile and density correlations in the canonical equilibrium distribution we outline the main steps of the calculation of Sandow and Schiitz (1994) and Schiitz (1997a). The normalized N-particle stationary states may be written IN*) ------[~])/ZL,N = q-N(L+l)q2~-'L=lknk[ N ) / Z L , N. (7.14) Using the commutation relation for Pauli matrices one then finds from (7.14) the relation s~-[ N ) = q-L-l+2k(1--nk) [ N -- 1 ) therefore, with (2.26), the recursion relation p~(N) = ( S [nk[ N* ) -- [ N ] q q - L - l + 2 k [ 1 - p ~ ( N - l ) ] / [ L - N + l]q for the stationary density profile. Iterating this recursion yields an exact expression for the density Pk in a finite system. From the step-function form of the grandcanonical density profile one expects for fixed N a step centred around k = L - N.
108
G.M. Schfitz
Thus we investigate the vicinity of this point by setting r = L + 1 - N + k. In the thermodynamic limit L, N ~ cx~ the recursion reduces to Pr ---- q 2 r ( 1 - - P r - l ) which with the boundary condition l i m r ~ - ~ / O r = 0 is solved by oo
19r = Z ( - - l ) n q
-n(n+l)+2r(n+l).
(7.15)
n=0
This density profile has a step of finite width cx 1/ (3 E), very similar to that of the uncorrelated distribution (Fig. 19). The difference between the canonical step distribution and the uncorrelated grand-canonical distribution appears more clearly on the level of correlation functions. One can derive exact expressions for all density correlation functions in the steady state in terms of the density itself by using (2.26) and the commutation rules of the s~-(q) for different k (Sandow and Schiitz, 1994). For the two-point function one finds ( nknl ) N =
qZk ( nk ) N -- qZl ( nl ) U qZk _ q21 "
(7.16)
Setting ri = L + 1 - N + k, r2 = L + 1 - N + l and subtracting Pr~Pr2 one realizes that unlike in the grand-canonical stationary distribution (7.4) the system has nonvanishing correlations in the domain-wall region even in the thermodynamic limit. To understand the relationship between these distributions we note that in an infinite system the reference point r -- 0 is arbitrary. Hence any normalized superposition of shifted canonical stationary distributions with density profile (7.15) is stationary. The uncorrelated grand-canonical distribution (7.4) is a special case of such a superposition of canonical distributions. Having understood the stationary properties of the system we can proceed to derive self-enantiodromy relations for the asymmetric exclusion process from the U q [ S U ( 2 ) ] - s y m m e t r y . Algebraically speaking, the essential feature in the derivation of the enantiodromy relations for density correlators in the symmetric exclusion process was the fact that nk is a spin-(l/2) operator. Unfortunately this is not the case in the U q [ S U ( 2 ) ] - s y m m e t r i c case of driven diffusion. Instead one has to consider another complete set of observables built by products of the operators Qk = q2Nk (7.17) or their normalized lattice derivatives Ok = (Qk - Q k - l ) / ( q 2 - l ) = q
2 N k - l nk
(7.18)
where Nk = Y]~=l n j is the integrated particle number up to site k. Some further nontrivial ingredients are necessary for the derivation of self-enantiodromy relations. Since (SI is not a factorized state with respect to the q-deformed
1 Exactly solvable models for many-body systems
109
local spin-lowering operators s + (q), the similarity transformation/3 = e s+ is not very useful for the asymmetric exclusion process. Instead, in order to obtain the analogue of (6.5), we first note that for an N-particle initial state any expectation value ( F ) is given by the average ( F ) = ( N IFI P ( t ) ) restricted to the Nparticle sector (because of particle number conservation). We recall the symmetry relation (7.12) for ( N I and calculate the commutator
S+)N
]
[U]q! ' Qk
(s+)N-I
(7.19)
= q N - l ( q 2 -- 1)QkS-~ [N _ l]q!
where S~- = Y~=] s+(q) 9Then, with (0 I(~k -- 0 and (7.12), (7.19), one finds
(NlOk
---- (01 =
S+)N ] IN]q! ' Ok
qN-I (k[
(S+) N-I
(7.20)
1]q!'
[Nand, by repeated application of (7.20),
(S+) N-m (Nlak, --- Ok,, = q m ( N - l ) ( k l . . . . . kml
(7.21)
[N - m]q!
The ki E Bm =-- {kl . . . . . km} are assumed to be pairwise different. With (7.21) the last steps in the derivation are straightforward. Multiplying (7.21) by exp ( - H t ) I P o ) , using the Uq[SU(2)] symmetry (7.11) of the time evolution operator and inserting a unit operator (4.29) gives the self-enantiodromy relations for the asymmetric exclusion process (Schlitz, 1997a)
(Ok,-" Ok>e0
-
(NI0k,"'Okme-Hrle0)
=
~---~(kl. . . . . k m l e - H t l n ) ( n l q -re(N-l) (s+)N-m IP0) n IN - m]q!
Z
(gl0k'~ "'" 0kL, IP0) • (kl . . . . . kmle-Htlk'l . . . . . k m) . (NIQk'I . . . . Ok;,, IP0) •
q2 S-'z..i=l(ki-ki).
l
(k'1. . . . . k m l e -
Ht
Ikl . . . . . km).
(7.22)
In the last step we have invoked detailed balance (7.2), giving the factor q 2~zim=l(ki-k~) inside the sum. This factor can be absorbed in a redefinition
11o
G.M. SchOtz
Ok ~ q-2k Ok. Relations involving correlators of Qk where some of the ki are identical, i.e., involving integer powers (Qk)n of Ok, can be obtained in the same way. Using the commutation relations (7.21) one may also derive relations for correlation functions involving different times. The late-time relaxational behaviour of the system can be studied by an analysis of the corresponding Bethe ansatz equations. One finds in the thermodynamic limit a volume-independent energy gap r -l = D e + DL - 2 x / D R D L which indicates exponential relaxation. This is not surprising as close to equilibrium particles move only in the finite domain wall region between the empty space and the fully occupied part of the lattice. Even in a very large system there are effectively only finitely many degrees of freedom and hence the relaxation is exponential. 7.1.2
Periodic boundary conditions
The introductory discussion of the effect of boundaries indicates that some care needs to be taken when taking the thermodynamic limit of the process. It is clear that the ASEP defined on a ring should have translationally invariant stationary states. So the thermodynamic limit of the stationary state of a periodic system is expected to be very different from that of the system with reflecting boundaries. Since in the periodic case there is neither global detailed balance nor any non-Abelian continuous symmetry it seems difficult to calculate the stationary state from first principles. However, the form (3.28) of the quantum Hamiltonian makes it easy to verify that the product measure (2.24) is stationary not only for the symmetric exclusion process, but also in the presence of a bias. The factorized form of the product measure and the locality of the action of hk on sites k and k + l yields hkl p ) = (DR -- DL)(nk -- nk+l)l P). (7.23) To prove this relation one has to consider only those factors in the tensor product which correspond to sites k and k + 1. This reduces the calculation to computing the action of the local 4 • 4 hopping matrix h = DR(n | v - s + | s - ) + Dr. (v | n - s - | +) on the two-site product state I P ) | ). Finally, taking the sum over k shows that I P ) is stationary. This translationally invariant product measure is also a stationary state of the infinite system (Liggett, 1977). The nonequilibrium nature of this distribution, i.e. the lack of detailed balance, is expressed in a nonvanishing stationary particle current. We derive from the equations of motion (2.17) for the local particle density pk(t) = ( n k ( t ) ) the continuity equation d ~-,ok - jk-1 jk (7.24) -
-
with jk = DR(nk(l -- nk+l)) -- DL((I -- nk)nk+l ). This expression is the
1 Exactly solvable models for many-body systems
111
difference between the current to the right across bond k (equal to the hopping rate DR times the probability of finding a particle on k - 1 and a vacancy on k) and the corresponding current to the left. Because of the absence of correlations in the steady state one finds the stationary current-density relation (7.25)
j ( p ) -- (DR -- D L ) p ( I -- p).
The current has a maximum jmax -- (DR -- D L ) / 4 at density p* = 1/2. Another signal of the importance of boundary conditions is seen in the volume-independent energy gap r -1 = DR + DL -- 2~/DRDL for the system with reflecting boundaries. By naive application of the spectral decomposition (4.23) of expectation values one might imagine that also in an infinite system the energy gap is finite and hence correlations would decay exponentially. On the other hand, in the periodic case the system does not evolve into the blocked equilibrium state that one has for reflecting boundaries. Indeed, just by looking at a single particle on a ring one realizes that there is no energy gap in the thermodynamic limit and hence one expects correlations to decay algebraically. This apparent paradox is resolved by noticing that initial states which have asymptotic density 0 to the left and 1 to the fight will evolve into the equilibrium distribution of the system with reflecting boundaries and hence show algebraic, relaxation at late times. However, if the asymptotic densities are different this blocked state is not approached and one expects different, generally algebraic, relaxation. To get more specific information we investigate here following Gwa and Spohn (1992a,b) the spectral gap of the ASEP on a ring with periodic boundary conditions. By applying the scaling argument one finds the dynamical exponent of the exclusion process in the finite-size scaling of the lowest energy gap in sectors corresponding to finite density. The Bethe ansatz goes through in the same manner as for the isotropic Heisenberg chain. One obtains the single particle 'energies' Ej = DR(I - e -ipj) -k- DL(I -- e ipj)
(7.26)
and as an analogue of (3.13) the boundary condition (DR -k- D L ) q J ( x , x q- 1) = DR qJ (x, x) + DL 9 (x + 1, x + 1) for the internal coordinates. This exclusion constraint gives rise to the scattering amplitude $21 = -
DR + D L e ipl+ip2 -- (DR + D L ) e ip2 DR + D L e ipl+ip2 -- (DR + D L ) e ipl
.
(7.27)
To investigate the finite-size behaviour of the ASEP we impose periodic boundary conditions on the Bethe wave function which leads to the system of Bethe ansatz equations (BAE) (3.23). The lowest energy gap in the N-particle sector is then
G.M. SchOtz
112
given by a certain set of solutions to this equation, to be determined below, through the expression E = ~ v = 1 ej. For DR ~: Dr. the late-time dynamics are dominated by contributions with small Pl, P2 like in the symmetric case, but there is no expansion of S21 around the noninteracting case S -- 1. This is the first mathematical indication that the dynamics of the asymmetric exclusion process are dramatically different from the symmetric process. To determine these solutions we outline the strategy of Gwa and Spohn (1992b), but for the calculational details that lead to the finite-size behaviour of E we refer the reader to the original paper. Consider the totally asymmetric process with D/~ = 1, DL = 0 at half-filling (N = L/2). In terms of the variables Zj = 2eipj - 1 the BAE (3.23) reduce to an equation
L/2 ZI
( 1 - z Z ) L / 2 = --2 L l-I
1
= Y
(7.28)
t=l Z l + l where, remarkably, Y is independent of j. For further analysis of the eigenstates it is useful to introduce the roots Ym of
.L/2
the equation Ym - - Y. A convenient parametrization of these roots is obtained by writing Y in the form Y = - ( a e i ~ L/2 with real-valued amplitude a and phase 0. In analogy to (3.24) this gives Ym = aeiO+4zri(rn-l/2)/L with the quantum numbers m = 1. . . . . L and leads to a parametrization Zm = s/1 - Ym of the shifted Bethe momenta Zm in terms of the integer quantum numbers m. In this way one realizes that each solution to the BAE (7.28) can be represented by a set of L/2 distinct quantum numbers m i E {1 . . . . . L} rather than by the momenta
Pi. The next task is to find the quantum numbers which represent the ground state and the lowest energy gap, then one calculates E for this set {mi}. The energy of a state characterized by quantum numbers mi is found by inserting the corresponding Ymi in (7.28). This yields an equation
(~)L/2
L/2 Zmj _ 1 : Hj =Zl m j + I
(7.29)
which determines the amplitude a and the phase 0 of a state defined by the specified set mi. In terms of the Zj the energy of the state is given by E = Y~.j(1 - Z j ) / 2 . By looking for the solution with the smallest real part of the energy for large L one obtains the finite-size scaling behaviour of the energy gaps and hence the dynamical exponent. Identifying the quantum numbers of the ground state and of the low-lying excited states is, in principle, very difficult. However, by analogy with the structure
1 Exactly solvable models for many-body systems
113
of BA solutions for the ground state of the Heisenberg chain (3.25), a natural guess is the choice mj = 1 , 2 . . . . . L/2 for the ground state and a minimal modification of this set for the low-lying excited states. This conjecture for the ground state is, in fact, easy to prove. The ground states, i.e. the stationary distributions of the ASEP with periodic boundaries, have by construction energy E = 0 and are the same constant distributions [ N* ) as for the symmetric hopping process. This implies a = 0 which is indeed the solution of (7.29) for the ground state quantum numbers. The conjecture regarding the quantum numbers of low-lying excitations has been confirmed by numerical investigation for small chains (Gwa and Spohn, 1992b). The last step of the calculation is largely technical. Taking the logarithm of (7.29) and representing the resulting sum by a contour integral allows for an analytical treatment of the equation for large L. We do not review the rather long calculation and merely quote the result. Asymptotically one finds E ~
c
L3/2
(7.30)
with c ~ 2.30134596 (Gwa and Spohn, 1992b). The energy gaps vanish for large system size and one reads off the dynamical exponent z = 3/2. The analysis was subsequently extended to the partially asymmetric process with arbitrary particle number N (Kim, 1995). One obtains for finite density z = 3/2, but for finite particle number E cx v/N/L 2. This corresponds to z = 2 for finite N with agreement with an earlier result (Henkel and Schlitz, 1994). A change in the dynamical exponent cannot be induced by the interaction of finitely many diffusive particles for which z = 2, even though, as shall be shown below, the quantitative effect of the exclusion interaction on the collective behaviour of just a few particles is rather large. Genetically, one expects to observe the dynamical exponent z = 3/2 in the scaling behaviour of late-time dynamics of local observables. However, the exponent z = 2 for configurations with finitely many particles demonstrates that the dynamical exponent is not a system-inherent quantity, but depends on the initial state. In fact, the occurrence of the dynamical exponent z = 2 is not restricted to initial states with finitely many particles; further below we shall see how the exponent z = 2 is related to the diffusive motion of shocks in manyparticle initial distributions. The self-enantiodromy relations derived above relate a property of the system with arbitrary initial state (including finite-density states where z = 3/2) to a problem involving only few particles where z = 2. For instance, the dynamics of the quantity ( Qk ) reduce to a single-particle problem involving a biased random walk and imply a dynamical exponent z = 2 (Section 4.2.4). This is another example where certain properties of the collective dynamics of a many-particle system reduce to a few-body problem. Hence the dynamical exponent that one
114
G . M . Sch0tz
observes in a system depends also on the quantity that one investigates.
7.1.3
Bethe ansatz in the infinite system
We consider now directly the infinite system with arbitrarily, but finitely many particles. A configuration of N particles may be specified by the set of coordinates ki in the physical domain Y = k l < k2 < . . . < k N. A convenient presentation of the ASEP is in terms of the master equation for the probability P(BN; t) of finding N particles on sites BN = {kl . . . . . kN} ~ Y at time t. We restrict ourselves again to the totally asymmetric exclusion process with DR = 1, DL = 0 (hopping only to the fight with unit rate). This involves no loss of physical content as far as universal behaviour is concerned, but leads to considerable gain in technical matters. The probability P (BN; t) satisfies the master equation d --
dt
P(kl . . . . . kN"
t)
P(kl - 1. . . . . kN; t) + . "
+ P(kl . . . . . kN -- 1; t)
- N P ( k l . . . . . kN; t).
(7.31)
This expression is obtained for the situation where there are no neighbouring particles. There are the N different gain terms for the possibilities of reaching the state I kl . . . . . k N ) from some other state and the loss term proportional to N for the N possibilities of changing from I kl . . . . . k N ) into some other state. To make this equation valid for all kl . . . . . kN ~ Z N (including the unphysical coordinate sets) it has to be supplemented by boundary conditions in Z N" if any two neighbouring arguments ki, ki+l are equal, P has to satisfy for all t > 0
P(kl . . . . . ki+l -- ki . . . . . kN; t) -- P(kl . . . . . ki+l -- k i + l . . . . . kN; t). (7.32) This boundary condition expresses the exclusion interaction. With specified initial condition AN -- {ll . . . . . lN} ~ Y, i.e.,
P(BN; O) = r
(7.33)
the probability P(BN" t) becomes the conditional probability P(BN; tlAN; O) and thus a complete solution of the problem. Notice that both as a function of the arguments ki and li the function P is a probability only in the physical domains kl < . . . < kN, Ii < . . . < lN, but is defined in the whole of Z N by the equations (7.31 )-(7.33). Following the strategy already employed above for the symmetric exclusion process we solve this master equation with the Bethe ansatz adapted to the anisotropic Heisenberg chain (Yang and Yang, 1966).
1 Exactly solvable models for many-body systems
115
(i) The case of a single particle has been solved in Section 4.2.4. It yields the dispersion relation (4.25) between energy and momentum. (ii) In the two-particle case we proceed as in the case of symmetric exclusion. The only difference (besides the expression for the single-particle energymomentum relation) is the boundary condition when two particles are on nearestneighbour sites. In the totally asymmetric case the corresponding scattering amplitude (7.27) reduces to $21 =
-
1 - eip2 ~ . 1 - eip ~
(7.34)
There is no bound state with a complex momentum. For the two-particle case we conclude that P], P2 6 [0, 2rr) and
P(Xl,
X2;
dp2 +~:p2)t f (Pl, Pz)(e iplxl +ip2x2 +$21 eip2x IWiplx2 ) t) -- f ~dpl f -~e-(Ep~
(7.35) is the general solution of (7.31) with boundary condition (7.32). In order to satisfy the initial condition (7.33) one has to determine f(p], p2) and discuss the pole resulting from the integration over S21. Assuming that the particles were initially at sites Yl, Y2 it turns out that choosing f(pl, p2) = e-iplyl-ip2y2 and defining the position of the pole in Szl by pl --+ pl + i0 gives the correct initial condition
P(xl,
X2; 0 ) =
•xl,Yl•X2,Y2 .
(ii) The absence of bound states and the simple form (7.34) of the two-body scattering matrix allow for a compact determinant representation of the N-particle solution of the master equation. Following Schiitz (1997b) we introduce the function
oo(
Fp(n" t) =- e -t y ~ k+p-I k=0
)tk+n
p-l
(k+n)i
(7.36)
where the binomial coefficient and the factorial are defined by the F-function, i.e. a! ~ F(a + 1) and (~) F ( a + 1) --= . (7.37) F(b + 1)F(a - b + 1) Here we need only p, n 6 Z and t 6 [0, cx:~). Let F(BN, AN; t) be the N x N matrix with matrix elements Fij = Fi-j(ki - lj; t). Then
P(BN; tlAu; 0) = det F(BN, AN; t)
(7.38)
is the solution of the master equation (7.31) with boundary condition (7.32) and with initial condition (7.33) (Schiitz, 1997b). To prove this result one uses general properties of determinants and some special properties of Fp (n; t): for integer p < 0, Fp reduces to a finite sum, IPl
tk+n
k=0
(k + n)!
Fp(n;t)=e-tZ(--l)k(IPkl )
(7.39)
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G . M . SchOtz
In particular, F0(n; t) -- the -t/n!. At time t = 0 one has
lil'nOFp(n't)= )\ ( - "p-I +P-I
(7.40)
which vanishes for n > 0. For the time derivative one finds
d
dtFp(n 9t)= F p - l ( n - 1; t ) = F p ( n - 1; t ) - Fp(n; t)
(7.41)
and for the integral
fotdtFp(n" t)
-- Fp+ l(n
+
1"
t)
-
() -n-l+p P
~
:
Fp(k; t) -
-n +p . (-Ip)
k=n+l
(7.42) In a very simple application of (7.38) one can see how the exclusion interaction affects the collective diffusion of two particles. Suppose two particles are placed at time t = 0 on lattice sites Yl = - 1 and Y2 = 1. The probability (nx) of finding a particle on site x at time t describes the diffusive broadening of the initially spatially concentrated density. The moments of this density distribution may be obtained from the Fourier transform ~(q) = Y~x e-iqx (nx) by taking derivatives with respect to q. Here we are interested in N
=
<x>
=
(X 2)
=
t3(0)= 2
(7.43)
i /5'(0) = 1
7.44) x
- ~ i ~,,(o) = ~l Z x 2 ( n x )
(7.45)
x
from which one obtains a collective drift velocity Vc and a collective diffusion constant Ac formally defined as in (4.15), (4.16), but with the initial state described above. A lengthy, but straightforward computation of these quantities from the two-particle conditional probability gives (Schlitz, 1997b) Vr
=
DR-
Ac
=
DR +
(7.46)
DL
+ 2(DR(DR--DL)2 DL)
-- --rrl) .
(7.47)
It is straightforward to show that in the case of noninteracting particles Vc = DR - DL, Ac = (DR + DL)/2. In the undriven system one has Ac = (DR + DL)/2 as in the noninteracting system. In the presence of the drift, however, we find that Ac increases to the value (7.47). It is remarkable that the crossover to a dynamical exponent z = 3/2 at finite density has its precursor already in a strong
1 Exactly solvable models for many-body systems
117
deviation of the collective behaviour from noninteracting particles in a system of only two particles. The solution of the master equation may also be used for a perturbative analysis of quantities in systems with finite density. Using the exact solution one can obtain an exact expansion of the structure function in powers of p where the nth power is obtained by solving the n-particle problem. This can be seen as follows: suppose one wants to calculate the time-dependent density profile Pk (t) up to second order in the background density p. The time derivative of the twopoint correlation function involves a three-point correlator which is of order p3 and which therefore may be neglected in the desired second-order approximation. Omitting the three-point correlator results in a differential-difference equation for the two-point correlator which is identical to the two-particle master equation (7.64) with boundary condition (7.67). Thus one can calculate (nknt) up to order p2 and then by summing up two-point correlators one gets (nk). For a thirdorder approximation one considers the three-point correlation function. If one neglects fourth-order correlators, it satisfies the three-particle master equation. Summing up three-point correlators yields (nk) up to order p3.
7.1.4
Diffusion and coalescence of shocks
Above we have seen that the ASEP defined on an infinite lattice has two very different kinds of stationary distributions: the equilibrium shock distribution where no particle current flows, and the translationally invariant product measures with current j = (DR -- DL)p(1 -- p). It is clearly of interest to ask what happens to an initial distribution which is a product measure with density Pl up to some point k - 1 on the lattice and a different density P2 from this point onwards. In an equilibrium system such a jump in the density would constitute a 'domain wall' connecting to equilibrium regions of different densities. Here (unless Pl = 0 and P2 = 1) the domain wall connects to stationary regions and we want to investigate how this domain wall behaves as time goes on. It is important at this stage to recall an early study by Rost of the totally asymmetric exclusion process (Rost, 1982) which describes the time evolution of the density profile from a step function initial state. Here the initial density decreases sharply from 1 to 0 at site k = 0 as one goes in the direction of motion (Fig. 20). Even though both stationary regions have vanishing current, it is clear that at the interface where the two regions meet a current will start to flow. The time evolution of the density profile on the hydrodynamical scale k, t --+ cx~ with the scaling variable u = k / t fixed can be calculated from the scaling solution p(u) of the continuum limit Otp = Oxj of the continuity equation (7.24), see Section 4.2.2. As time proceeds, the density profile p(u) = (1 - u ) / 2 interpolates linearly on length scales of order t between the high-density region and the low-
118
G.M. SchOtz
density region. Locally the system remains uncorrelated. The initially sharp domain wall ( . . . 1 1 1 0 0 0 . . . ) is unstable and smears out in the course of time.
p=l
p-O
t
Fig. 20 Large-scale time evolution of the density profile in the totally asymmetric exclusion process from a step-function initial profile (dashed line). Particles hop to the fight, and the step smears out as time proceeds. A forward domain w a l l - which is understood to mean a sudden i n c r e a s e of the density in the direction of the flow (Fig. 21) - behaves very differently. Remarkably, this domain wall remains a localized object (De Masi e t a l . , 1989; Ferrari e t a l . , 1991), a phenomenon only too well known from traffic jams. It constitutes a shock which performs a random motion but does not smear out in the course of time. We shall refer to such an initial distribution as a shock distribution.
p=l
oOeeeeeeooeeoeooeeoeoeeoooeOeeeeeoeeeeeeeoeeeoeeeooe
| ooooooooooooooOoeoooooooooooeoooeoooooooo~
0=0
Fig. 21 Motion of second-class particles (black circles) in a background of ordinary particles. The sudden increase of density (dotted curve) at site k constitutes a shock (= domain wall) in the system. To the left of the domain wall particles are distributed homogeneously with an average density PL < 1/2 and the second-class particle travels with positive velocity in direction of the shock. To the fight of the domain wall the background density is #R > 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).)
1 Exactly solvable models for many-body systems
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)
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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 << t << L the scaling form implies Dc = t ~ D* (0). On the other hand, for the stationary case 0 << L << t the collective diffusion coefficient depends only on L. Hence the scaling form implies an asymptotic divergence of the scaling function D* (x) ~, x a which is necessary in order to cancel the time-dependence of Dc in this limit. Thus, for times large compared to system size, Dc ~ L za. Since the exact result shows De ~ L 1/2 (Derrida and Evans, 1994) and z = 3/2, one reads off ct = 1/3.
7.1.6
Stability o f a shock
The motion of small perturbations in a stationary background provides a clue as to why the upward domain wall in the ASEP constitutes a stable shock whereas the downward domain wall dissolves. One may imagine that by a small fluctuation a certain amount of mass detaches from the shock and forms a perturbation at a small distance from the shock position. Equation (7.67) shows in the case of the upward domain wall that for all shock densities PR,L L
R
v c > Vs > Vc.
(7.68)
In the moving reference frame of the shock the excess mass drifts back to the position of the shock and hence stabilizes it. On the other hand, in the case of the downward domain wall the excess mass moves away from the shock. Therefore this domain wall is not stable against fluctuations, in the course of time the shock smears out. Clearly, this is not a rigorous argument. But equation (7.68) may be used as a heuristic criterion for the stability of a shock in a lattice gas. Support for this picture comes from the hydrodynamic limit. The collective velocity is then nothing but the speed of the characteristics of the corresponding hydrodynamical equation OtP = Oxj resulting from the continuum limit of the continuity
1 Exactly solvable models for many-body systems
125
equation (7.24). In this limit, the criterion (7.68) becomes the defining property of a shock discontinuity (Lax, 1957). It asserts that the characteristics are moving into the shock. For current--density relations which are not globally convex one decomposes a single shock into subsequent small shocks and then applies (7.68) to these minishocks in order to decide on stability. By taking the limit of infinitesimal shocks one recovers in this way the scaling solution of the hydrodynamical equation discussed in Section 4.2.2 with the appropriate shock discontinuities (Ballou, 1970). An interesting lattice gas model in this context is the k-step exclusion process where particles hop to the closest empty site within a distance of k lattice sites (Guiol, 1999). The current--density relation has an inflection point which leads to a rich variety of shock- and scaling solutions (Guiol et al., 1999).
7.2
7.2.1
TASEP with open boundaries
Stationary bulk density
It is well-known that in one-dimensional equilibrium systems with short-range interactions no phase transitions associated with an onset of long-range order take place at nonzero temperature. Furthermore, the bulk state of the system is independent of the boundary conditions. None of this needs to be true if the system is maintained in a nonequilibrium state through the action of external forces which impose a current on the system (Krug, 1991). Various types of phase transitions may occur as a consequence of the interplay of particle transport with a localized defect or inhomogeneity, represented, e.g. by the presence of a boundary. Indeed, a recurrent problem in the investigation of many-body systems far from equilibrium is posed by the coupling of a driven particle system with locally conserved particle number to external reservoirs with which the system can exchange particles at its boundaries. In the presence of a driving force a particle current will be maintained and hence the system will always remain in a nonequilibrium stationary state characterized by some bulk density and the corresponding particle current. While for periodic boundaries the density is a fixed quantity, the experimentally more relevant scenario of open boundaries naturally leads to the question of steady-state selection, i.e. the question which stationary bulk density the system will assume as a function of the boundary densities. In the topologically simplest case of quasi one-dimensional systems there are two boundaries and bias of particles towards one of the boundaries. The issue of steady-state selection in this set-up is of importance for the understanding of many-body systems in which the dynamic degrees of freedom reduce to effectively one dimension. Examples are the diffusive systems discussed in the introduction.
126
G.M. Sch0tz
The best studied model of this kind is the totally asymmetric exclusion process coupled to a reservoir of fixed particle density PL = ct at the origin (site 0) and fixed density ,oR = 1 - / ~ at the right boundary site L + 1. This coupling leads to an injection rate a of particles at site 1 and to an absorption rate/3 at site L of the system. For a first (and preliminary) understanding it is instructive to discuss the mean field approximation to the phase diagram originally obtained in MacDonald e t al. (1968). The mean-field approximation is obtained by using the (exact) stationarity condition j = (nk(1 -- nk+l) ) = c~(l -- ( n l ) ) = /~(nk ) and approximating (nk(l - n k + l ) ) = (nk)(l - - ( n k + l ) ) by neglecting all correlations. This leads to an exactly solvable recursion for the stationary density profile Pk = (nk). One finds three distinct bulk phases (Fig. 22). When particles are supplied at the left end with rate c~ > t5, and removed at the fight end not too fast,/3 < 1/2, there results a high-density phase for which particle extraction is the dominating process; the average density Pbulk in the bulk equals the fight boundary density l - / 5 > 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
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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_
with the initial condition
X Lp, p + l
= ot~YL-1 , p+l
for 1 -<- p -<- L 9
(7.75)
These recursions are linear and have constant coefficients. However, the boundary conditions make a solution with standard generating function techniques difficult. They were solved (Schlitz and Domany, 1993) by explicit calculation up to L = 5, then guessing the pattern behind the integer coefficients of the monomials c~t/3 m (see Appendix C for the YL,k) and finally proving that the result satisfies the recursion relations. This strategy is possible since by looking at the coefficients appearing in the expressions for YL,k one realizes that they are all differences of binomial coefficients. For writing down the solution of the recursions it is useful to introduce the function M-l
GML,k(X)= Z bL'k(r)xr
(L _> 1)
(7.76)
r--O
with
bL,k(r)=(k-2+r)
k - 2
-
(k-2+r) L
(7.77)
and with the definitions b0,1(0) = G 0l. 0 -- G 0l, 1 - 1 9 Using the properties of the binomial coefficients one proves that the function k-2
YL,k(ot, j3) =/3LG~,k(a) + )---~eft-s/3 t'-k+l+sr:s+l --t.,t (~) s--0
solves the recursion relations (7.71) with the initial conditions (7.73).
(7.78)
130
G.M. SchQll:z
The structure of the coefficients in the solution for the X L,k p is more complicated. Except for the initial condition (7.75) they satisfy the same recursion as the )'I.,/,. This suggests that these coefficients are built from linear combinations of products of binomial coefficients. By investigating the pattern that arises from the explicit solution of the recursion for small L it turns out that relations (7.74) with initial condition (7.75) are satisfied by
X L,k p (Or,~)
L-p Z bL-p'k-p(r)otr+l ~r+l YL-r-l,p+l (Or, ~) -+r--0 k-p-2
~ L+l-p-r(7.k-p-l-r (fl)yp l+r,p+l(Of,f) " L-p.L-p -
~L-k+2 Z r--0
(7.79) From these expressions one obtains L ZI. = YL,L+I = Z ~ s=0
s~
(7.80)
and after some computation, involving relabelling of indices, we get
L-p r,.,
=
--
(7.81)
L_p,L_p (fl) YL_s_l,p+l (or, fl) s=0
and hence the exact stationary density profile for any L > 1, 1 < p < L. Equations (7.80) and (7.81) provide also an exact expression for the current j = (nk(l - nk+l) ) = fl(nL ) = Ct( 1 -- n l ) which because of stationarity is spaceindependent. In order to analyse the density profile it is more convenient to study the lattice derivative tt. (p) = (Tt.,p+l - Tt.,p)/Zt. of the density profile (p # L). From (7.81 ) follows (7.82) tL(p) = Fp(ot)FL-p(~)/ZL with Ft. (x) = x - L - I G~,t. (x)
(7.83)
(see Appendix C) and
Z~L
ZL (1 --or --/~)aL+l/~L+l
=
FL (/~) -- FL (c~) a ( l -- a) --/~(1 --/~) F~(f)
i-5-
a#f,l-f 1
(7.84)
1 Exactly solvable models for many-body systems
131
where the prime denotes the derivative with respect to ft. For a -- 1 - fl one obtains tL (p) = 0, i.e., the profile is constant on this curve (Derrida et al., 1992), as derived in a different way above. With TL,t. ---- a Z L - l which gives the right boundary density one can reconstruct from (7.82) the full density profile. From (7.82) we learn that up to the amplitude 2, the derivative tL (p) of the density profile can be written as a product of two functions; one of a and the other of/~: tL(p) cx F p ( a ) F L _ p ( ~ ) . This fact has important and surprising consequences. It implies that nonanalytical changes in the p-dependence of the density profile can occur on two kinds of lines: ct = ac and a n y ~, or/3 =/~c and any a. Hence if a phase transition is predicted to occur on the fl > 1/2 segment of the line a - 1/2 (the mean field transition to the maximal current phase), then the transition must extend in some way to the fl < 1/2 regime as well! Indeed, analysis of the function F p ( x ) for large L reveals that its dependence on p changes nonanalytically at x = 1/2 (Schiitz and Domany, 1993). Similar considerations hold for the line fl = 1/2 and a < 1/2, which separates the low density phase A into two distinct regimes (Fig. 22). These transitions are not found by the mean-field calculation. Another unexpected consequence of the separability into a product is the existence of two independent length scales in the model, one determined by the injection rate ct, the other one by the absorption rate/~. This is surprising, as one might believe that only the larger of these two quantities determines the behaviour of the system. In fact, as long as the system is not in the maximal current phase, this indeed is the case as far as the current j = (nk(1 -- nk+l) ) is concerned: in the continuum limit one has j =/~(1 - / ~ ) for o~ >/~,/~ < 1/2, and j = t~(l - a ) for/~ > ct, ct < 1/2 (whereas j = 1/4 if both a and/~ are larger than 1/2). Since in the mean-field calculation the shape of the density profile is determined by only the current, phase transitions are seen neither at a = 1/2,/~ < 1/2, nor at/~ -- 1/2, a < 1/2. Prior to presenting an explanation for the unexpected structure of the phase diagram, we study the density profile in the thermodynamic limit L ---> c~. We want to discuss the density profile of a large system (L >> 1) as a function of the space coordinate p, at large distances from both ends, i.e., we consider p >> 1 and r = L - p >> 1. We define a length scale ~o by ~-I = _ In (4o(1 - tr)).
(7.85)
As tr reaches 1/2, ~o diverges. For the various phases AI--C one finds the following results (Fig. 23): High-density phase BI This phase is defined by the region/3 < ~ < 1/2. One finds an exponential decay of the density profile with a localization length
-~ = ~-~ - ~ - ~ ,
(7.86)
G. M. SchQtz
132
(a)
~=t3
= i
(b)
u = 0.3, /3 = 0.2
(d)
a = 0.2, fl -- 0.3
J
(c)
o~ -- 13 = 0.3
.f
Fig. 23 Exact stationary density profiles for a lattice of 100 sites in the low-density phase AI (a), in the maximal current phase C (b), on the coexistence line (c) and in the highdensity phase B1.
and
tL(p) = (1 -- 2 t r ) ( 1 - - e - l / ~ ) e -p/~.
(7.87)
The density approaches its bulk value ,Obulk = 1 -- fl from below. One has j fl(l - / 3 ) and (nl) = 1 - / 3 ( 1 - ~ ) / a < 1 - fl = (nL). T r a n s i t i o n line f r o m BI to BII On approaching ce = 1/2 from below in the region /3 < 1/2, we find that the ~a diverges but ~t~ remains finite, and the localization length ~ depends only on/3. For a = 1/2 the slope of the profile becomes (1 --2/3) 2 p-l/2e-P/~. (7.88)
tL (p) =
2~/-~
1 Exactly solvable models for many-body systems
133
The current and the boundary densities are given by the same expressions as in the high-density phase I.
High-density phase BII On crossing the phase transition line into the highdensity phase BII defined by ct > 1/2 and/3 < 1/2 one obtains tL(p) = (1 -- ~ -- fl)(~ -- /3) p -3/2 e (1 -- 2ct) 2 4%-
P/~l~.
(7.89)
The power-law correction to the exponential decay is different from that on the transition line. The current and the boundary values are given by the same expressions as in the high-density phase I, but notice that the slope of the profile changes sign on the curve ~ = 1 - / 3 where the density is constant,/9bulk = 1 - / 3 for 1 < i _5< L. For ct > 1 - 13 the slope is negative.
Transition from Bu to the maximal current phase C When /3 reaches the critical value 1/2 in the region ct > 1/2,/3 < 1/2, then also ~t~ diverges. The slope of the profile is given by
tL(p)--
1 ( p)-l/2p_3/2 4v/-if- 1 - - ~ .
(7.90)
Near the origin (1 << p << L) we can neglect the piece with p / L in (7.90), so the slope is dominated by p-3/2. In the boundary region (p = L - r, 1 << r << L) the shape of the profile is determined by r -l/z, but the amplitude of tL (r) is only of order 1/L. Therefore, up to corrections of order l/L, the profile near the boundary is flat, whereas it decays as p-l/2 with the distance p from the origin to its bulk value Pbulk = 1/2. The current reaches its maximal value j -- 1/4 and one finds (n t.) = P b u l k = 1/2 and (n 1) = 1 - 1/(4a).
Maximal current phase C
If/~ > 1/2 and a > 1/2, the localization length remains infinite. The derivative tL (p) depends on leading order in inverse system size neither on ct nor on/3 and one has algebraic behaviour of the slope:
ta(p)
1
= -~(1 4 x/-ff-
-
p/L) -3/2p-3/2 .
(7.91)
Therefore the density approaches its bulk value Dbulk = 1/2 as p-l~2 with the distance p from the origin from above and a s r - 1 / 2 with the distance r - L - p from the boundary from below. The current takes its maximal value jmax = 1/4 throughout the phase and one obtains (nL) = l/(4fl) and ( h i ) = l - - 1 / ( 4 o ~ ) .
134
G.M. SchQtz
Low-density phase AI This phase is defined by ct < ~ < 1/2 and is related to the high-density phase BI by a particle-hole symmetry. Therefore the decay is exponential, (7.92) tL(p) = (1 -- 2,6)(1 - - e - l / ~ } e -r/~, /
\
\
/
with localization length ~ - l = ~d-l _ ~ - l
(7.93)
and r = L - p >> 1. The density approaches its bulk value/)bulk = t~ from above. The current is given by j = ot(l - t~) and (nL) = a ( l -- u ) / ~ > t~ = (nl).
Low-density phase All The profile in this regime (/~ > 1/2, t~ < 1/2) is obtained from (7.89) by exchanging t~ and ~ and substituting p by r. This is a result of the particle-hole symmetry of the model. In the same way one obtains the profile on the phase transition lines from AI to All and from All to C out of the profiles on the phase transition lines from BI to BII and BII to C respectively. Coexistence line AI/BI If a = fl < 1/2 both ~c~ and ~t~ are finite, but since ~a = ~ one gets a divergent localization length ~-1 0. The density profile is linear with a positive slope =
tL(p) -- (1 -- 2t~)/L.
(7.94)
The current is given by j = a(1 - a ) and one has (n l) = a and (nt,) = 1 - ct.
7.2.3
Domain wall dynamics and overfeeding
In order to intuitively understand the shape of the density profile in phases AI and BI and hence the first-order phase transition we follow SchLitz and Domany (1993) and Schlitz (1993b) and approximate the stationary distribution by assuming that the density profile is built up by a superposition of shock profiles (Fig. 21) with a constant density PL ---- t~ up to the domain wall position x0, followed by constant density RR 1 - ft. The picture we have in mind for this scenario is that particles injected with rate a at the origin move with constant average velocity f) = j / p = 1 - a > l / 2 until they hit the domain wall where they get stuck and continue to move only with velocity fl < 1/2. This region of high density is caused by the blockage introduced through the connection to the reservoir of high density l at the boundary. Such a scenario is plausible, since constant densities u < l / 2 starting from the origin and 1 - fl > 1/2 connected to the fight boundary are both stationary states of the system. We postulate a probability p ( x ) cx exp ( - x / ~ ) of finding the domain wall at position x. Since PR = 1 - fl > ct = PL for all times this postulated domain wall, once established, cannot disappear from the system. =
1 Exactly solvable models for many-body systems
135
If ct < 13 (low-density phase) then particles are absorbed with a higher probability than they are injected. The incoming current is less then the outgoing current and the probability of finding the domain wall decreases exponentially with increasing distance r = L - x from the fight absorbing boundary. On the other hand, in the case where ct > t3 (high-density phase) the situation is reversed. Since the incoming current exceeds the outgoing current the domain wall is most likely to be found near the left boundary and p(x) decreases with increasing distance from the origin x = 1. Averaging over all such profiles with the weight p(x) leads to an exponential decay to the respective bulk value on the length scale ~ which we identify as the localization length satisfying ~ - l = ~-1 _ ~ - l (see (7.87) and (7.92)). This picture provides also a natural explanation of the linear profile on the transition line a = /3 where the absorption and injection probabilities are equal. Here ~ diverges and the probability of finding the domain wall at x is independent of x. Averaging over step functions with an equal weight for every position of the step gives a linear profile. This is a result of fluctuations in the incoming and outgoing currents which cause the domain wall to perform a random walk. The fixed boundary densities prevent the domain wall from disappearing from the system, but in the bulk there is no mechanism that could localize the position of the domain wall. We also conclude that in the phases AI and Bl the current is given by the expression j = min (~(1 - a), fl(l - / 5 ) ) ,
(7.95)
in agreement with the exact result. It is worth noting that the mean-field calculation which neglects fluctuations singles out the constituent step function with the domain wall located in the centre. In order to get some insight regarding the second-order phase transition and the transition within the high-density phase, we consider the transition from the low-density phase Ail to the maximal current phase on the line a = 1/2 but 13 > 1/2. In the maximal current phase C the bulk density and the way it is approached does not depend on a whereas in the low density phase An ct does determine the bulk density and how the profile decays to it. This may be understood by assuming that as ct exceeds 1/2 the particles close to the origin block each other rather than flowing away freely with maximal average speed ~ = jm/P* = 1/2. As a result, injection attempts fail frequently and an increase in the injection rate does not lead to an enhanced current penetrating into the system. Clearly this overfeeding effect does not depend on the absorption at the fight boundary and is therefore also applicable to the transition from phase Bl to phase Bn. We conclude that these transitions are caused by reaching the maximal transport capacity of the system at the origin (or boundary) and result in the divergence of the corresponding length scale which determines the shape of the profile. These arguments are supported by the observation that in the maximal current phase the current j -- 1/4 becomes
136
G.M. SchQll:z
independent of both boundary densities. No change of the boundary density is transported into the system. '~
7.3
More on the origin of domain-wall physics
Much of the dynamical behaviour discussed above can be verified in a simpler exactly solvable exclusion process introduced by SchLitz (1993b). In this model one can explicitly calculate time-delayed correlation functions in the stationary state and hence without making any assumptions directly study the dynamical behaviour of the system. The time evolution is discrete and consists of two half-time steps. In the first half-step we divide the chain with L sites (L even) into pairs of sites (2, 3), (4, 5) . . . . . (L, 1). If both sites in a pair are occupied or empty or if site 2x is empty and site 2x + 1 occupied, they remain so at the intermediate time t' = t + 1/2. If site 2x is occupied and site 2x + 1 empty, then the particle moves with probability 1 to site 2x + 1. These rules are applied in parallel to all pairs except the pair (L, 1). In this pair representing the boundary (site L) and the origin (site 1) respectively, particles are absorbed and injected according to the following stochastic rules. If site 1 was empty at time t then it remains so with probability 1 - a and becomes occupied with probability ct at time t'. If site 1 was occupied at time t then it remains occupied with probability 1. These two rules are independent of the occupation of site L. On the other hand, if site L was occupied at time t it remains so with probability 1 - / ~ and becomes empty with probability ~. If site L was empty, it remains empty with probability 1. These two rules are independent of the occupation of site 1. In the second half-step t + 1/2 ~ t + 1 the pairing is shifted by one lattice unit such that the pairs are now (1,2), (3,4) . . . . (L - 1, L). The hopping rules are applied in all these pairs, there is no injection and absorption in the second half-time step (Fig. 24). One notices that the bulk dynamics are completely deterministic. At densities less than 1/2 particles move ballistically without ever colliding. The randomness comes purely from the stochastic injection and absorption at the boundaries. This makes the model ideal for disentangling the effects of boundary processes and bulk randomness.
*As a result of the particle-hole symmetry of the problem, the discussion of the transition from the low-density phase AI to the low density phase All and from the high-density phase BII to the maximal current phase C proceeds along analogous lines.
1 Exactly solvable models for many-body systems
t
I"
'1
[9
137
tt
t
Fig. 24 Updating sequence for sublattice parallel updating. In each pair of sites the configuration is updated in each half-time step t ~ t' and t t ~ t + 1. In the first half-step a particle has been injected at site 1. 7.3.1
Construction o f the stationary state
This model was originally solved by constructing the stationary state explicitly for small lattices up to 14 sites and then guessing the general solution for arbitrary length L (Schlitz, 1993b). The result is a set of recursion relations for the stationary distribution of a system with 2L sites in terms of the distribution for a system with 2(L - 1) sites. Various consistency requirements strongly suggest validity of the result obtained in this way. Later a proof for these recursions was obtained using a stationary matrix product ansatz adapted to discrete-time dynamics (Hinrichsen, 1996). From the resulting rules for the construction of the stationary state, or more straightforwardly from the matrix solution, one obtains the exact stationary density 1 _ (~)2x
ct -t-- (1 - ct)
(n2~)
I -- l-ct ( ~ ) L + I
2x
ct + (1 --Ct) 2
ct -7(=/3 O~=
1 + L ( 1 - ct)
1-
(1 - - ~ ) ( n2x-I ) (1 --Or) 2
1-
l-oe
L+I
2x - 1
(7.96) -k-
1 + L ( 1 - ct)
a ( l - a) 1 + L(1 - a)
ct=fl.
As a consequence of the parallel updating mechanism the particle current (see (A. 12)) is related to the sublattices densities by j - ( n2x ) - ( n2x-I ). One finds the current-density relation J(P)=
2p 2(1-p)
for p _< 1/2 f o r p > 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).)
1 Exactly solvable models for many-body systems
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~
(7.100)
This length is composed of two individual length scales ~ - l = In cr in a similar way as in the usual ASEP. For large systems one obtains from (7.96) the density profile up to corrections of order exp ( - L / ~ ) , (n2x)
=
ce + (1 - / 3 ) e -(L+l-2x)/~
(n2x-l)
=
(1 - fl)e -(L+2-zx)/~.
(7.101)
In the low-density phase the profile decays exponentially with increasing distance _(even)
_(odd)
from the boundary to its respective bulk values Pbulk = ce and Pbulk = 0. In the high-density phase ce > 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
1 Exactly solvable models for many-body systems
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) and/5 > 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 << r < t . Notice that the two scales r and ~ are not independent quantities but related through r - l = 2 1 n c o s h ( ~ - l / 2 ) . As/5 approaches 1/2, r and ~ diverge and are asymptotically related through r ~ 4~2 indicating a dynamical exponent z = 2. This is surprising at first sight since one expects z = 3/2 for the exclusion process. To understand this result we consider the scaling form of the time-dependent correlation function in the scaling region of large 2~ < t 1/2
G* (x, x + 2y" t) = e -R/~
W/~_~te- [Y+t/t(2~)12
(7.112)
1 Exactly solvable models for many-body systems
143
where R = L - x measures the distances ofx from the boundary. G*(x, x + 2 y ; t) is invariant under the scaling transformation R --+ ~.R,
y --+ ~.y, t ---* k2t.
~ ---* k~,
(7.113)
The form (7.112) of the correlation function has a simple interpretation in terms of the constituent profiles discussed in the preceding section. The timedependent joint probability ( dx+2yTtdx ) gives 1 ifx is in a region of high density at some time to and x + 2y is in a region of high density at time to + t and zero otherwise. Since we are studying the stationary correlation function to --+ oo the probability that x is in the high-density region is the stationary probability ( dx ) = e - R / ~ given by (7.101). This explains the amplitude of the correlator (7.112). From the y-dependent part one reads off the random-walk nature of the motion of the domain wall. It moves with shock velocity Us = 2(2/3 - 1) in agreement with the generally valid expression (7.65) and fluctuates around its mean with diffusion constant D = 1, confirming the general picture developed above. The dynamical exponent z = 2 originates in the domain wall diffusion. In this model there is no maximal current phase with a power behaviour of the density profile and no phases corresponding to the phases AII or BII (for a > 1/2 or/3 > 1/2) where the shape of the density profile is determined by a product of a power-law behaviour with an exponential decay. Phase transitions to such phases cannot occur here because of the deterministic nature of the dynamics which do not allow for the overfeeding effect: the system reaches its maximal transport capacity at average density 1/2, but here this corresponds to a completely filled even sublattice and an empty odd sublattice. An average density of 1 on the even sublattice at the origin can only occur if particles are injected with probability 1. Thus there can be no mutual blockage near the origin. (Similar arguments can be used for a discussion of the dependence of the phase transitions on the absorption rate/3 by exchanging particles with holes and studying the injection of holes at the boundary).
7.4 Theory of boundary-induced phase transitions In the absence of detailed balance stationary behaviour cannot be understood in terms of a free energy, but has to be derived from the system dynamics. Generally speaking this is a very difficult task. However, after having achieved a quantitative picture of the phase transitions of the exclusion process in terms of the underlying shock dynamics we are in a position to develop an approach which allows us to predict the selection of stationary states and the associated boundary-induced phase transition for generic driven lattice gases.
144
G . M . Schfitz
This problem was first considered in general terms by Krug (1991) who postulated a maximal-current principle for the specific case where the density PR at the right boundary to which particles are driven is always kept at zero. This principle asserts that independently of the details of the dynamics the system tries to maximize its current j in the sense that j = maxp~[0,m~] j (p). The validity of the maximal-current principle is supported by phenomenological stability arguments and Monte Carlo simulations for the totally asymmetric simple exclusion process (TASEP) and for a generalized exclusion process with nearest-neighbour interaction (Katz et al., 1984). Here we describe the approach of Kolomeisky et al. (1998) and Popkov and Schiitz (1999) and show that the current obeys the extremal principles j
=
max
j (p) for PL > PR
(7.1 14)
j (p) for PL < PR.
(7.1 15)
pE[PR,PL]
j
-
min PE[PL,PRI
This implies that the phase diagram can be constructed from the knowledge of the macroscopic current-density relation. In particular, the structure of the phase diagram depends only on the number of extrema of the current. The microscopic details of the interaction matter only in so far as they may determine whether or not there are several maxima and minima. For systems with a single maximum in the current the phase diagram has the same structure and features as the ASEP, and only the location of the second-order transition in terms of the boundary densities and the shape of the first-order transition line depend on the precise form of the current-density relation. For systems with two (or more) maxima in the currentdensity relation a novel phase of rather unexpected nature appears: in a certain range of boundary densities the steady state carries the minimal current between the two maxima even though both boundary densities support a higher current. Below we shall refer to this phase as the minimal current phase.
7.4.1
Mechanisms for steady-state selection
The previous discussion has pointed to the importance of the shock motion in the exclusion process. It is natural to think of the shock as a domain wall separating stationary regions of different densities in analogy to a domain wall in equilibrium which is a localized region where the order parameter interpolates between degenerate ground states. We adopt a coarse-grained description of the domain wall motion which allows us to understand the behaviour of the finite system coupled to boundary reservoirs. Since we are studying a finite ergodic system, the stationary state is uniquely defined and the initial conditions are immaterial. Following the arguments of Kolomeisky et al. (1998) we find that there are two
1 Exactly solvable models for many-body systems
145
different types of domain walls whose structure and motion can be characterized to a large extent. The stationary states, the nature of the phase transitions and some aspects of the system dynamics can then be fully understood in terms of the domain wall picture. We first return to the usual TASEP where we only need to study the region ct >/~. From the exact solution we realize that understanding the origin and the physical meaning of the localization lengths ~,t~ holds the key to understanding the phase diagram of the system. We start from two examples that provide insight into the physics of the TASEP. (1) Let us assume (Derrida et al., 1995) that a and 13 are very small (ct L << 1,/~L << 1) and that the initial distribution of particles is far from the true stationary state. The particles will travel to the fight end where they get stuck. At late times there will be a low-density region at the left and a high-density region at the right (which we can present schematically as 00001111), with a domain wall between the low (0)- and high (l)-density segments. The subsequent late-stage evolution of the system can be interpreted in terms of the motion of this domain wall. The argument can be made rigorous by taking the bulk hopping rate to infinity using the infinite-rate formalism of Section 4.3. When a particle exits the system, the remaining particles rearrange themselves so that the whole filled region shrinks by one lattice unit, and the domain wall moves one step to the fight. The conditions a L << 1,/~L << 1 guarantee that while this rearrangement takes place, no extra particle enters or leaves the system. Similarly a particle entering the system from the left causes the domain wall to move one unit to the left. As in the Zel'dovich theory of kinetics of firstorder transitions (Lifshitz and Pitaevskii, 1981), the domain wall motion can be understood as diffusion of the 'size of the high-density segment'. The 'elementary processes' that change the length of the filled region consist of motion of the domain wall to the left at rate ct or to the right at rate t3, so that the domain wall does a biased random walk with drift velocity Vs - 13-ct and diffusion coefficient D = (a +/3)/2. As a result, three physically different situations are possible. If ct < /3, then the domain wall is drifting to the fight and will eventually reach the end of the system; thereafter the system is in the low-density stationary state. If a > fl, the domain wall is travelling to the left, leading to the highdensity stationary state. When a = /~, the domain wall position fluctuates with no net drift, and its r.m.s, displacement increases with time as x/-b-~. Hence, at large times it can be anywhere in the system, resulting in a linearly increasing stationary density profile as suggested above for finite rates ~,/~ on an intuitive basis. We note that following the dynamics of the wall also explains why the phase transition between the cases a > 13 and ct 1/2 and that only/~ is very small. We start from an empty lattice. After a while, but before the true high-density stationary state is reached, the system consists of two visually different segments which
146
G . M . Sch~z
can be presented schematically as mmmml 111. Near the left end of the system the high entering rate causes the formation of a region closely resembling the maximal-current phase (m), while on the right there is a high-density region (1) dominated by the small exit rate/~. The expansion of the high-density segment is again a biased random walk with some drift velocity and diffusion coefficient, determined below. We argued above that the domain wall picture exhibited in the two examples is not specific to the case that one of the boundary rates is small; everywhere in the low-/high-density phases there are two kinds of domain walls" the (011) wall (for c~ < 1/2) connecting the high-density stationary state to the low-density state (as in example 1), and the (m Il) wall (for a > 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)
1 Exactly solvable models for many-body systems
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-
1 Exactly solvable models for many-body systems
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 Exactly solvable models for many-body systems
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-
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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)
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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
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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 <_ x < L / 2 and x < y < L / 2 and any right eigenvector I A) of the transfer matrix. This simplifies the construction of the stationary state considerably: if in a state I Xl . . . . . XL ) one of the xi is odd, then it has a nonvanishing weight ~L (Xl . . . . . XL) only if all even x j with xi < x j <_ L are also contained in the set {Xl . . . . . XL }. The reason for the occurrence of this relation is the existence of an absorbing domain in state space. It easy to see that states with a particle on an odd site and a vacancy on an even site cannot be accessed in the course of the time evolution. Proper consideration of the role of such 'garden of Eden' states, a term coined by Schadschneider and Schreckenberg (1998) for the same phenomenon in traffic flow models, also leads to considerable improvement of mean-field approximations. (iii) The ASEP on a ring in the presence of a blockage, i.e., a bond in the lattice where particles hop with rate r # 1 has attracted considerable attention (Wolf and Tang, 1990; Janowsky and Lebowitz, 1992" Tang and Lyuksyutov, 1993; Janowsky and Lebowitz, 1994; Hinrichsen and Sandow, 1997; Janowsky and Lebowitz, 1997; Kolomeisky, 1998). Based on numerical results for the phase diagram and stationary density profile Janowsky and Lebowitz have suggested understanding the behaviour in terms of a bulk randomness and a boundary randomness. The exclusion process with deterministic sublattice bulk dynamics (hopping probability 1), but stochastic hopping across a defect bond (with probability p) has been solved exactly (Schlitz, 1993a). Both the stationary distribution and the Bethe ansatz equations for the spectrum of the transfer matrix have been derived. The stationary phase diagram is qualitatively the same as for the fully stochastic ASEP, thus confirming the picture of bulk- and boundary randomness. It is remarkable that the same exact expressions that describe the density profile and other quantities appear also in a fully stochastic continuous-time TASEP with a moving blockage (Lee et al., 1997), studied also in Mallick (1996).
1 Exactly solvable models for many-body systems
161
Section 7.4: (i) The extremal principles (7. 114), (7.115) for the current can be obtained by generalizing the phenomenological approach of Kn~g and Spohn (1991) to systems with several maxima in the current. To describe stationary flow in a regime with nonconstant density (caused by the coupling to the boundaries) one adds to the 'systematic' current given by the current-density relation a phenomenological diffusive current proportional to the local density gradient. Solving for the resulting stationary differential equation for the density with the proper boundary conditions yields the extremal principles and hence the correct phase diagram. Related arguments involving the anomalous scaling of the collective diffusion coefficient as well as renormalization group arguments (Krug and Spohn, 1991; Janssen and Oerding, 1996; Oerding and Janssen, 1998) predict the correct power-law decay of the density profile in the extremal-current phases. (ii) The stationary distributions of various simple traffic flow models have been calculated exactly using the stationary matrix product ansatz, thus providing detailed insight particularly in density profiles in open systems and in bulk correlations of various models (Chowdhury et al., 1999, 2000; Rajewsky et al., 1998, Evans et al., 1999; Klauck and Schadschneider, 1999).
G. M. SchQtz
162 8
Reaction-diffusion processes
We have seen already in earlier sections that reaction-diffusion systems have a wide range of applicability. Here we restrict our considerations to a prototypical class of systems, viz. one-species reaction-diffusion mechanisms with nearestneighbour interaction of the form (3.34). After a general survey of this class of models we focus in Section 9 on one-dimensional free fermion models which have received the most attention in the past. These models show much of the behaviour genetic for diffusion-limited reactions in one dimension. In this way contact can be made with actual measurements on polymer systems. Our main interest in the study of these processes is the dynamical behaviour. The stationary distribution for the general reaction--diffusion system (3.34) is not known. We merely point out that there is a submanifold in parameter space where the stationary distribution is a product measure (2.24) with some parameter dependent density p. This submanifold is identified by solving the equation hktl p ) = c(nk - nt)l P ) with an arbitrary constant c, provided that by taking the sum over all distinct pairs of sites (k, l) the local divergence terms nk nt add up to zero. Otherwise the submanifold is given by the solution of this equation with c = 0. In either case the submanifold is easy to identify, as this is only a set of four linear equations in the reaction- and hopping rates tOij. On another submanifold the stationary distribution of the one-dimensional model corresponds to the equilibrium distribution (3.3) of the one-dimensional classical Ising model. Also this manifold can be calculated explicitly by making an ansatz of the form (4.4) and finding transition rates Wij such that the action of the local transition matrix (3.34) on this vector reduces to diagonal divergence terms. These observations, however, should not be taken as an indication that correlations in this class of models are always short ranged. The contact process (see comment in Section 4.2) is a not exactly solvable counterexample.
8.1
Enantiodromy relations
One may apply a general factorized similarity transformation (5.4) and investigate the possibility of equivalences and enantiodromy relations. This is of interest not only to prove that there are equivalences, but also since the enantiodromic process may be much simpler than the original process. An example is the symmetric exclusion process where self-enantiodromy relates properties of N-particle systems to the same system with k < N particles. Here we discuss three enantiodromy relations for spin-relaxation processes involving the transformation 13 = e s§ (1) In one dimension the voter model (3.38) defined by the equal-rate
1 Exactly solvable models for many-body systems
163
transitions A0, 0A ~ (30, A A is identical to zero-temperature Glauber dynamics. Defining/-) = e s§ H Vme-S + one obtains for this process the enantiodromy relation 151T = H OLF, (8.1) with diffusion-limited fusion AO ~ OA, A A -+ AO, OA. This process is related by the similarity transformation (5.46) to diffusion-limited pair annihilation AO ~ OA, A A --+ 00 (Krebs et al., 1995; Simon, 1995). Hence in one dimension with nearest-neighbour interaction zerotemperature Glauber dynamics and diffusion-limited pair annihilation are not only equivalent by domain-wall duality (Section 3.3.4) (R~icz, 1985; Santos, 1997a) but also related by enantiodromy (Henkel et al., 1995; Schlitz, 1995a; Sudbury and Lloyd, 1995). (2) One may add to the voter model a diffusion term given by the isotropic Heisenberg Hamiltonian (3.26) (Droz et al., 1989; Krapivsky, 1992, 1996). In three dimensions this is then a simple biological toy model of growing tissue cell populations (Drasdo, 1996). The branching process describes cell mitosis, while the death process kills both the original cell and its offspring during the mitosis. The diffusion term represents diffusive motion of the cells in their environment. Specifically in one dimension, one may alternatively interpret these dynamics as a model for nonequilibrium spin-relaxation dynamics (Droz et al., 1989): spins relax according to the zero-temperature rules described by Glauber dynamics. In addition to that there is a coupling to an infinite-temperature heat bath, which causes spinexchange events according to Kawasaki spin-flip rules, i.e., as described by the Heisenberg chain. The question of interest is whether the disordering high-temperature dynamics or the ordering low-temperature dynamics win in the long run, i.e. whether the system approaches an ordered or a disordered state. Because of the self-enantiodromy of the symmetric exclusion process the enantiodromy relation (8.1) for this combined process reads /~r = HDLF + HSeP. The enantiodromic process remains DL F (Schiitz, 1995a; Sudbury and Lloyd, 1995), but with an increased hopping rate not tuned to the fusion rate. As a simple calculation shows, the spin correlation function which describes the ordering of the system is given by the diffusion-limited fusion process with just two particles in the initial state. It is then without further calculation immediately clear that in the infinite-time limit the system evolves into an ordered state. (3) This enantiodromy relation can be extended to the biased voter model with a branching rate A0, OA ---> A A different from the death rate A0, 0A --+
G. M. SchQtz
164
1313. In spin language this means that no creation of domain walls is permitted (zero-temperature dynamics), but spin flip events which increase the magnetization M = Y]k sk have a different rate from those that decrease M (due to a very small magnetic field). In the biological interpretation as a growth model this corresponds to independent death and birth rates. The enantiodromic process is diffusion-limited fusion A0 ~A, A A ~ AgJ, OA with a nonvanishing branching rate AO, 13A ~ A A (Sudbury and Lloyd, 1995). Through a suitably chosen factorized similarity transformation this process is equivalent to the pair-annihilation-creation process (3.39), but with no constraint on the hopping rate (Simon, 1995). So in one dimension the biased voter model (without extra diffusion term) describes zero-temperature Glauber dynamics in a weak magnetic field, while the pair-annihilation-creation process (3.39) is related by domain wall duality to Glauber dynamics in zero field, but at finite temperature (R~icz, 1985; Santos, 1997a). Combining these relations we arrive at the somewhat surprising conclusion that both Glauber processes are enantiodromic.
8.2
Decoupling of the equations of motion
After these general considerations of equivalences between various processes we consider how one can get information on their dynamical properties. We shall from now on phrase all results in particle language, even though some of our results include the dynamics of spin systems discussed in the previous section. The equations of motion for correlation functions are the differentialdifference equations (2.17) d dt ( nkl
... nkm ) =
--(
nk~
... nkm
H )
(8.2)
obtained from the master equation. Genetically, the r.h.s, of (8.2) contains correlation functions of higher and lower order than m, thus leading to an infinite hierarchy of equations. Fortunately, there are some exceptions from this rule. These fall into two classes, decoupling from higher-order correlations and decoupling of string-expectation values. In the case of decoupling from higher-order correlators the r.h.s, of (8.2) contains only correlators of equal (or lower) order. The general conditions for decoupling in processes of the form (3.34) was investigated in SchLitz (1995a). By inspecting the equations of motion it turns out that on the 10-parameter submanifold //324
--
1031 -~- 1/)41 q - 1/)13 q - 1/323 - - t 0 3 2 - - t 0 4 2 - - 1/)14
(8.3)
1/334
=
1/)21 +
(8.4)
1/341 d'- 1/312 -k- 1/)32 - - 1/)23 - - 1/343 - - //314
165
1 Exactly solvable models for many-body systems
of the 12-parameter space defined by the rates llOij there is coupling only to lower order correlators. This allows, in principle, for an iterative solution of the problem in terms of solutions of linear equations. This decoupling is independent of the underlying lattice and the range of interaction. Similar results for multi-species models were obtained in Fujii and Wadati (1997). It appears difficult to develop a qualitative understanding why such a decoupling should appear. Rather, from a physical viewpoint, it seems to be a coincidence without direct implications on the qualitative features of the dynamics. Mathematically this decoupling is of course rather fortunate. It can be understood by performing a factorized similarity transformation which at the same time provides information on the relaxational behaviour of the system. The transformation/-) = e s§ H e -s+ yields transformed transition matrices
hi,j
=
0 0 0 0
0 -bl b3 0
0 b2 -b4 0
0 0 0 -b5
+
0 al a2 a3
0 0 0 a4
0 0 0 a5
0 0 0 0
(8.5)
In the first part one recognizes the interaction matrix of a generalized particlenumber conserving anisotropic Heisenberg Hamiltonian (3.6). One may write H in the form 17-1= H x x z + H (8.6) where H - has the property that it connects sectors with particle number N to sectors with particle number N + 1 and N + 2, but not to sectors with a lower particle number. In a similar context it was noted by Alcaraz et al. (1994) that the spectrum of a Hamiltonian of this form does not depend on H - . Because of the particle number (S z) conservation of the Heisenberg Hamiltonian, the whole matrix H can be brought into a block-triangular structure. H x x z gives rise to block matrices defined by the quantum number N -- L / 2 - S z on the diagonal. In addition to that there are block matrices on the lower off-diagonal resulting from H - . The characteristic polynomial of/-) does not depend on these off-diagonal entries and therefore the characteristic polynomials (and hence the spectra) of the stochastic Hamiltonians H , / 4 and the Heisenberg Hamiltonian HXXZ are identical. Thus, with the reservations discussed in Section 4.2.5, one can read off the relaxation times of the 10-parameter process defined by (8.3) from the spectrum of the generalized Heisenberg Hamiltonian. Proceeding in a way analogous to the proof of the enantiodromy relations of the symmetric exclusion process one finds for the density relaxation ( nk(t) ) po
(k le-Jqtl0 )
-k- Z m
(nm (0))Po ( k le-/4/[ m )
(8.7)
G. M. SchQtz
166
and for the two-point correlation function
( nk(t)nj(t) ) po
(k, j ]e-~qtl 0)
+Z
(nm(O))t0(k, j le-Jqtl m )
m
"b y ~ l,mES
(nm(O)nl(O) ) Po (k, j
le-/qtl m , l ).
(8.8)
Generally, since/4 couples only to sectors with equal and higher particle number, a matrix element of the form (kl . . . . . km le-/4tl/l . . . . . IN ) with N > 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
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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)t-3/2e-(•
(8.26)
y ( l - Po) = 1"
Ap(t) -- A ( y ) t - l / Z e -(•
(8.27)
y ( l - P o ) > 1"
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 -
1 Exactly solvable models for many-body systems
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,
1 Exactly solvable models for many-body systems
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.
1 Exactlysolvable models for many-body systems 9
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.
1 Exactly solvable models for many-body 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
1 Exactly solvable models for many-body systems
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.
1 Exactly solvable models for many-body systems
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.
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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).
1 Exactly solvable models for many-body systems
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).
1 Exactly solvable models for many-body systems
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 .
1 Exactly solvable models for many-body systems
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)
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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
1 Exactly solvable models for many-body systems
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)
1 Exactly solvable models for many-body systems
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 _
-
<sl ~"
l+cot--~--bp,
b_p, (9.87)
+ <sl ~
The product and sum respectively over p, p' run over p = 1/2, 3/2 . . . . . (L 1)/2 (even sector), and over p' = 1,2 . . . . . L / 2 - 1 (odd sector). This in turn implies 1
E CO rrpT
=
P
c,,, c '2.
E
(9.88)
1
Hence
12N)
=
~1
(
Zcot(
rrp )Bpt -/--
)N
10)
(9.89)
p
Ip) even
=
2 1+(1 -
2p)LI-I( ( 1 - p
)2
+
p2
COt(7rp--L--)BPt) 10)
p
(9.90) The completely full lattice is simply given by IL) = H Bpf I0). p
(9.91)
G.M. SchQtz
202
The pair creation and annihilation operators the Pauli matrices, with
Bp and Bpt satisfy the algebra of
2Cp = [Bp, Btp] = b_pbt_p - btpbp playing the role of the cr~ matrix.
Ip =_ btpbp d- b-pbt_p
(9.92) commutes with all
Btp, Bp, Cp
and acts as unit operator on this subspace. It satisfies IpXp = Xplp = Xp f o r Xp = Bp, Btp, Cp. For different p all Xp commute with each other. These relations are easy to verify by using the anticommutation relations for btp and bp. It is useful to note that L
Z +ksk+' +s
=
2p~>oSin(-~)Bp
(9.93)
=
2~-"~sin(-~)Btp
(9.94)
k=l L
Y']s k sk+ l k =l
p>0
L
+ i sin (2zrp
=
(9.95)
L
k~=lSkS++, = p~>o[2COS (~-~-)Np - i sin (2rrp----~)(Ip - 1)]
(9.96)
L
--
2
k=l
Nr
(9.97)
p>0
where we have introduced the number operator Np = 1/2 - Cp. Consider now the subspace 1; generated by the action of the operators Bpt (9.86) on the vacuum state 10 ). This subspace is of interest for three reasons. Firstly, H can be written in terms of these operators as H -- Y~.p Hp with:
Hp=2D
l-cos
~
Np-2Dsin ~
Bp-irlsin
(/p-l).
(9.98) Therefore V is an invariant subspace of H. Secondly, a physically important class of initial conditions, namely random initial conditions, including the steady state and the fully occupied lattice, are in this subspace. Finally, some physically important expectation values are given by operators constructed from Bpt and Bp, e.g. the number operator (9.97). The previous considerations provide all the essential ingredients for proving the following important results.
1 Exactly solvable models for many-body systems
203
The subspace V of dimension 2 L/2 generated by Btp acting on the vacuum state 10) is an invariant subspace of H. On this subspace Hd = O, i.e., the driving has no effect on any correlation function if the system is at time t = 0 in an initial state which is contained in )2 (e.g. random initial conditions). That this is correct can be seen by observing that Ip is the unit operator on this subspace which gives lid = 0. As a result, the state at time t does not depend on 17which in turn implies that no correlation in that state can depend on the bias.
The time evolution of operators F build from operators Btp and Bp (e.g. the density operator) does not depend on the driving, irrespective of the initial condition. This is again obvious since Hd commutes with any such operator. Applying this result to powers N m of N gives the corollary:
The probability P(N; t) of finding precisely N particles at time t in the system does not, for any initial condition, depend on the bias. The same applies then of course also for the moments (N m (t)) of this distribution. The beauty of these results lies in the fact that they are derived by purely algebraically means, without any concrete calculation. But also concrete calculations in this subspace are very simple. Since the matrices Btp and Bp all commute among each other for different p and satisfy the relations of Pauli matrices for equal p, they may be represented by two-dimensional Pauli matrices. Hence the problem of calculating correlation functions in this system with 2 L/2 states reduces to exponentiating two-by-two matrices.*
9.5.5
Density and density fluctuations
Having clarified the role of the hopping asymmetry we consider from now on the undriven case r/ = 0. Throughout this subsection sums or products over p run over the set p = 1/2, 3/2 . . . . . (L - 1)/2 (even sector). Following the strategy employed in the preceding section the calculation of the total particle number and its fluctuations from a random initial condition becomes straightforward. One may use the representation
0
0
'
, (01 0) 0
Bp-
(9.99)
*From (9.94) it is clear that also the Hamiltonian with pair creation leaves V invariant. Hence the previous general results hold also in the presence of pair creation.
204
G.M. SchQtz
for the annihilation and creation operators acting in the subspace labelled by the momentum quantum number p. From this one calculates Np -- BtpBp and the matrix H r defined by (9.98) in this representation. Exponentiating Hp gives
e-npt - ( 1 c~ ~2(e1-2~pt - e-2E't)
.
(9100).
Both the vector (s[ even and the initial state [p)even factorize in the subspace V into tensor products of vectors ( s 'P leven-(l
'
cot 2p )
'
l ((l-p)2) [p)even = __ P Af p2 cot
(9.101)
where the p-independent normalization ./V" is given by (9.90) and p is used as a shorthand for 2zrp/L. For the practical calculation of expectation values it is convenient to use the decomposition 1 - - ( S i p ) even = .A/" 1 H ( ( 1 - p ' 2 + p2 cot 2 ~P )
(9.102)
p
and to introduce
p2 YP= p 2 + ( l - p )
2tan 2s
(9.103)
The behaviour of the total number of particles N as a function of time is determined by the cumulant function
C(a) = In ( e it~N )
=
In (s[evene2i~ ~P Nt'e- ~t, Hpt [p)even.
(9.104)
Using the factorization in V one finds C(a) = Z
In (1 - yp(1 - e2ia)e -2~t't)
(9.105)
p
for initial density p in the even sector. The behaviour of the cumulant function can be analysed for 1/p2 << t << L2 in terms of an expansion in l/~/t-. For times t >> l/p 2 only p << p make a leading contribution to the product and the dependence on the initial density becomes a subleading contribution in the time-dependence. Thus to leading order in time the cumulants become independent of the initial density. For very large systems and large times the sum in (9.105) can be replaced by an integral, ~ p L/(2zr) fo dp with y p , ~?p now considered as functions of the real variable p. Expanding the logarithm gives the asymptotic behaviour of the cumulant function as
C(c~) =
L oo (1 -- e2i~) m 4~/2rrDt = m3/2 .
(9.106)
205
1 Exactly solvable models for many-body systems
The cumulant expansion C(ot) - y'n~__o Xn(iOt) n / n ! then yields the cumulants
Xn
_
L 4~/zr D t
s
k" m . v
2n
(_1) k-l m--I
(9.107)
m3/2kV(m _ k) v .
k=l
"
"
From this one reads off the mean xl - ( N ) and the variance K2 -- ( N 2 ) - ( N )2 of the particle number distribution (N)
=
L
(9.108)
,/8~r Dt ( N 2 ) - ( N )2
=
(2 - ~/2)( N ).
(9.109)
For large times the density decays proportional to 1 / 4 7 (Bramson and Griffeath, 1980; Torney and McConnell, 1983; Toussaint and Wilczek, 1983). For initial density ,o = 1 and ,o = 1/2 one can obtain simple exact expressions for the cumulants for all times. For ,o = 1 and L ~ c~ differentiating (9.105) with respect to c~ at c~ = 0 yields (Lushnikov, 1987) (9.110)
( N ) = L e -4Dt l o ( 4 D t ) .
(Fig.
31)
and (NZ)-(N)
2 -- 2 L [ e -4Dr l o ( 4 D t ) - e -8Dt l o ( 8 D t ) ] .
(9.111)
The same strategy can be employed to calculate the two-time correlation function ( N ( t w + t ) N ( t w ) ) and show that the system ages (Murthy and Schlitz, 1998).
l
__
p(t)
'
0
5
10
Dt
15
Fig. 31 Density decay in DLPA from initial density 1. The exact expression (9.110) (full curve) and the asymptotic approximation (9.108) (dotted curve) which are both shown here are practically indistinguishable for Dt > 3.
G.M. SchQll:z
206
We explore now some of the consequences of these results for other free-fermion systems and for a special correlated initial distribution. Using the similarity transformation to the annihilation-fusion process, one finds for the system with both annihilation and fusion ( N ) b = (1 + b ) ( N ) o (5.6) where b = e a - 1 parametrizes the branching ratio of fusion and pair annihilation (Section 9.3). Thus also in this system the density of particles decays proportional to l/x/7. Renormalization group calculations (Lee, 1994) predict that the amplitude A = l/x/8rr/9 of the density decay is independent of both the initial density and the reaction rate in the case of short-range interactions. The independence on the initial density is confirmed by (9.108). One expects (9.108) then also to be the correct amplitude for the pure annihilation process away from the free-fermion point. Peliti (1985) has argued on the basis of RG-arguments that also the pure fusion process should be in the same universality class as the pure annihilation process, with a universal amplitude A'. The equivalence between the two processes confirms this claim and determines A ' / A = 1 + b for the mixed process (Krebs et al., 1995; Simon, 1995). The particle number fluctuations for the mixed system are given by ( N 2 )b = (1 + b)2( N 2 )0 - b(l + b)( N )0 in terms of quantities of the pure annihilation process. This yields
(N2)b
_
(N)6
(N) b 2
= (1 + b)(2 - v/2) - b.
(9.112)
as a measure for the branching ratio of an annihilation-fusion process. This quantity depends neither on the time scale set by the hopping rate of the particles nor (for sufficiently long times) on the initial density. Hence by measuring the particle number fluctuations one measures the branching ratio of a mixed process. An interesting question arises from the study of the decay of nearest neighbour spin-spin correlations S ( x , t ) = (1 - (cr~(t)tr~+l(t)))/2 in zerotemperature Glauber dynamics. At zero temperature the duality transformation of Section 9.3.3 turns this quantity into the particle density (nk(t)) of the pair annihilation process. Choosing uncorrelated random initial conditions in the pair annihilation process leads to the universal function (9.108) also for the correlator S of Glauber dynamics. However, choosing the same random initial state for Glauber dynamics (i.e., an initial state with random magnetization) and performing the duality transformation results in a density with a nonuniversal amplitude which does depend on the initial density (Family and Amar, 1991). The reason for this peculiar correspondence becomes clear in terms of the similarity relations discussed above. Duality transforms an uncorrelated state I P) into a state V I p) which has correlations. To see this, define a state
1 Exactly solvable models for many-body systems
I m) = ( I m + ) + } m _ ) ) / 2
where
L [1-t-m Im•
207
= l--I
2
lq:m +
k=l
2
x] a/,
105
(9.113)
are the uncorrelated initial states for Glauber dynamics with magnetization +m. The superposition I m ) is invariant under global spin flip (i.e. it is in the even sector of the process). The transformation V+ (9.52) transforms this state into the initial state proportional to l u ) = e ~ y]LI (akrakX+l--1)}0 )
(9.114)
with m = e x p ( - 2 a ) . Thus ( S I ( I - cr[(t)cr~+l(t))/2lm) - (Slnk(t)la) where the time evolution of the l.h.s, of this equation is under Glauber dynamics, whereas the r.h.s, evolves under annihilation dynamics. For the calculation of the initial density and density correlation one makes use of the factorization properties of lc~ ). One has ( S lexp [ct(tr/,x O'/rx - 1 ) ] = (SI and nt commutes with each factor exp [ct(crffo'/~+lX _ 1)] for l # k, k + 1. This yields the initial density in terms of m as P0 = (nk) - (1 - m2)/2. The two-point correlation functions are given by (nknk+]) -- (nk)2 = /90/2 -- p2 and (nknt) - p2 = 0 for Ik - 11 > 2. Remarkably, the initial density-density correlations extend only over one lattice site. It remains to calculate the expectation value p(t) = (nk(t)) of the density at time t > 0 for this initial state or to calculate the correlator S(t) for Glauber dynamics. This has been done first by Glauber (1963) by solving the differentialdifference equation resulting from the master equation for this correlator. However, up to taking the exponential of a 2 x 2 matrix, the result is implicit in the calculations performed above for the density. The initial state I ct ) has a simple expression in terms of the operators Bp, Btp and Np and is therefore contained in
Ap with Ap = 2(Bp+Btp) sin p+4Np cos p - 2 . As above p is a shorthand for 2rrp/L and
the subspace V. Using (9.93)-(9.96) gives )--]~=l (tr~tr/,x+l - 1) -- Y~p>0
the sum extends over p = 1/2, 3/2, (L - 1)/2. Hence Ic~ ) factorizes into states l a )p = exp (c~Ap)l 0) with normalization Zp = ( S I Ctp ) = exp (2a cos p). Calculation of the time-dependent density requires computing
(Np(t) ) = --
(S INpe-npte~ P0(l
+ COS
p)e -(~p+~-p)t
(9.115)
which follows from exponentiating the 2 x 2 matrix A p and using (9.100), (9.101). In the infinite volume limit the summation over the allowed values of p turns into an integral from 0 to rr and finally results in
p(t) = poe -4Dt (lo(4Dt) + I1 (4Dt))
(9.116)
G.M. Schfitz
208
The surprising content of this result is the observation that an initial distribution lot ), where the two-point density correlation function C(r) = ( n k n k + r ) -- p2 is nonzero only for microscopic lattice distances r = 0, -t- 1, leads to a nonuniversal, initial-density dependent amplitude for the time-dependent density expectation value p(t), whereas the uncorrelated random initial distribution I P ) (2.24) leads to a universal amplitude which does not depend on the initial density.
9.5.6
Local properties
The previous calculations give no information about local properties of the system. At present there are no experimental data of this nature available, but we argued above that an understanding of density correlations is of theoretical interest for the role of inefficient diffusive mixing in one-dimensional systems. First we study the time evolution of the local density P k ( t ) = (nk(t)) for arbitrary, not translationally invariant initial states. For the calculation of the local density one uses the Fourier decomposition (9.82) together with the solution (9.79), (9.80) for the time dependence of the Fourier components. It is helpful to note {sleven ( b ; + cot ( - ~ ) b _ p ) = 0 (9.117) which may be verified using the momentum space representation (9.87) of (s[ even. In particular this gives ( S levenbtp(t) = exp ( - e _ p t ) ( S [evenb; and thus .
pk(t) = ~ ~
!
e 2rr'x(p-p )/Le-(r
( b;bp, )
(9.118)
p,p' f
For factorized initial states it is convenient to reexpress ( bpbp, ) in terms of
fnx'-I
((l--nk)[lU=x+l(1-2nl)
( ctkck , ) -
] nk' )
(nk) -(nk,
t=x,+l(l--2nl) ] (1--nk)) In x-'
x < x' x --x'
(9.119)
x>x
Here we consider an inhomogeneous factorized random initial distribution in an infinite undriven system. At time t = 0 the negative half space is completely empty, while at all positive sites x >_ 0 the system is completely filled. In the course of time particles will diffuse into the negative half space, but this diffusion is suppressed by the annihilation process and leads to a nontrivial density profile with an amplitude decaying as l/x/7. Therefore we define the scaled profile pk(t) - 8 x / D t ( n k ( t ) ) and calculate the exact expression for the local density using (9.119) (Santos et al., 1996). In the scaling limit k ---> oo, t --> oo with
1 Exactly solvable models for many-body systems
209
the scaling variable u = k / x / 2 D t fixed one obtains the universal scaling function (Fig. 32) 2
~3(u) = x/~(l + erf(x/2u)) + e -u (1 - eft(u))
(9.120)
where the error function with negative argument is defined by e r f ( - u ) = - e f t ( u ) . The density profile is universal in the sense that it does not depend on the initial density and, presumably, not on possible short-ranged interactions in the system. The maximum of the diffusive current ( n k ( t ) ) - (nk+l (t)) moves into the negative region which was empty at t = 0 with a velocity ~ l/x/7. This is in contrast to a purely diffusive situation where under the same initial condition the maximum of the current remains at x - 0.
-
)
U
-2
0
2
Fig. 32 Exact scaled density profile fi(u) for DLPA in an infinite system with an initial step profile. Another question of interest is the decay of a localized perturbation in an initially uncorrelated and homogeneous environment. The calculation proceeds in analogous manner. The width of the perturbation increases diffusively, but the amplitude decays very fast cx t -e (Schiitz, 1996b). The similarity transformation to diffusion-limited annihilation yields similar behaviour for this process. In the presence of branching the finite energy gap manifests itself in a mixed algebraicexponential decay of the amplitude cx t - 2 e - t / r (de Oliveira, 1999). The broadening of the perturbation remains diffusive. Now we turn to the calculation of correlation functions. We recall that in the Smoluchowski approximation which leads to the correct exponent for the decay of the total density it is implicitly assumed that C(r, t) = (nknk+r(t) ) -- ( n k ( t ) )2 < 0. In particular, on length scales r is small compared to the diffusive length scale x/7 one expects almost never to find two particles at the same time, reflecting the existence of a depletion zone. Hence one should find C(r, t) -~ - p e ( t ) for
r_
210
G.M. Sch0tz
all these parts together yields after some calculation in the infinite volume limit L ---~ oo C(r,t)
1 fo 2n dpl fo 2n dp2 cot ( 2 )
=
4n.2 -
~
dpypeirp-2~p t
P2
tan (-~-) Fp2
eir(p2-pl)-2(Epl q-Ep2)t (9.121)
.
For times t >> 1/,o 2 the dependence on the initial density may again be neglected. For r 2 << t also the space dependence disappears to leading order in time and yields C(r, t) = -(8rr D t ) -1 + O ( t -3/2)
for r << x/r/
(9.122)
in accordance with what one expects from the Smoluchowski argument. For distances comparable to V~-, i.e. in the limit r, t ~ oo with u = r 2 / ( 8 D t ) fixed one finds the asymptotic scaling function (Alcaraz et al., 1994) e-tt C(u) -
8zrDt (e-u - v/-~--fferfc~)
(9.123)
which is indeed negative for all values of u.
9.5. 7
D o m a i n - w a l l duality revisited
The most striking property of the equations of motion for the density is that it is given by a description of just two site variables, the time evolution of the coordinates of the two annihilation operators ck, ct. These are the quantities which enter the time derivative of the density. This is implicit in the exact solution (9.82) or in (9.119).* If one imagines these coordinates to be the positions of two particles then their time evolution corresponds to diffusion with annihilation if both particles hop onto the same lattice site. Hence the stochastic dynamics of a system of just two particles reappears in the equations of motion for the density of the many particle system. This is the reason why properties of the density expectation value can be understood in terms of a two-particle picture as is done in the Smoluchowski approach. Mathematically this observation can be understood by exploiting domain wall duality and similarity. We have discussed above how the duality transformation can be expressed as an algebra automorpohism hk ~ hk+l of an associated affine *In the IDPF approach for the fusion process these are the two coordinates of the probability (nk(1 - n k + l ) . . . (1 -
nl_l)n ! ).
1 Exactly solvable models for many-body systems
211
Hecke algebra (Levy, 1991). For stochastic generators of the form (3.35) the transformation hk ---* hk+l reduces (up to boundary terms) to a lattice translation. However, the result of this operation is nontrivial for the one-species pair annihilation-creation process H ~ (3.39) which can be written in the form H = 2 Z [ ( 1 - h2j)(~. - (~. - D L ) h 2 j - I - (~. - D R ) h 2 j - 2 ) ] J
(9.124)
in terms of the Temperley-Lieb quotient (3.42), (3.43) of the Hecke algebra H M ( q ) with q + q - l ___ 1 (Santos, 1997a). t Applying this automorphism to DLPA yields an interesting result. Since the Hamiltonian for the pair annihilation process is real (as any stochastic Hamiltonian must be) and since the transformations ~ and V (9.53) of Section 9.3 are unitary one finds H = TC.V - l H T V ' ~ - 1 , (9.125) i.e. H is self-enantiodromic (SchLitz, 1997c). Exploiting this property for the calculation of the expectation value for the density and using the transformation laws (9.54), (9.55) for the ground states one finds for the even particle sector (Slnke-Htlll
.....
12k) = 1 -- (SI H ( I leQ
- - 2 n l ) e - H t l k , k + 1)
(9.126)
where the set Q = {ll + 1, l] + 2 . . . . 12; 13 + 1,13 h- 2 . . . . . /4; . . . " 12k-I . . . . . 12k} contains all intervals 12j-1 -t- 1,12j-1 d- 2 . . . . . 12j. Since the particle number only decreases in time, the two-particle dynamics is entirely sufficient to obtain the density of the full system with arbitrary initial state. Notice that the expectation value of the expansion of the product over the set Q in density operators n / t e r m i nates at the level of two-point correlation functions. Similarly, self-enantiodromy relates m-point correlators to 2m-point correlators of a system containing initially 2m particles. This relation holds also for systems with space-dependent hopping rates, e.g. in the presence of disorder or for reflecting or absorbing boundaries (SchLitz, 1997c" SchLitz and Mussawisade, 1998; Kafri and Richardson, 1999" Richardson and Kafri, 1999; Richardson and Cardy, 1999).
9.6 Open boundaries It is not known which constraints one obtains on boundary fields for free-fermion models with open boundary conditions. Such models are less well understood, particularly in the presence of drift. We have seen in Section 8.4 for the fusionbranching process that in nonperiodic systems with reflecting boundaries a driving tAs seen above, one may write the same process in the form (3.35). However,the corresponding matrices h k belong to a different quotient H M ( q ) , s e e (3.47).
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G . M . Schlitz
force has a significant impact even in the free-fermion case. It is therefore interesting to understand the role of an external force field in the non-translationally invariant pure fusion process. Hinrichsen et al. have calculated the exact stationary density profile of the biased fusion process with open boundary conditions (Hinrichsen et al., 1997). For this purpose it turns out to be convenient to consider the empty-interval probabilities S k , r ( t ) ~- ( (1 - nk)(l - n k + l ) . . . (1 ilk+r) ). They satisfy the bulk equations ( 8 . 1 1 ) , ( 8 . 1 2 ) f o r k # l , k + r # LwithA = C =q-1 B = D=q and E = 2(q + q - i ) where q is the hopping asymmetry q = ~ / D R / D L and the time scale is normalized such that D = ~/DRDL = 1. Particle injection at the boundaries results in separate equations. In the thermodynamic limit (semiinfinite system) the exact solution for the finite system can be represented in terms of an elliptic integral which allows for an asymptotic expansion of the stationary density profile for large distance k from the boundary. The decay of the density from the boundary depends on the bias. In the absence of bias one finds p~ c~ 1/k, in case of bias to the fight p~ cx 1/~/k" and with bias to the left, i.e. towards the injecting boundary, p~ cx q4k. Mean-field predicts similar asymptotic behaviour but different exponents. The correct exponents can be obtained from a Smoluchowski-type improved mean-field theory (Cheng et al., 1989). -
Comments Section 9:
Not surprisingly, there is a vast amount of literature on what we call freefermion systems and we refer the reader to Chapters 1, 2, 4, 5, 11, 20 and 21 in a recent collection of reviews (Privman, 1997) for an in-depth review of some of these models. Here we try to complete the picture that emerges from this work by discussing the advances obtained from the free-fermion representation of the process. The most comprehensive current survey of equivalences (Henkel et al., 1997) recovers most of the known relations between free-fermion systems, but there is neither a complete list with respect to factorized similarity transformations nor a general discussion of enantiodromy and domain-wall duality in the existing literature. Section 9.1" Free-fermion systems with alternating reaction-diffusion rates (= spin flip rates in the language of spin-relaxation models) have been considered e.g. in Droz et al. (1986) and Stinchcombe et al. (1998). One finds a nonuniversal dynamical exponent as in other, not exactly solvable spin-relaxation processes (Haake and Thol, 1980). Most of the methods discussed in this chapter can be extended with little modification to freefermion models with non-constant rates.
Section 9.2" The bilinear form of H does not seem to allow for systems with open boundary conditions with particle injection and absorption at boundary sites 1, L represented by boundary fields b i (3.31). However, such processes can be expressed in this
1 Exactly solvable models for many-body systems
213
bilinear form by the following trick (Peschel, personal communication 1996): one embeds the system in a larger chain with additional sites 0, L + 1 where panicles are created or destroyed with equal rate whenever an attempt is made to inject or absorb a particle at the original boundary sites 1, L. By putting the virtual boundary sites into the stationary state this modification does not change the original process taking place on sites 1. . . . . L. However, the off-diagonal part of the left boundary matrix of the modified process reads -u(s~- + So)S 1 -/~(s~- + So)S~-. The diagonal parts do not change. This new boundary matrix is bilinear in the creation-annihilation operators. The right boundary matrix is modified analogously.
Section 9.3: (i) A subtle issue left out of the discussion of equivalences is the question of boundary conditions. In Sections 9.3.1 and 9.3.2 we have considered only periodic boundary conditions even though the free-fermion Hamiltonian H f f (9.10) admits the possibility of antiperiodic boundary conditions. The reason was that under a transformation 13, antiperiodic boundary conditions would inevitably have led to a nonstochastic Hamiltonian. This is not true under the duality transformation. Depending on whether one considers the even sector with eigenvalue Q = 1 of H f f or the odd sector, the duality transformation D+ may change the periodicity properties of the transformed Hamiltonian. This is discussed in detail by Santos (1997a) and can be read off the transformation properties (9.53). Choosing as representatives of type I and type II processes the Hamiltonians HI'A for diffusion-limited creation-annihilation and n II'A for Glauber dynamics one finds the duality relations between these processes for the two sectors and boundary conditions. The nonstochastic, antiperiodic version of H t'A is related to the sector of Glauber dynamics with negative spin-flip parity C = - 1 , whereas the stochastic periodic sector is the dual of the positive spin-flip sector. This includes both periodic and antiperiodic boundary conditions for Glauber dynamics, which both give rise to well-defined stochastic processes. In this sense the duality transformation does give a new equivalence, since antiperiodic boundary conditions for Glauber dynamics cannot be obtained by a factorized transformation/3. (ii) The properties of the duality transformation have been derived on a purely algebraic level, using properties of affine Hecke algebras and of the representation (9.124) of zerotemperature Glauber dynamics in terms of the Temperley-Lieb matrices hj (9.46), (9.47). It would be interesting to investigate the possibility of stochastic representations of the generators h j which correspond to multi-species analogues of the pair annihilation process and its transformed equivalent stochastic processes. The duality transformation could then be applied to obtain further equivalences between multi-species processes. (iii) In the pure fusion process AA ~ AO, OA as described here we do not keep track of the number of fusion events each particle has experienced in the course of time. Viewing fusion as a coagulation event between sticky particles this number corresponds to the mass each particle has acquired up to time t. The probability of finding a particle of mass k has been studied and calculated in Kang and Redner (1984) and Spouge (1988). Section 9.5: (i) In the presence of pair creation the equations of motion for the Fourier modes of the fermion operators can be solved in an analogous way. The only difference is a term proportional to b t- p (t) in (9.78). This gives two coupled differential
214
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equations for btp(t) and b_p(t) which are solved by a canonical transformation, i.e. by introducing suitably chosen linear combinations dp = apbtp(t) + flpb_p(t) and dp = apbp(t) + ~pb-p(t) which are eigenstates of d/dt and which satisfy {dp, dq} = 3p,q. From these linear combinations one can then either reconstruct btp(t) and bp(t). The main -
t
physical difference a nonvanishing creation rate makes is the appearance of an energy gap which leads to an exponential decay of all correlations even in the infinite volume limit. (ii) The calculation of density correlation functions quickly becomes tedious with increasing order. A field-theoretic free-fermion approach (Santos et al., 1996; Santos, 1997b; Bares and Mobilia, 1999) involving Grassmann variables yields a Wick theorem for the zero-time correlators ( b pt ... bpk ). The contraction rules reduce these COl'relators to sums of products of two-point correlators (bpibpj). This simplifies the calculation considerably. For instance, for random initial states one has in the even sector ( bpibpj ) = Spi ,-pj sign(p/)YPi" Section 9.6: As discussed in Section 5.3 the cumulant function (9.104) for the (unbiased) fusion process with infinite injection rate at the boundary in a finite system gives the number of spins PL (q) that have never flipped under Glauber dynamics for the q-state Potts model. By exploiting the free-fermion properties of the system Derrida et al. (1996) found an exact expression for this quantity which decays in system size with a nontrivial power law PL(q) ~" L-2Oq. The exponent Oq = 2[1/rr arccos ( 2 - q)/(q~/-~)]2 _ 1/8 is reminiscent of the exponents of the two-dimensional equilibrium Potts model. It would be interesting to understand the link between these two rather different sets of problems. Nontrivial algebraic behaviour appears also in the context of domain statistics (Krapivsky and Ben-Naim, 1997) and in the survival probabilities of tagged particles (Frachebourg et al., 1998). In principle it should be possible to compute these exponents exactly, but so far 0nly numerical and heuristic results are available.
1 Exactlysolvable models for many-body systems 10
215
Experimental realizations of integrable reaction-diffusion systems
As a conclusion of our account of integrable stochastic processes we return to some of the experimental realizations outlined in the introduction, the motion of entangled polymers (Section 10.1), then the kinetics of biopolymerization on nucleic acid templates (Section 10.2) and finally the exciton dynamics on TMMC chains (Section 10.3).
10.1 Dynamics of entangled DNA 10.1.1
Gel electrophoresis
A widely used and simple method for the separation of DNA fragments is gel electrophoresis in a constant electric field. The DNA mixture to be separated is introduced into a gel matrix. Since the DNA is charged, it will move in a constant electric field E with velocity o(E, L) where L is the length of the fragment. After some time fragments of different length will have travelled different distances in the gel and can therefore be separated. Clearly it is desirable to have a quantitative understanding of the motion of DNA in gels. For small field strength the drift velocity is given by the NernstEinstein relation v = D F / ( k T ) where F cx E L is the force acting on the polymer chain. Since reptation theory predicts D cx 1/L 2 one obtains a length-dependent drift velocity v cx E / L . For sufficiently large field E or polymer length L experiments show a phenomenon called 'band collapse' - all polymer fragments travel with the same velocity, irrespective of their length. No separation is possible in this regime. Rubinstein (1987) and Duke (1989) introduced a simple model for the motion of a polymer in a gel matrix. In this model the gel is idealized by a regular cubic lattice where the cells are the pores of the gel through which the polymer reptates. The polymer itself is represented by a string of reptons which represent units of stored length. These reptons hop stochastically from pore to pore according to rules based on the mechanism of reptation and assuming local detailed balance. Since in electrophoresis only the average velocity of the centre of mass in field direction is of interest, one can project the motion of the reptons onto this direction. The motion perpendicular to the field is diffusive with a diffusion constant (Widom et al., 1991; van Leeuwen and Kooiman, 1992; Pr~ihofer and Spohn, 1996) D = l / ( 3 d L 2) in d dimensions. This quantity can be calculated in this model using the Einstein relation (van Leeuwen, 1991; Widom et al., 1991; Kolomeisky and Widom, 1996). Some mappings that we do not describe here lead to a lattice gas model
216
G . M . Sch~tz
representing the relative motion of all reptons in field direction. In this model there are two kinds of particles, A and B, moving on a lattice of L sites and each site can be occupied by at most one particle, A or B. A-particles hop to fight (left) with rate q ( q - l ) if the site is unoccupied. Here In(q) is the energy gain when a repton moves into a pore in field direction. On site 1 of the chain A-particles are created (annihilated) with rate q (q-l), while on site L they are annihilated (created) with rate q (q-I). For the B-particles the same rules hold, but with q and q - I interchanged. The average drift velocity v(E, L) of the DNA is the difference between the stationary current j A ( E , L) - jB (E, L) of A particles and B particles. The stationary distribution of the system with q ~ 1 (E ~ 0) is not known except in the periodic system (van Leeuwen and Kooiman, 1992) which does not have an interpretation in terms of polymers moving through a gel. However, extensive Monte Carlo studies (Barkema et al., 1994) have provided a reliable knowledge of v(E, L) in the framework of the model. The surprise is that these results are in excellent agreement with experimental data (Barkema et al., 1996; Barkema and Schlitz, 1996). This gives confidence that despite all its simplifications, the Rubinstein-Duke model captures the essential physical processes involved and allows for reliable predictions in real gel electrophoresis. The stochastic time evolution of the system is given by the Hamiltonian of a three-state quantum chain (Barkema and Schiitz, 1996)
L-I
ui(q)
H(a, q) = bl (c~, q) + bL(ot, q -1) %-Z
(10.1)
i=1 where bi (or, q) - c~q(1 - n a - a +
-
b i ) %-
o~q-I (1
-
n B
-
ai
-
b +) and Ui (q) =
q(nAnO+l" %-nOniB+l--aia-~-+l--b?bi+l)%-q-l(nOnA+1%-nBi ni+lO _ a ? a i + l _ b i b ? + l ). Here n A -- E~ l, n/B = E/33 and n O - - l - n A -n/B = E~ 2 are projection operators on states with an A-particle, vacancy and B-particle respectively on site i. The operators ai = E 21, a + ---- E~ 2, b i E2i 3, and b + - E 32 are annihilation and
creation operators for A- and B-particles. E/k is the 3 • 3 matrix with matrix elements (E/k)a,~ -- 3j,a3k.f acting on site i. The factor c~ takes into account the possibility of a different mobility of the end-reptons compared to those in the bulk. Nothing is known about the integrability of the model in a nonzero field. In a zero field (q : l) the model has an integrable subspace with a spectrum which is identical to that of the isotropic Heisenberg chain with nondiagonal, symmetrybreaking boundary fields. This can be shown by using a similarity transformation on H and projecting on one of its invariant subspaces.
1 Exactly solvable models for many-body systems
10.1.2
217
Relaxation of stretched DNA
The theory of reptation has become a basic concept in the understanding of static and dynamical properties of entangled polymers in concentrated polymer solutions or networks (Doi and Edwards, 1986). While traditionally quantities such as the diffusion coefficient, the viscosity or the drift velocity in the presence of an electric field have been measured as averages over ensembles of many polymers, more recently the investigation of single entangled polymer chains has become possible (K~is et al., 1994; Perkins et al., 1994) and thus a direct experimental verification of some of the key assumptions of reptation theory could be achieved. Nevertheless, despite considerable experimental and theoretical effort there is still a number of rather perplexing open problems including the length dependence of the viscosity, the influence of disorder, and dynamical properties of reptating polymer chains. In order to obtain a clear picture of the role of the many different factors determining the behaviour of entangled polymers it is tempting to try to address some of these questions by using simple model systems like the Rubinstein model which try to capture the essential mechanisms of reptation and then to add to such models more refined details if experimental evidence is not consistent with predictions from the model. Here we consider the dynamics of the tube using the lattice gas model described in the introduction, i.e. the symmetric exclusion process with open boundaries studied in Section 6. In quantum spin language such boundary conditions correspond to boundary fields. Of course, one cannot keep track of the actual position of the DNA within the gel by using this model, only the mass distribution of the polymer within the tube and the tube length are accounted for by the particle distribution along the chain. Also dynamical properties which are typical for time scales less than r cannot be described with this model, but this is not the purpose of the investigation of the tube dynamics. For a polymer of length M the number of lattice sites L = M / 6 corresponds to the individual polymer segments (reptons) of unit length 3 which we identify with the mean entanglement distance (pore size) of the network. In equilibrium the polymer is not fully stretched but a certain amount of excess mass is stored inside the tube, hence giving rise to a fluctuating equilibrium tube length A* less than the actual polymer length M. Even though this tube is a rather theoretical construct, the implied tube diameter (related to the mean entanglement distance 3) has been measured in neutron scattering experiments (Richter et al., 1990). Indeed, the tube may be seen as a coarse-grained polymer contour and hence be identified with the experimentally observable visual contour of fluorescencemarked polymer chains. Following Sch/itz (1999) we define the hopping time r0 which sets the scale for the elementary diffusion processes within the tube as the mean first passage time of the motion of one repton to a neighbouring pore of the tube. We stress
218
G.M. SchOtz
that in this description of the dynamics along the tube backbone a 'hopping event' is defined as the first passage of a repton to its neighbouring pore along the tube, independently of what happened between two consecutive hopping events. This eliminates the unphysical requirement that bulk polymer units may not (temporarily) move transverse to the tube direction by developing microscopic, transient hernias. To describe the interaction of the polymer with the network and with itself we impose the weaker constraint that the consecutively labelled reptons may not pass each other within the tube. End reptons may move freely to new pores but are of course not allowed to detach from the polymer. In this mapping the tube length is given by A = N& where N is the total number of particles on the chain. The end-point dynamics of the tube correspond to injection and absorption of particles at the boundaries as if the particle system was connected to a reservoir of fixed density p* = ( N )*/L = A * / M where particles are injected with rate a = p*/r0 and absorbed with rate y = (1 p*)/r0. In the experimental setup discussed below one end of the chain is kept fixed in the network. This is described as one reflecting boundary with no particle exchange. An initially fully stretched polymer chain corresponds to an initially fully occupied lattice. The dynamical particle number N (t) yields the tube length at time t. It is convenient to consider the experimentally measurable relaxation function A(/) - A* (N(t))-(N)* R(t) _----(10.2) A0-A* (N)0- (N)* which depends only on L and r0. The strategy of Section 6.5.3 yields the density dynamics for the symmetric exclusion process. For a homogeneously stretched polymer of averaged initial tube length A0 -- ( No )& one obtains the exact expression L-I
R(t) = L(2L + 1)
cot 2 ~
e -~'//r~
(10.3)
with Pn = rr (2n+ I)/(2L + 1) and the inverse relaxation times En - 2(1 - c o s Pn). For early times in the range r0 << t << t* with the crossover time t* defined by R(t*) = 1/2 one extracts from (10.3) the universal relaxation law (Schiitz, 1999)
R(t)- 1
7t.3/2
(10.4)
where the Rouse relaxation time r = 4M2r0/(zr2& 2) is the only fitting parameter. It is interesting to note that using continuum reptation theory (Doi, 1980) one obtains the L ---> o0 limit (with M = L& fixed) of (10.3). We arrive at the conclusion that the symmetric exclusion process represents an exactly solvable lattice gas model for reptation. Given the
1 Exactly solvable models for many-body systems
219
rather different salient features of the two approaches this agreement is both surprising and encouraging as it indicates a certain robustness of predictions of reptation theory with respect to the precise realization of the process. The relaxation of an initially stretched entangled DNA has been measured by the observation of the visual length of a single fluorescence-marked DNA strand moving in a dense, monodisperse solution of shorter, unmarked DNA (Perkins et al., 1994). The DNA was prepared in a stretched initial conformation by attaching a polystyrene bead to one end and pulling it with optical tweezers through the solution which provides the entanglement network. After some time the bead is stopped and the relaxation of the attached DNA is recorded with a video camera. Comparison of the theoretical prediction (10.4) to the data presented in Fig. 4 of Perkins et al. (1994) gives experimental confirmation of the relaxation law. Figure 33 shows good agreement between the predicted initial power-law behaviour of the relaxation function (10.4) and the experimental data of Perkins et al. T h e only adjustable parameter is the combination (A0 A*)/~/r- ~ 5 . 9 ( 1 ) # m / ~ .
0.5
1
1.5
2
Fig. 33 Relaxation of the visual length of an initially stretched DNA. The full curve shows the power law increase of AO - A ( t ) (in #m) as a function of time (in s) as predicted from (10.4). The dots are experimental data taken from Fig. 4 of Perkins et al. (1994) in the range to = 0.6 s < t < 2.8 s. The time is measured relative to to where reptation sets in. (From Sch/itz, 1999.)
220
10.1.3
G.M. SchQtz
Other b o u n d a r y conditions
If both ends of the polymer are fixed in the network, then H (10.1) for the Rubinstein-Duke model reduces to tracer diffusion (Sections 3.3.1 and 3.4) with reflecting boundaries. This quantum chain is integrable (Alcaraz and Rittenberg, 1993) as can be seen by verifying that the hopping matrices for both the tagged particle process and the usual symmetric exclusion process satisfy the same Temperley-Lieb algebra h 2 = 2hk, h k h t + l h k = hk, [hk,ht] - 0 for Ik - II >_ 2. In this case the model describes the internal random fluctuations of the polymer within the fixed tube. Using the Bethe ansatz one can compute the relaxation of the DNA to equilibrium where each configuration is equally probable. There are other boundary conditions that are interesting. Remarkably the model predicts that there is no band collapse if one pulls only at one end of the polymer rather than at each repton (Barkema and Schlitz, 1996). The velocity is then always length dependent and asymptotically given by the exact expressions v = E / 3 N 2 for E --, 0 and v = 1 / ( 3 N - 5) for E ---, oo. In this case H consists of the integrable zero-field bulk part and boundary terms bl (1, q) + bL(1, 1). Band collapse disappears also if a force acts at a polymer segment in the bulk (Richardson and Schlitz, 1997). Also the opposite situation where the motion of the polymer is hindered at one of its ends has been investigated, both experimentally (Ulanovsky et al., 1990) and theoretically (Aalberts, 1995). In the case of two very slow end reptons spontaneous symmetry breaking in the orientation of the polymer occurs in the presence of a field (Aalberts and Leeuwen, 1996). Whether any of these variants of the boundary conditions leave the model integrable is not clear. It would be interesting to ask generally which integrable boundary conditions one can obtain for the model with vanishing bulk field (Sklyanin, 1988).
10.2 Kinetics of biopolymerization In the process of biopolymerization the particles are the ribosomes, moving along the chain of codons which is the messenger RNA. In the modelling by the ASEP one should properly describe the process in terms of particles of size r ~ 2 0 . . . 30 in lattice units. Even though this model is integrable (at least with periodic boundaries) (Alcaraz and Bariev, 1999) and has a product measure as stationary distribution there is no exact solution for the relevant case of open boundaries (see Introduction). In the idealized case r = 1 corresponding to small ribosomes of similar size as the codons the stationary state was first studied using a meanfield approach (MacDonald et al., 1968). In a following paper (MacDonald and Gibbs, 1969) the generalized case r > 1 was studied numerically. It turned out
1 Exactly solvable models for many-body systems
221
that the phase diagram for general r is similar in the sense that there are three distinct phases, a low-density phase, a high-density phase and a maximal-current phase. These observations are not very surprising in view of the general theory developed in Section 7 since the current--density relation j (p) = p r ( l - p) has a single maximum just as the usual TASEE This feature encourages us to use the TASEP as a simple model for this biological system. Indeed, experimental data (Dintzis, 1961; Winslow and Ingram, 1966) from which one can deduce the stationary density distribution of ribosomes along the chain were found to be consistent with the mean-field results obtained from the model with q = 0 and ot --/~ < p / 2 . This is the first-order phase transition line from the low-density phase to the high-density phase where both the mean-field analysis and numerical calculations predict a region of low density of ribosomes from the beginning of the chain up to the shock where the density jumps to the high-density value. '~ In the light of the exact solution discussed in Section 7.3 which came only more than 20 years after the work by MacDonald et al. and independently of it this interpretation of the experimental data is in need of re-evaluation. The exact solution reproduces the three phases predicted by mean field, but gives a linearly increasing density profile rather than the sharp step predicted by mean field. Above we argued that this exact shape of the stationary density profile can be explained by assuming that at any given moment a sharp shock exists, but, due to current fluctuations, performs a random walk along the lattice. Therefore, if one waits long enough, the shock will have visited each lattice site with equal probability. What one therefore expects for an experimental sample is indeed a region of low density of ribosomes followed by a sharp transition to a region of high density of ribosomes as found experimentally. However, this point of rapid increase can be anywhere along the m-RNA with a probability distribution given by the effective initialization and release rates c~, ft. If c~ = /3 the distribution of shock position would be constant over the lattice. This provides a new insight into the behaviour of the position of the change of specific activity in so far as there should not be a well-defined point where the specific activity changes, but (on average) a gradual slowing down. To verify this picture one has to inspect the experimental data (Dintzis, 1961; Winslow and Ingram, 1966) considered by MacDonald et al. Indeed, the data points are too few and have too strong fluctuations to definitely point to a step function profile. One realizes that the hypothetic activity curve derived from a linearly increasing density profile is also consistent with the experimental data *This description of the stationary mean-field density profile describes correctly the situation for r = 1, but disregards a morecomplicatedsublattice structure for r > 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 <_ x <_ L / 2 -
1)
(A.12)
a straightforward calculation yields (x -7r L / 2 ) :
T, n2x-I ]
= -
[T, n2x ]
T (n2x-I - (1 - v2x-2V2x-l)n2xn2x+l) T .odd _ :even (A.13)
=
T
:
T
( 1)2x- 2 U2x- 1 ( / :even .odd
1
~J2x
1) 9
-- J2x-
- - n 2x n 2x + 1 ) -
1)2x )
Exactly solvable models for many-body systems
229
There is no proof that the system is integrable even though, given the factorized structure of the boundary matrix TL,I, this seems likely. In order to prove integrability one has to adapt the approach by Sklyanin (1988) to the diagonal-todiagonal transfer matrix.
230
G.M. SchQtz
Appendix B: Universality of interface fluctuations Among the motivations behind the study of the one-dimensional simple exclusion process is its relevance to the behaviour of interface fluctuations in twodimensional systems. A typical example of interest is the existence of large spatial fluctuations of the interface location in the magnetization profile between coexistent phases in models of subcritical uniaxial ferromagnets (and their analogues). Phenomenologically speaking, these fluctuations are generated by capillary waves described in a continuum theory of an interface which is a sharp dividing surface between oppositely magnetized phases, controlled by a somewhat arbitrary shortrange spatial cut-off (Buff et al., 1965). Against this, one has to set the density functional theories going back to van der Waals and Maxwell (Fisk and Widom, 1969) which do not give divergent spatial interface fluctuations. It is fortunate that both exact and rigorous results are available for the planar Ising model which resolve the conflict in favour of the large spatial fluctuations (Abraham and Reed, 1974), but which do not resolve the problem of local interface structure. The conclusions of the exact calculation were recaptured in a Helmholtz fluctuation theory for the phase separating surface (Fisher et al., 1982) which brings in the concept of interfacial stiffness and which is valid in a spatially coarse-grained sense, much as in Widom scaling theory (Widom, 1965). These fluctuations are an essential ingredient in the statistical mechanics of a number of surface phase transition phenomena (Fisher, 1984). The availability of exactly solvable interface models allows one to go beyond the equilibrium description and to attempt to understand the dynamical behaviour and local characteristics of the interface such as the roughness given by its height difference fluctuations ( (h(y, t) - h(x, t) 2 ). First, we recall an equilibrium result which helps motivate the model described below and an earlier phenomenological treatment. Suppose in a zerofield planar lsing ferromagnetic model, the interface between coexistent phases is established and localized in laboratory-fixed axes by specifying the boundary spins as shown in Fig. 37. The straight line of length L = 2N connecting the spin-flip points is the Wulff shape for the interface. It is convenient to define coordinates parallel and perpendicular to this line, with origin at its centre. Let the thermodynamic limit of infinite strip length be taken first. Then, denoting the magnetization at (x, y) by re(x, y/N), Abraham and Reed (1974) and Abraham and Upton (1988) have shown (for - 1 < ct < 1) rh(ct, 13) = ~v~oolimm(aN, fiN~/N) = m*erf
b(O, T)fi ) x/1 - ct2
( 3 - 1/2)
(B.I)
where m* is the spontaneous magnetization, and b(O, T) = or(O) + cr"(O) is the surface stiffness. (Here tr(O) is the angle-dependent surface tension.) On the other hand, one has rh(a, 13) = 0 (m'sign(13)) for 6 < 1/2 (> 1/2).
Exactly solvable models for many-body systems
231
+
Fig. 37 Two-phase Ising system on a strip at zero temperature. The interface (straight line of length 2N) separates a domain of positive magnetization (bottom) from a domain of negative magnetization (top).
In order to generalize to the dynamical case one needs to define a model with dynamics appropriate to the physical system described above. To motivate our model, which goes back to a mapping between interface dynamics and the exclusion process introduced independently by Meakin et al. (1986) and Plischke et al. (1987), note that as T --+ O, b ( O ) > 0 (strictly) provided 0 g: 0, • f o r O = 0 , + z r / 2 , b ( O , T ) ~ O a s T ~ 0. This is a primitive example of faceting. At T = 0, the Peierls contours reduce to paths connecting ( - N , 0) to (N, 2N tan O) which either step to the fight or upwards (tan O > 0); all such paths are degenerate. They separate regions of opposite magnetization; since T = 0, this magnetization is of unit magnitude. Let us now concentrate on the case tan 0 = 1, with an initial sawtooth configuration as shown in Fig. 38. For later convenience we have rotated the system by 45 ~ and we shall consider periodic boundary conditions along the rotated axis. One does not expect this simplification to be of physical relevance for the bulk properties of the system. At T = 0, any minimum energy path can be represented as a sequence S = {n-/v+l,n-N+2 . . . . . n N } of 2N binary numbers nk where nk = 0 if the kth segment of the interface steps upwards (in an angle of 45 ~ and nt, = 1 if the steps go downwards (Fig. 39). One can think of nk as an occupation number which is related to the interface height hk in the rotated system by 1 - 2ni, = hk - h k - l . The configuration at any time t is then given as a time-slice {m,(t) : k = - N + 1 .....
N}.
232
G. M. SchQtz
r/ I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0
t
~
t
0
t
~
t
0
t
=
:
0
t
~
t
Fig. 38 Microscopic initial state corresponding to a fiat interface and corresponding particle occupation in the associated lattice gas model.
h-5
h=O
I I I I I I I I I I ~--o--+--o--+--o '
9 ' 0
I I I I I : =
I I I I I -- : 0
Fig. 39 The mapping between the restricted interface and the particle exclusion process" we show a possible interface configuration and the corresponding particle occupancies on a lattice with sites labelled by k. The indicated flips in the interface correspond to particles hopping on the lattice, marked by horizontal arrows. (From SchLitz, 1997a.)
As dynamical rules for the time evolution of the system we allow for independent spin-flip events along the boundary of the system which do not increase the energy of the system. This translates into particle-hole exchange on neighbouring lattice sites for which we choose a rate of occurrence D -- 1/2. In order to retain
Exactly solvable models for many-body systems
233
connectivity of the interface, we leave pairs of particles or holes unchanged. This relates the interface model to the simple symmetric exclusion process. Hopping of a particle to the right corresponds to growth of the surface height at a local minimum, while hopping to the left corresponds to shrinking of the height at a local maximum (see Fig. 39). The initial state is the N-particle state Iv) = (I 1,0, 1,0 . . . . ) + 1 0 , 1,0, 1. . . . ))/2.
(B.2)
The displacement of the interface at (x, t) relative to the arbitrarily chosen reference point ( - N, t) is thus Ah(x,t) =
)_~
(1 -- 2nk(t))
(B.3)
k=-N+l
with zero mean by spin-reversal symmetry, but with fluctuations x
W2(x, t) -- ( ( A h ( x , t ) ) 2 )v = 4
Z
( n k ( t ) n t ( t ) )v - (x + N) 2.
(B.4)
k,l=-N+l
Using the duality relations (6.4) one finds M
w2(x,t)=2
Z P(k,l'tlp, k,l=-N+l p.q---N+l
q'O)+2(x+M)-(x+M)
2, (B.5)
p-qeven
which reduces the calculation of the interface fluctuations to the solution of a two-particle problem. We study first the infinite-time limit, thus deriving the static magnetization m ( x , y) = 1 - 2(| - y)) where | is the Heaviside step function. If the model is to be consistent with the physical situation described above, one should recover (B. 1) in the scaling limit x = c~N and y = / 3 N ~/ N. The (normalized) Nparticle stationary state of the system is I N* ) (Section 3.1.1). A short calculation then yields y-1
m(x,y)
= sign(y)
Z k=-y+ 1
Bin(N-x,
N-x --~
k) Bin
(N + x , - ~ + k)
Bin(2N, N) (B.6)
with the binomial coefficient Bin(K, M ) -- K ! / ( M ! ( K - M)!). In the scaling limit x -- c~N and y = /3N ~/N one obtains again the scaling form (B.1) with surface stiffness b = 1 and m* -- 1. Thus the dynamics of the model indeed lead to the desired equilibrium magnetization.
234
G.M. SchQll:z
Now we turn to the dynamics. For the calculation of w 2(x, t) one observes that because of translational invariance only two-particle states 1
N
~N +l S~S++r]O) (1 < r < N ) 1 0 ' r ) = 2~/2-Nk=-
(B.7)
of total momentum 0 contribute to the exact expression of the two-particle conditional probability. This reduces the full solution to a diffusion problem involving only the relative coordinate r. Note that Ir), [ - r) and Ir + 2N) are identical and (rlr) = l + 6r, N. With these conventions one can write x-l
N-l
w2(x, t) = 2 ~
y ~ (N + x - r)(O, rle -Ht lO, 2R) + 2(x + N) - (x + g ) 2 R--l
r---N+l
(B.8) The matrix elements cR(r, t) -- (0, rl exp ( - n t ) [ 0 , R) satisfy the differentialdifference equation
CR(r,t)
--
cR(r + l , t ) + c t c ( r - - l , t ) - - 2 c R ( r , t )
kR(r, t)
=
cR(r + l, t) -- cR(r, t)
(r > 1)
(r=l)
(B.9)
with initial condition cR(r, 0) = 6r, R(l -q- 6 R , N ) . Taking into account the periodicity and reflection properties of the states 10, r) one finds oo
cR(r, t) -- e -2t
~
(Ir-R+(2N-l)m(2t) -k- Ir+R-l+(2N-l)m(2t)).
(B.10)
m=-cx~
and we finally obtain with M = 2N - 1 (Abraham et al., 1994) w2(x, t) = ~ 4 ~ - ~ ( 1 - ( k=l
l)tC~ cos [Jrk/M]
l-e-2(l-c~ 1 - cos [2rrk/M] (B.I 1)
We have neglected here an irrelevant quantity e (x) = 0 (1) for x even (odd). It is instructive to compare this exact result with that obtained from a phenomenological approach based on a Langevin description where the dynamics of the interface are assumed to be described by the following additive noise Langevin equation 1
Othx -- -~(hx-i -k- hx+l - 2hx) + rig(t)
(B.12)
where fix(t) is the delta-function correlated Gaussian white noise with disorder average (rlx(t)) = 0 and (rlx(t)lTy(S)) - - S x , y ~ ( S -- t ) . Integrating the
Exactly solvable models for many-body systems
235
inhomogeneous equation (B. 12) gives
hx(t)
--
[
Z Z hy(O)e - t l x - y + 2 m N ( t ) km=-cX~ y = - N + l d r r l y ( r ) e r - t I x _ y + 2 m N ( t --
f0t
r)]
(B.13)
For the initial condition h y(O) = 0 which corresponds to the sawtooth configuration of the microscopic lattice model one finds m !
2(x, t) = ~-7 1 - ( 1 cos[2rrkx/M'] 1 -- e -2(l-c~ 4 k~l( __)k ) = 1 -cos[2rrk/M'] (B.14) with M' = 2N. The similarity between the two expressions (B.II) and (B.14) is striking, considering the vast simplifications inherent in the phenomenological Langevin model. In the scaling limit x, t, N ~ c~ with c~ _= x / N and u =_ Jrt/N 2 fixed both expressions are identical. The exact results confirm scaling arguments which predict for an interface in the Edwards-Wilkinson universality class (B.12) a scaling form wZ(x, t) = NCb(x/N, t / N 2) (Krug and Spohn, 1991" Halpin-Healey and Zhang, 1995). We find the scaling function O0
1 - e -zruk2
(B.15) k=l
For arbitrary c~ and u --+ c~ we have the following asymptotic forms of the scaling function" 1 - c~2)_ ~-~(1 2 9 (a, u) = ~(1 + cosrrc~ )e_~r u (B.16) and for c~ = 0 and u << 1 we have 9 (0, u) -- ~ .
(B.17)
In the first limit corresponding to t >> 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 << t << N 2. Having shown that in the scaling limit both the static magnetization m(x, y) and wZ(x, t) coincide, suggests that the simplified Langevin dynamics with additive noise represent a qualitatively and quantitatively adequate approach to this problem in large but finite systems in the scaling region. To actually show this one
236
G . M . SchQtz
has to derive a full dynamical version of the magnetization (B.6) for our model, as was done by Abraham and Upton (1989) for the Langevin dynamics with additive noise. To this end, one has to calculate all higher even moments of the height fluctuations. (The odd moments all vanish due to spin reversal symmetry.) Since, to leading order in time, finitely many identical exclusion particles behave like noninteracting identical particles and since the moments of the height distribution of order 2k are determined by the behaviour of 2k particles, we conclude that the height fluctuations of the infinite system are universal, i.e. identical in the microscopic model presented here and in the coarse-grained Langevin description.
Exactly solvable models for many-body systems
237
Appendix C: Exact solution for empty-interval probabilities in the ASEP with open boundaries Iterative solution of the recursion relations (7.71) with initial condition (7.73) up to five sites gives the following polynomials:
k=l" YL
I -"
9
k-2: Y~(2) Y2(2)
= =
/3+or /32(1 +c~) +/3c~ 2
Y3(2)
=
/33(1 4- c~ 4- c~2) 4- fl2c~3
Y4(2)
--
/:14(1 4- ot 4- c~2 4- ~3) 4- f13ot4
I"5(2)
=
/35(1 --]--o,~ 4-0 '2 4-G 3 4-o~ 4) 4- flaot5.
k--3" Y2(3)
=
/32(1 + C~) + flC~(1 + 0~) + c~2
Y3(3)
=
/33(1 + 2C~ + 20t 2) 4-/~2C~2(1 4- 2C~) + ~Ot3
Y4(3)
=
,t34(1 + 2C~ + 3C~2 + 3C~3) 4- fl3C~3(1 4- 3C~) 4-/32C~4
I"5(3)
=
/35(1 + 2C~ + 30t 2 4- 40~3 4- 40t 4) 4-/340t4(1 4- 4C~) 4- fl30~5.
k--4: Y3(4)
=
/33(1 + 2c~ + 2c~2) +/J2c~(l + 2c~ + 2o~2) +/3c~2(1 + 2c~) + c~3
Y4(4)
=
f14(l + 3o~ + 5c~2 + 5c~3) + fl3~2(1 4- 3c~ 4- 5o~2) 4-f120t3 (1 4- 3or) 4- flog4
I"5(4)
=
fls(1 + 3c~ + 6~ 2 4- 9c~3 + 9c~4) 4- fl4c~3(1 4- 4c~ + 9c~2)
4-fl30~4 (1 + 4c~) + fl20~5. k=5"
Y4(5) =
f14(l + 3c~ + 5c~2 4- 50t 3) 4- fl3c~(l -F 3c~ + 50t 2 --t- 50~3) 4-fl2o~2(1 4- 30t + 51y2) + flO~3(1 4- 3c~) + O~4
Y5(5)
=
/35(1 + 4c~ + 9c~2 + 14~ 3 + 14~ 4) + fl4c~2(1 4- 4~ 4- 9c~2 + 14c~3) +/33c~3(1 + 4c~ + 9c~2) 4- f12ff4(l 4- 4c~) 4- flc~5.
238
G. M. Sch~tz
k=6: 115(6)
=
f15(l 4-4~ + 9a e + 14~ 3 -+- 14~ 4) +fl4a(1 + 4~ + 9a e + 14~ 3 + 14o~4) -k-/33a2(l 4-4~ 4- 9a 2 + 14o~3) + ]32a3(1 4-4~ 4- 9a 2) +/~ot4(1 + 4~) + ot5.
In the integer coefficients one recognizes differences of two binomial coefficients. In conjunction with the pattern of the powers of ~, t3 this leads to the conjecture (7.78) for the general structure of YL,k. It is then straightforward to prove that this expression indeed satisfies the recursion relation (7.71) with initial condition (7.71 ). The function FN(x) introduced in (7.83) may be written in terms of a hypergeometric function (Gradshteyn and Ryzhik, 1981) as follows: l
_
FN(x) = 2 [ x ( l - x)] N+l + 2
(2) 2F1(1, N + 1/2; 1/2; ( 1 -
2x)X).
(C.1)
From the asymptotic form of this function for large N one obtains the density profiles discussed in Section 7.2.2.
239
Exactly solvable models for many-bodysystems Appendix D: Frequently used notation
D.1
Single-site basis vectors and Pauli matrices
1. The vector 10 ) represents the empty lattice, or, in spin language, the state with spin up. The vector I1 ) represents the state with a particle (or spin down). We use the representation 10)=
(,) 0
'
2. We use s • - ( o x 4- t y Y ) / 2 , s z = o z / 2 , (1 + t r z ) / 2 with the Pauli matrices crX=(0
1 ) 1
0
(0 '
o "y =
-i i
(o)
I1)-
1
n --
(1 - ~rz)/2, v = 1 - n =
)
0
z cr
'
,o,,
9
( 1 =
0
0
) .
-1
(D.2)
3. AT is the transpose of a matrix A. The row vectors (. I are the transposed vectors of the column vectors I. ).
D.2
Tensor products
1. The tensor product C = A | B of an n l x n2 matrix A with a m l x m2 matrix B is a Pl x P2 matrix with Pl,2 -- n l,2ml,2. It can be written as a matrix where the matrix elements are themselves matrices B and which has the following form:
a | B =
allB
al2B
...
a21 B .
a22B . .
999
aln2B a2nz B
anjlB
an12B
...
anln2 B
.
(0.3)
In particular, for 2 x 2 matrices
A| B =
allbll
allbl2
al2b]l
allbZl a21bll a21b21
allb22
alebel a22bll a22b21
a21b12 a21b22
al2bl2 a12bz2 a22b12 a22b22
(D.4) "
Also the n x m matrix [a ) (b I represents a tensor product of the column vector l a ) (an n x 1 matrix) with the row vector ( b I (which is a 1 x m matrix with m = n in our applications).
240
G. M. Sch(itz
2. The expression X ~t" represents the L-fold tensor product of the matrix or vector X with itself. 3. The vector I n l, n2 . . . . . nL ) -- In l) | In1 ) Q . . . @ I nL ) represents the state with local occupation numbers nk -- 0, 1. An N-particle state with particles on sites k I . . . . . kN is denoted I k l . . . . . kN ). 4. For a lattice of L sites we use 10 ) also for the state 10)| empty lattice.
representing the
5. The L-site summation vector ( S I is defined by ( S I (1, I)| For two sites: (SI = (1, 1 ) ~ ( 1 ,
((01 + ( 1 I) |
(D.5)
1) = (1, 1, 1,1).
6. The L-site product measure ] p ) is defined by I P) pll)) | For two sites:
=
=
((1 - P ) I 0 ) +
(1 -- ,o) 2
p
p
p(1 - p ) p2
9
(D.6)
7. The local operators Ak act nontrivially only on that factor in a tensor product which represents site k in a lattice. In a one-dimensional chain this is the kth factor. In tensor notation these matrices are given by the tensor product
Ak--I|174174174
(D.7)
where 1 is the m x m unit matrix (for m-state models). By construction, matrices Ak, At commute for k ~ I. An exception from this notation is the two-site interaction matrices h k which act nontrivially on sites k, k + 1, and t the fermionic operators Ok, c~, bp, bp which anticommute for different k or p respectively. 8. (( W I, I V )) are vectors in the unspecified vector space on which the matrices D, E, S, T act. These are understood as matrix representations of the time-dependent reaction-diffusion algebra (6.36). The expression I V ))10 ) stands for a tensor product.
Exactly solvable models for many-body systems
D.3
241
Other notation
k,l,m,x,y Wij c~, ~, y,~ bi Sk 11 X
1,1) (,11 (Plq) AB p(t) pk(t) J 13 Tr
In(t) eft(x) erfc(x) ASEP DLF DLFPA DLPA DW IPDF SSEP TMMC VM
Lattice sites Reaction rate (see Table 1) Boundary injection and absorption rates (see (3.31)) Stochastic boundary matrix for site i = 1, L. Classical Ising spin variable on lattice site k Configuration of a particle or spin system State space or corresponding vector space of a particle system Canonical basis vector of vector space associated with X Canonical basis vector of dual vector space associated with X scalar product Zi Piqi matrix product C = A B with Cij = Y~-k Aik Bkj Expectation value of space-averaged particle density Expectation value of local particle density Current expectation value Factorized similarity transformation (5.4) Trace over matrices Modified Bessel function of order n (4.27) Error function (see (6.28)) Complementary error function 1 - erf(x) Asymmetric simple exclusion process Diffusion-limited fusion process Diffusion-limited fusion/pair annihilation process Diffusion-limited pair annihilation process Domain wall annihilation/citation process Inter-particle distribution function Symmetric simple exclusion process Tetramethylammonium manganese trichloride (CH3)4N Mn C13 Voter model
242
G.M. SchQtz
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Schadschneider, A. and Schreckenberg, M. (1993). J. Phys. A26, L679. Schadschneider, A. and Schreckenberg, M. (1998). J. Phys. A31, L225. Schmittmann, B. and Zia, R. K. E (1995). Phase Transitions and Critical Phenomena (eds. C. Domb and J. Lebowitz), p. 1. Academic Press, London. Schreckenberg, M., Schadschneider, A., Nagel, K. and Ito, N. (1995). Phys. Rev. ESI, 2939. Schulz, M. and Trimper, S. (1996). Phys. Lett. A216, 235. Schulz, M., Trimper, S. and Kimball, J. (1997). Phys. Lett. A235, 113. Schiitz, G. (1993a). J. Stat. Phys. 71, 471. Schlitz, G. (1993b). Phys. Rev. E47, 4265. Schlitz, G. M. (1995a). J. Stat. Phys. 79, 243. Schlitz, G. M. (1995b). J. Phys. A28, 3405. Schlitz, G. M. (1996a). cond-mat 9601082, Proceedings of a Satellite Meeting of Statphys 19 on Statistical Models, Yang-Baxter Equations and Related Topics and of the 7th Nankai Workshop on Symmetry, Statistical Mechanical Models and Applications (eds. M.-L. Ge and E Y. Wu), World Scientific, Singapore. Schlitz, G. M. (1996b). Phys. Rev. E53, 1475. Schlitz, G. M. (1997a). J. Stat. Phys. 86, 1265. Schlitz, G. M. (1997b). J. Stat. Phys. 88, 427. Schiitz, G. M. (1997c). Z Phys. BI04, 583. Schlitz, G. M. (1998). Eur. Phys. J. BS, 589. Schlitz, G. M. (1999). Europhys. Lett. 48, 623. Schlitz, G. and Domany, E. (1993). J. Stat. Phys. 72, 277. Schlitz, G. M. and Mussawisade, K. (1998). Phys. Rev. E57, 2563. Schlitz, G. and Sandow, S. (1994). Phys. Rev. E49, 2726. Siggia, E. (1977). Phys. Rev. BI6, 2319. Simon, H. (1995). J. Phys. A28, 6585. Sire, C. and Majumdar, S. N. (1995). Phys. Rev. Lett. 74, 4321. Sklyanin, E. K. (1988). J. Phys. A21, 2375. Spitzer, E (1970). Adv. Math. 5, 246. Spohn, H. (1983). J. Phys. AI6, 4275. Spohn, H. (1991). Large Scale Dynamics of Interacting Particles. Springer, Berlin. Spouge, J. L. (1988). Phys. Rev. Lett. 60, 871. Stinchcombe, R. B. and Schlitz, G. M. (1995a). Europhys. Lett. 29, 663. Stinchcombe, R. B. and Schiitz, G. M. (1995b). Phys. Rev. Lett. 75, 140. Stinchcombe, R. B., Grynberg, M. D. and Barma, M. (1993). Phys. Rev. E47, 4018. Stinchcombe, R. B., Santos, J. E. and Grynberg, M. D. (1998). J. Phys. A31,541. Sudbury, A. and Lloyd, P. (1995). Ann. Prob. 23, 1816. Sutherland, B. (1967). Phys. Rev. Lett. 19, 103. Sutherland, B., Yang, C. N. and Yang, C. E (1967). Phys. Rev. Lett. 19, 588.
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2
Polymerized Membranes, a Review*
K. J. Wiese Fachbereich Physik, Universit#t GH Essen, 45117 Essen, Germany 1 Introduction and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
256
2 Basic properties of membranes . . . . . . . . . . . . . . . . . . . . . . . . . .
261
2.1
Fluid membranes
..............................
2.2
Tethered (polymerized) membranes
261
2.3
Crumpling transition, the role of bending rigidity, and some approximations 265
....................
..........................
262
2.4
Stability of the flat phase
2.5
Experiments on tethered membranes . . . . . . . . . . . . . . . . . . . .
273
267
2.6
Numerical simulations of self-avoiding membranes
276
2.7
Membranes with intrinsic disorder . . . . . . . . . . . . . . . . . . . . .
............
3 Field-theoretical treatment of tethered membranes . . . . . . . . . . . . . . . .
278 279
3.1
Definition of the model, observables, and perturbation expansion . . . . .
279
3.2
Locality of divergences . . . . . . . . . . . . . . . . . . . . . . . . . . .
282
3.3
More about perturbation theory . . . . . . . . . . . . . . . . . . . . . . .
284
3.4
Operator product expansion (OPE), a pedagogical example . . . . . . . .
286
3.5
Multilocal operator product expansion (MOPE)
291
..............
3.6
Evaluation of the M O P E coefficients . . . . . . . . . . . . . . . . . . . .
292
3.7
Strategy ofrenormalization . . . . . . . . . . . . . . . . . . . . . . . . .
297
3.8
Renormalization at one-loop order . . . . . . . . . . . . . . . . . . . . .
298
3.9
Non-renormalization of long-range interactions
303
4 Some useful tools and relation to polymer theory
..............
................
305
4.1
Equation of motion and redundant operators . . . . . . . . . . . . . . . .
305
4.2
Analytical continuation of the measure . . . . . . . . . . . . . . . . . . .
309
*The author has won the Physics Prize of the Academy of Science in G6ettingen PHASE TRANSITIONS VOLUME 19 ISBN 0-12-220319-4
Copyright (~ 2(X)I Academic Press Limited All rights of reproduction in any fl~rm reserved
254
K.J. Wiese 4.3
IR regulator, conformal mapping, extraction of the residue, and its universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
310
4.4
Factorization for D = 1, the Laplace-De Gennes transformation . . . . .
313
5 Proof of perturbative renormalizability . . . . . . . . . . . . . . . . . . . . . . 5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Proof
5.3
Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
318 318
....................................
6 Calculations at two-loop order
319 334
..........................
340
6.1
The two-loop counterterms in the MS scheme . . . . . . . . . . . . . . .
6.2
Leading divergences and constraint from renormalizability
340
6.3
Absence of double poles in the two-loop diagrams . . . . . . . . . . . . .
345
6.4
Evaluation of the two-loop diagrams . . . . . . . . . . . . . . . . . . . .
346
6.5
RG functions at two-loop order . . . . . . . . . . . . . . . . . . . . . . .
347
7 Extracting the physical informations: extrapolations . . . . . . . . . . . . . . .
347
........
342
7.1
The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
347
7.2
General remarks about extrapolations and the choice of variables . . . . .
349
7.3
Expansion about an approximation . . . . . . . . . . . . . . . . . . . . .
351
7.4
Variational method and perturbation expansion
352
7.5
Expansion about Flory's estimate . . . . . . . . . . . . . . . . . . . . . .
354
7.6
Results for self-avoiding membranes . . . . . . . . . . . . . . . . . . . .
355
..............
8 Other critical exponents, stability of the fixed point and boundaries . . . . . . .
357
8.1
Correction to scaling exponent t.o . . . . . . . . . . . . . . . . . . . . . .
357
8.2
Contact exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
357
8.3
Number of configurations: the exponent y . . . . . . . . . . . . . . . . .
359
8.4
Boundaries
361
9 The tricritical point
................................. ................................
363
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Double e-expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
364
9.3
Results and discussion
368
10 Variants
363
...........................
......................................
10.1 Unbinding transition
371
............................
10.2 Tubular phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Dynamics
374
.....................................
11.1 Langevin dynamics, effective field theory
371 377
.................
377
11.2 Locality of divergences . . . . . . . . . . . . . . . . . . . . . . . . . . .
380
11.3 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
381
11.4 Inclusion of hydrodynamic interaction (Zimm Model) . . . . . . . . . . . 12 Disorder and nonconserved forces
........................
12.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
385 388 390
2
255
Polymerized membranes, a review
12.2 Field-theoretical treatment of the renormalization group equations
....
392
12.3 Fluctuation-dissipation theorem and Fokker-Planck equation . . . . . . .
393
12.4 Divergences associated with local operators
395
................
12.5 Renormalization of disorder (divergences associated with bilocal operators) 397 12.6 The residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
399
12.7 Results and discussion
402
...........................
12.8 Long-range correlated disorder and crossover from short-range to longrange correlated disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 13 N-coloured membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
407 408
13.1 The O ( N ) - m o d e l in the high-temperature expansion
...........
409
13.2 Renormalization group for polymers . . . . . . . . . . . . . . . . . . . .
412
13.3 Generalization to N colours
........................
420
13.4 Generalization to membranes . . . . . . . . . . . . . . . . . . . . . . . .
421
13.5 The arbitrary factor c ( D ) . . . . . . . . . . . . . . . . . . . . . . . . . .
427
13.6 The limit N ~
427
oo and other approximations . . . . . . . . . . . . . . . .
13.7 Some more applications . . . . . . . . . . . . . . . . . . . . . . . . . . .
429
14 Large orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
434
14.1 Large orders and instantons for the SAM model . . . . . . . . . . . . . .
435
14.2 The polymer case and physical interpretation of the instanton . . . . . . .
439
14.3 Gaussian variational calculation
442
......................
14.4 Discussion of the variational result . . . . . . . . . . . . . . . . . . . . . 14.5 Beyond the variational approximation and l i d corrections
445 ........
449
15 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
450
Appendix A: Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
451
Appendix B: List of symbols and notations used in the main text . . . . . . . . . .
453
Appendix C: Longitudinal and transversal projectors
454
................
Appendix D: Derivation of the RG equations . . . . . . . . . . . . . . . . . . . . .
455
Appendix E: Reparametrization invariance . . . . . . . . . . . . . . . . . . . . . .
458
Appendix F: Useful formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
459
E1
Momentum space integrals . . . . . . . . . . . . . . . . . . . . . . . . .
459
E2
Calculable manifold integrals . . . . . . . . . . . . . . . . . . . . . . . .
459
E3
Distributions
460
................................
Appendix G: Derivation of the Green function . . . . . . . . . . . . . . . . . . . .
460
Appendix H: Exercises with solutions
461
H. 1 Example of the MOPE H.2
........................
...........................
Impurity-like interactions . . . . . . . . . . . . . . . . . . . . . . . . . .
461 463
H.3
Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
464
H.4
Tricritical point with modified two-point interaction . . . . . . . . . . . .
464
H.5
Consequences of the equation of motion . . . . . . . . . . . . . . . . . .
466
256
K.J. Wiese
H.6 Finitenessof observables within the renormalized model . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
466 467
Introduction and outline
One of the most challenging ideas in modem physics is the concept of universality: certain properties of physical systems do not depend on microscopic details and furthermore are equivalent for seemingly unrelated problems. This is epitomized by systems undergoing symmetry-breaking continuous phase transitions. The most powerful tools to reveal these relations are delivered by quantum field theory, which has celebrated an overwhelming success in nearly all areas of physics. The study of the O(N)-model, which is a field theory for the statistics of N-component spins with short-range interactions, has shown that their critical behaviour is described by a set of exponents which are completely characterized by the dimension and the underlying symmetry (the number of components of the order parameter). Universality is ensured since the microscopic details are averaged out, and do not affect the large-scale fluctuations. A variety of techniques has been developed to examine the critical behaviour of this model; possibly the most successful one is the renormalization group procedure (Wilson and Kogut, 1974) which analytically justifies the concept of universality. The technically most convenient implementations are field theoretical methods, e.g. the e-expansion about the upper critical dimension of 4, an expansion about the lower critical dimension of 2, and exact resummations in the large N limit. (For a review of these techniques, see Zinn-Justin, 1989.) The best studied method is the e-expansion about the upper critical dimension of 4, where calculations have been performed up to fifth order. Together with resummation techniques which take care of the large-order behaviour known from instanton calculus, this is a very powerful tool for extracting critical exponents. On the other hand, field theories have strong connections to geometrical problems involving fluctuating lines. For example, the motion of particles in spacetime describes a world-line. Summing over all world-lines, weighted by an appropriate action, is the Feynman path integral approach to calculating transition
Fig. 1 Budding of fluid membranes (from Kaes et al., (1993).)
2
Polymerized membranes, a review
257
probabilities, which can alternatively be obtained from a quantum field theory. The latter can be extended to string theory, generalizing the sum over particle trajectories to the sum over trajectories of lines. Another example is the hightemperature expansion of the Ising model. The energy-energy correlation function can be expressed as a sum over all self-avoiding closed loops which pass through two given points. Self-avoidance is necessary in order not to overcount configurations. We face an important new theoretical concept, which is the subject of this review: parameterizing the loop by its length, different parts of the loop interact with each other irrespective of their distance. Treating such phenomena in the framework of field theory demands an enlargement of the concept of local field theories to m u l t i - l o c a l ones. The first such direct approach was developed in the context of self-avoiding polymers, which are formally equivalent to the loops appearing in the high-temperature expansion of the Ising model, by Edwards and des Cloizeaux (Edwards, 1965; des Cloizeaux, 1981; des Cloizeaux and Jannink, 1990). In this approach, hard self-avoidance is replaced by a soft short-range repulsive interaction upon contact of the monomers. This interaction is then studied perturbatively by expanding about ideal random walks. Here too, the perturbative expansion can be reorganized into an expansion about the upper critical dimension of 4, which was shown (De Gennes, 1972) to be equivalent to the perturbation expansion of q~4-theory in the limit N ~ 0. This equivalence provides two apparently different approaches for calculating the same exponents. There is much work in the field theory community on generalizing results for fluctuating lines to entities of other internal dimensions D. The most prominent example is string theory, which describes D = 2 world sheets (Polyakov, 1986a,b; Green et al., 1987; Polchinsky, 1996). An earlier example is provided by the correspondence between gauge theories and random surfaces (Kogut, 1979; Savit, 1980). The low temperature expansion of the Ising model in d dimensions also results in a sum over surfaces that are (d - 1)-dimensional. For d = 3, the surfaces are made out of plaquettes, the basic objects of lattice gauge theories. All these objects share the common property that not only fluctuations of shape but also topology changes occur and have to be summed over in the partition function. The biologically relevant representatives of this class of membranes are fluid membranes, which in general are formed by a lipid bilayer. In contrast to fluid membranes are 'tethered', polymerized surfaces (Kantor et al., 1986, 1987), which have a fixed internal connectivity, and are thus simpler than their fluid counterparts. Experimental realizations are, e.g. the network formed by spectrin in red blood cells or graphite monolayers. These systems may be found in three quite different phases: a collapsed compact phase, a flat phase and an intermediate crumpled swollen phase with fractal dimension of about 2.4. Experimentally, the situation is still under debate (cf. Section 2.5). In numerical simulations (cf. Section 2.6), generically flat membranes are found (see Fig. 2). The reason why eventually no crumpled swollen phase may be observable is that the rigidity of
258
K.J. Wiese
Fig. 2 Polymerized tethered membrane in the flat phase (from Abraham and Nelson, 1990a). tethered membranes is - in sharp contrast to fluid membranes - strongly enhanced by the effect of shear waves. Technically, integrating out these degrees of freedom renormalizes the rigidity, and if the initial rigidity is beyond a certain threshold, the membrane will become flat (see Section 2.4). Intuitively this is analogous to a crumpled sheet of paper, which is much more rigid than an uncrumpled one. Numerically, it has been observed that tethered membranes seemingly are always flat, even when starting with self-avoidance only, as can be seen in Fig. 2. This can be traced back to the effective (entropic) bending rigidity which is always present in such models. However, since the largest membranes simulated so far consist of only 75 x 75 atoms in the simplest spring and bead model, which has the inconvenience of being rather rigid, and of about 25 x 25 atoms in the more sophisticated plaquette models, simulations are far form being conclusive. These general physical, including numerical and experimental considerations, are presented in more detail in Section 2. For theoretical analysis, it is convenient to further generalize to membranes of arbitrary (inner) dimension D, interpolating between polymers for D = 1 and membranes for D = 2. Simple power counting indicates that self-avoidance is relevant only for dimensions d < dc = 4 D / ( 2 - D), making possible an e = 2D - d(2 - D ) / 2 ,~, ( d c ( D ) - d)-expansion, which was first carried to one-loop order about an arbitrary point on the line e = 0 in Kardar and Nelson (1987), Kardar and Nelson (1988), Aronovitz and Lubensky (1987), Aronovitz and Lubensky (1988), Duplantier (1987). To obtain results for polymers or membranes, one then has the freedom to expand about a n y internal dimension D, and the corresponding upper critical dimension of the embedding space (Hwa, 1990). This freedom can be used to optimize the calculation of critical exponents (Wiese and David, 1997; David and Wiese, 1996).
2
Polymerized membranes, a review
259
A major breakthrough in the understanding of these nonlocal field theories is the proof by David, Duplantier and Guitter (1993a,b, 1994, 1997), that the fieldtheory of a D-dimensional self-avoiding tethered membrane is renormalizable to all orders in perturbation theory. The main technical tool is the multilocal operator product expansion (MOPE), generalizing the concept of (local) operator product expansion (OPE), introduced into field theory a long time ago by Wilson (1969) and Kadanoff (1969), to the multilocal situation. We shall present this technique in Section 3. A collection of useful tools is given in Section 4, and a condensed version of the above-mentioned proof in Section 5. These general arguments have been checked by explicitly going to two-loop order (David and Wiese, 1996; Wiese and David, 1997). The calculation is technically difficult but it is valuable to understand the underlying principles. We therefore review these calculations in Section 6, suggesting to the reader more concerned with applications to skip this section as well as Section 5 with the discussion of the proof of perturbative renormalizability. The most important physical prediction of this calculation is that there exists a crumpled swollen phase with fractal dimension of about 2.4. Another important question is whether nonleading terms play a role for the critical behaviour of tethered membranes. This is certainly the case at the tricritical point, which separates the crumpled swollen from the compact phase, and which is analysed in Section 9. In contrast to polymers, whose tricritical behaviour is dominated by the three-point self-repulsion (which formally punishes triple intersection of the polymer with itself), in the case of the membrane (D = 2), this role is played by a modified two-point interaction, not proportional to a g-interaction, but to its second derivative (Wiese and David, 1995). Subdominant operators may also play a role at the self-avoiding fixed point, at finite e, i.e. well below the upper critical dimension (Wiese and Shpot, in preparation). It is well known that different dynamical models can lead to the same static behaviour (Hohenberg and Halperin, 1977). In the case of polymers, people have paid most attention to purely diffusive dynamics (Rouse model, model A) eventually including the effect of hydrodynamics (Zimm model). For a long time, the question whether these dynamical models are renormalizable, stayed open. As discussed in Section 11, the methods mentioned above finally allowed to settle this question (Wiese, 1998a,b). Somehow surprisingly, the same kind of model also applies to the dynamics of an extended elastic object, be it a polymer or a membrane, in quenched disorder. Technically, averaging over disorder generates nonlocal interactions on the polymer, with interactions proportional to the disorder correlations. The latter may be taken to be g-distributions. In this respect, it is worth recalling that self-avoidance can also be generated by averaging over all realizations of an (imaginary) random potential, in which the polymer or membrane is fluctuating. In Section 12, we review the analysis of a D-dimensional membrane (with D = 0 for a particle,
260
K.J. Wiese
D = 1 for a polymer and D = 2 for a membrane), in a quenched random force field with both potential and nonpotential parts. In contrast to the pure potential case, this situation is accessible perturbatively (Le Doussal and Wiese, 1998; Wiese and Le Doussal, 1999). As is well-known, string theory is defined as the sum over all closed manifolds with arbitrary topology. Excluding from this sum self-intersecting configurations is a formidable task beyond current technical capabilities. For polymerized membranes, i.e. with nonfluctuating metric, this sum can indeed be taken, generalizing the high temperature expansion of the O(N)-model mentioned above from a gas of self-avoiding loops of fugacity N, to a similar gas of closed fluctuating manifolds of internal dimension D (Wiese and Kardar, 1998a,b). As will be discussed in Section 13, this generalization is not unique, leaving space for adaptation of the model to the situation in question. Among others, the model contains a novel mechanism not present in standard field theory, which turns first-order transitions into second-order ones ('reverse Coleman-Weinberg mechanism'). The model further contains a one-loop fixed point for the random bond Ising model and finally allows for an intriguing conjecture regarding the nature of droplets dominating Ising criticality. So far, these models have only been treated via perturbative techniques. An important question is whether the theory is meaningful beyond perturbation expansion. This is a difficult issue, which so far is only partially answered for the case of self-avoiding polymers. A little bit easier to answer is the question, whether the perturbative series is well defined. For the case of the O(N)-model, it has been shown by Lipatov (1977a,b), that the series is divergent, but can be resummed using a Borel-transform. For tethered membranes, the situation is difficult, since the usual instanton methods do not apply. In Section 14 we show what the analogue of the instanton for the q~4-theory is, and why this implies that the perturbation series is also Borei-summable (David and Wiese, 1998). Finally let us point out that even though the primary aim of this review is to present from a unified viewpoint the theoretical concepts of multilocal field theories, an effort is made to motivate the physical models and experimental relevance. On the other hand, the real progress which goes beyond today's interest lies in the fundamental technical achievements, and the author feels that skipping technically important details would render this review much less useful. In order to keep the text readable, the central ideas are given before embarking on technical calculations, and wherever this is possible, we try to sketch how the techniques developed will be useful later. The general structure of this review is therefore organized so that relevant material, which is necessary to place the following more technical parts in the physical context, is collected in Section 2. The next section is devoted to the necessary elementary technical tools. The following sections are more specialized and can mostly be read independently, only necessitating Section 3, and eventually 4.
261
2 Polymerizedmembranes, a review 2
2.1
Basic properties of m e m b r a n e s
Fluid membranes
Let us start by characterizing the different possible types of membranes. One very popular class of membranes are fluid membranes (Fig. 3). We all know of soap bubbles from childhood days. Biologically more relevant are bilayers of lipid molecules that are composed of a hydrophilic head and two hydrophobic chains. As shown in Fig. 4, in water the hydrophobic chains group together and form a lipid bilayer. This is the basis of most of the biologically relevant membranes. For an analytical description, one needs the coordinate 7(x) of the membrane as a function of an internal parameter x, characterized by the mapping r'x
E R2
> 7(x) ~ I~a
(2.1)
and by the induced metric g~
(2.2)
-- 0 ~ 7 3 ~ 7 .
We are now looking for the statistical weight of a membrane configuration. Since the lipid molecules in the membrane are free to move around, the energy, i.e. 'Hamiltonian' of the membrane has to be invariant under coordinate transformations. This is achieved by the Canham-Helfrich Hamiltonian (Canham, 1970; Helfrich, 1973) 7-~[;] =
j
d2x ~
E
r + ~ (H(x) - H0)
9
(2.3)
d2x q ~ is the invariant volume element of the membrane, r its surface tension, and x the bending rigidity, which is coupled to the square of the mean curvature 1 (R.~-
1 )
,
(2.4)
Fig. 3 Fluid membranes with higher topology. From left to right: a l-torus (Michalet and Bensimon, 1995) and a 2- and 4-torus (Michalet et al., 1994).
262
K.J. Wiese
Fig. 4 Model of a fluid membrane: bilayer of lipid molecules that are composed of a hydrophilic head and two hydrophobic hydrocarbon chains.
where R l and R2 are the two curvature radii. H0 is a spontaneous curvature, present in the case of symmetry breaking between the two sides of the membrane. Physically, rigidity is explained by the finite thickness of the membrane. RGcalculations indicate that bending rigidity should be irrelevant at large distances (Peliti and Leibler, 1985; Peliti, 1996); this, however, has recently been criticized by Helfrich (1998). Experimentally, fluid membranes offer a wide range of interesting and complex phenomena. Let us only mention the budding of a fluid membrane, as given in Fig. 1 and the appearance of higher genus objects (Fig. 3). For a general review about fluid membranes, see Peliti (1996); Nelson et al. (1989); Lipowsky (1992) and Seifert (1997). Interestingly, the Hamiltonian (2.3) with K = 0 also plays a central role in string theory. Here, one of the inner coordinates
.
X--
(x,) X2
is identified as i x time, and the other one as length on the string. Equation (2.3) is then the action generating the motion of the string. Further generalizations use a metric ga# independent of the embedding space (Polyakov, 1986a,b; Green et al., 1987; Polchinsky, 1996). Strings are considered as one of the most promising candidates for unifying all fundamental interactions.
2.2 Tethered (polymerized) membranes In this review, we shall concentrate on another class of membranes, which have a fixed and constant internal metric: g.~ = ~.~.
(2.5)
These membranes have not yet found applications in high-energy physics, but are realized in experiments (see Section 2.5). They are either called solid, tethered or polymerized membranes.
2
Polymerized membranes, a review
263
Fig. 5 A tethered membrane (spring and bead model) (from Kantor et al., 1987).
A microscopic model is given by the so-called 'spring and bead model' (see Fig. 5), which consists of balls (beads) which are connected by springs and form a regular lattice. The model membrane is called 'self-avoiding' since the beads cannot intersect each other. We will discuss Monte Carlo simulations of this model in Section 2.6. A simpler situation occurs when self-intersections are allowed ('phantom membrane'). Simulations as well as renormalization group calculations (David and Guitter, 1988; Aronovitz et al., 1989; Paczuski and Kardar, 1989) indicate that such a membrane is crumpled for weak bending rigidity, K < Kc and flat for K > Xc. At the phase-transition point x = xc, the membrane is in another critical (or more precisely tricritical) state with a fractal dimension df in between the dimensions of the crumpled and fiat phases. A mean-field treatment of this so-called 'crumpling transition' is given in Section 2.3. Contrary to intuition, the fiat phase is not destroyed by fluctuations. This is demonstrated in Section 2.4, where also the tricritical state at K = Kc is discussed. On the other hand, in the small-rigidity phase, phantom membranes will have a fractal dimension of infinity. For physical (self-avoiding) membranes which can not intersect themselves, this is clearly impossible, and one expects the physical bound
df _< d induced by self-avoidance.
(2.6)
264
K.J. Wiese
A continuous model to describe a self-avoiding membrane is
7/[r]-
f
dDx -~ (V;(x)) 2 -k- ~
'
dDx
dDy 6a (-~(x)
--~(y)).
(2.7)
It has first been proposed by Edwards (1965) to describe polymers (D = 1). In that case, it is equivalent to scalar 4~4-fieid theory in the limit of N - 0 components (De Gennes, 1972). In 1986 the model has been generalized to membranes (D = 2), independently by Kardar and Nelson (1987, 1988) and by Aronovitz and Lubensky (1987). They observed that a direct calculation at D = 2 is impossible, but that one can make an analytic continuation from D < 2. In contrast to polymers, with their equivalence to scalar field theory, renormalization is not evident. At leading order, renormalizability has been verified by Duplantier et al. (1990). For the general case, an important step was achieved by David et al. (1993a,b) who showed renorrnalizability of the theory 7-/[~1 =
/
dDx
1
-~ (V;(x)) 2 -k- g
/
dDx ~d (?(x)),
(2.8)
which describes a phantom (nonself-avoiding) membrane in interaction with a single point (an impurity). The proof is based on a generalization of the forest algorithm introduced by Zimmermann (1969) to 6-like interactions. Their last step was to prove the renormalizability of the full model (David et al., 1994, 1997), which we shall describe in Section 5. To extract numerical predictions from the e-expansion is a tedious task. One of the problems is that since one cannot start from D = 2, an analytical continuation has to be performed starting at any point (D, d) on the critical curve, which will be defined in Section 3.1. The first calculations which tried to fix the expansion point via a minimal sensitivity scheme at one-loop order were performed in Hwa (1990). The result of df ~ 3.5 for membranes in three dimensions even violated the geometric bound of 3 discussed above. It therefore became necessary to perform two-loop calculations, not only to test the renormalization proof, but also to obtain more reliable values for the fractal dimension. This task was accomplished in Wiese and David (1997) and David and Wiese (1996), and we review the main steps in Section 6. For membranes in three dimensions these calculations predict a fractal dimension of about 2.4, eventually seen in some experiments and numerical simulations (see Sections 2.5 and 2.6). It is interesting to note that the model (2.7) can also be used to study selfavoiding fractal objects like Sierpinsky gaskets (Levinson, 1991). (But attention: one has to be careful in distinguishing the fractal and the spectral dimensions of the membrane.) Let us mention still another class of membranes, namely hexatic membranes. They play an intermediate role between tethered and fluid membranes. For a
2
Polymerized membranes, a review
265
review see Nelson et al. (1989); Nelson and Peliti (1987); Guitter and Kardar (1990) and David et al. (1987). In the rest of this section, we review some simple arguments for tethered membranes, as well as experiments.
2.3 Crumpling transition, the role of bending rigidity, and some approximations Let us start by studying the different terms appearing in a mean-field description of membranes. Let 7 " x ~ IK ~
~ -d(x) ~ I~d
(2.9)
be the coordinates of a D-dimensional manifold embedded into a d-dimensional space. For D = 1, this represents a polymer, for D = 2 a membrane. Suppose that the underlying lattice is regular and that after integration over the fast degrees of freedom the effective model becomes translationally invariant. An expansion h la Landau then leads to an effective free energy or 'Hamiltonian' (Paczuski et al., 1988)
tr 7-L[F(x)] = f dDx ~(OaOc~;) 2 4- ~t (Ou;) 2 4- tt (Oct;Off;) 2 4- 0 (Oc~;Oa;) 2 4--~
dDx
tiDy 3d(7(x) -- -d(y)).
(2.10)
The last term, a self-repulsion upon contact, is a nonlocal interaction in the internal coordinates x, but local in the membrane position 7(x). The local terms are the different contributions to the elastic energy. The coefficients t, u and 0 weight the elastic and inelastic harmonic energies, whereas a: measures the bending rigidity. The analogy to the usual 4)4-theory becomes apparent upon identifying the tangents t,~ "-- 0or7 as order parameter. However, this analogy is only valid at the mean-field level, and will be destroyed by fluctuations. Mean-field theory suggests a phase transition at t -- 0, where the parameter t is equal to T - To, the difference in temperature T to the critical temperature Tc. At high temperature, t is positive due to entropy and the correlation between the tangential vectors decays exponentially fast. The membrane is in a crumpled phase. For negative t, the terms proportional to (Or) 4 restore positivity of the action, provided that u 4- 0 > 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)
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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
Polymerized membranes, a review
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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)
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Polymerized membranes, a review
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)
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Polymerized membranes, a review
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)
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Polymerized membranes, a review
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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.
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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).
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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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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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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
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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.
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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)
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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
D-e
D-e
- Le-D),
(3.97)
where A is a high-momentum UV regulator and L a large distance regulator. For e ~ 0 this is UV-divergent but IR-convergent. The simplest way to subtract this divergence is therefore to replace the dipole operator by x
--
=
y
~
x
--
-- y
-
x
- ......... 9
y'
(3
98)
where x . . . . . . y. . I x - y l -~a. This amounts to adding to the bare Hamiltonian (3.1) the UV-divergent counterterm
Ix-yl -~d, x
(3.99)
y
which is a pure number and thus does not change the expectation value of any physical observable. We next consider marginal operators: in the MOPE of (3.94), the integral over therelativedistanceoffx_y(x~yl~+~)~+
~ is logarithmically divergent at e
0. In order to find the appropriate counterterm, we use dimensional regularization, i.e. set e > 0. An IR cut-off L, or equivalently a subtraction momentum scale = L -1, has to be introduced in order to define the subtraction operation. As a general rule, let us integrate over all distances appearing in the M O P E coefficient, bounded by the subtraction scale L = / z -1. Defining
f tx yl.+.)
(3.100)
Ix-yl
x
(3.101)
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subtracting explicitly the divergence in the integrals, as discussed in the last section. The reader is invited to verify this explicitly in Exercise 6 (Appendix H) on the example of the expectation value of O e iklr(s)-r(t)], as given in (3.27). Since the angular integration in (3.100) reduces ,~+~ to -+-, we can replace (3.101) by the equivalent expression =
+),f +x
(3.102)
x
for which the numerical value of the diagram is calculated as L
1
f C yI+) =
2D
Ix-yl
f
J
dx 2D-vd --X
x
1
Ls
-~--. 2D s
(3.103)
0
We can now subtract this term in a minimal subtraction scheme (MS). The internal dimension of the membrane D is kept fixed and (3.103) is expanded as a Laurent series in e, which here starts at e - l . Denoting the term of order e P of the
by ( ] )~,
Laurent expansion to be
of(] )L for L
-- 1, the residue of the pole in (3.103)is found
-+.....
~
. . . . 2D
.
(3.104)
e
We shall also frequently employ the notation for the residue .... +
=
2)9"
(3.105)
It is this pole that is subtracted in the MS scheme by adding to the Hamiltonian a counterterm
x
Note that by going from (3.101) to (3.106), we have reduced the integral counterterm to a number. We recall our initial remark that if one wants to check that this counterterm renders the theory finite, one should think of it as its defining integral (3. l0 l), and verify that in the resulting perturbation theory, the first-order divergence is absent. Similarly, the divergence arising from the contraction of two dipoles to a single dipole is subtracted by a counterterm
x
y
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K.J. Wiese
with
(,:i~;:1- o),: f f (':~::1" --).
(3.108)
Ixl
Reducing this integral counterterm to a number, we subtract the residue of the single pole of
Ixl
=f
f(Ixl2"+lyl2~) -d/2 9
(3.109)
Ixl
Note that the regulator L cuts off both integrations. One can now either utilize some simple algebra or show by the methods of conformal mapping (see Section 4.3) that the residue is obtained by fixing one distance to equal 1 and by freely integrating over the remaining one:
(,i~;:1o -)~ f~x OO
=
X
D
(l+x
2-D)
.
(3.110)
0
(Recall that d/2 = 2D/(2 - D) + O(e).) The above is easily related to Euler's B-function and reads
,
(2_-00) 2
As a result, the model is UV-finite at one-loop order, if we use in the renormalized Hamiltonian (3.93) the renormalization factors Z and Zb,
z _ 1_
2_o , ( 0....
+
lb - + O(b2),
=
-
+ O(b2).
(3.112) (3.113)
r
Note that due to (3.92) no counterterm for _ ~ is necessary. The renormalized field and coupling are re-expressed in terms of their bare counterparts through ....
ro(x) = Zl/2r(x),
bo = bZbZd/2# e.
(3.1 14)
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Finally, the renormalization group functions are obtained from the variation of the coupling constant and the field with respect to the renormalization scale #, keeping the bare coupling fixed. (For a derivation, see Appendix D.) The flow of the coupling is written in terms of Z and Zb as
fl(b) "- #
1 b-b0
= -eb+
-eb . 1 + b~b In Zb + d b ~ In Z
2-D
F[2D'~ +
4D
"
Similarly, the full dimension of the field (the exponent entering into the correlation function) is obtained as
v(b) "=
2-D 2
1 +] 2#
In Z =
2-D 2
1 Oln z 2 fl(b) 0--b
bo 2-D 2-D[l+b 2
~
1 ]
(3.116)
+ O (b2)"
Note that minimal subtraction is used on the level of counterterms or equivalently Z-factors. Since Z enters as Z d into the/~-function, the latter also contains a factor of d in the one-loop approximation, i.e. Z a is not minimally renormalized. In order to calculate the leading order in e, the factor of d can be replaced by dc = 4 D / ( 2 - D). The fl-function has a nontrivial fixed-point with fl(b*) = 0, which has positive slope and thus describes the behaviour of the model at large distances: b* ---
e
2-
D
+ O(e
+1
2).
(3.117)
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K.J. Wiese
The anomalous dimension v* "= v(b*) becomes to first order in e
2-D
e 1 --1-~ 20
2
1 l
)2 r(2_DD
2-D
c ( 22-~Do)
+ 0(82).
(3.118)
+1
For polymers, this result reduces to the well-known formula
v*(D -- 1) --
3.9
1
+
4-d 16
+ O ( ( 4 - d)2).
(3.119)
Non-renormalization of long-range interactions
Long-range interactions are in general not renormalized (David et al., 1994). This is very useful, as it immediately enforces scaling relations among the critical exponents, which in some cases are already sufficient to determine these exponents. Let us explain the non-renormalization by analysing the long-range interaction
(~ > 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)
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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
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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)
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Polymerized membranes, a review
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)
308
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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_~<~] + )L =-2---DL 1 fx x D_vd• ( x - L ) = - 2 - - - D 1 L e
tThis section may be skipped upon first reading.
"
(4.25)
312
K.J. Wiese
So the residue of (4.24) is 0
1
(4.26)
We can apply this recipe to the second one-loop counterterm: ((~)[-"
--)t. = fx
[~"
--)'
(4.27)
since it is also proportional to L E. We thus have to calculate
0
o>,_L
OL
[fx
+f ] (x2u+y 2v)-d/2
We now introduce a general method which is very useful to manipulate and simplify such integrals. It relies on (global) conformal transformations in position space and is called conformal mapping of sectors. It has first been introduced in Wiese and David (1995), where a geometric interpretation can be found. We will explain the method on a concrete example and then state the general result. Let us consider the second integral on the r.h.s, of (4.28)
Lf,<x:L(X2V--t-y2v)-d/2 =
L
fo
cx:)-dx- x D x
xS(x - L ) |
my fo ~ dy y
D (x2V _k_y2V)-d/2 (4.29)
< x).
Now two changes of variables are performed: the first one, x
> 2,
(4.30)
x = x y L -1,
leads to
L I-D+vdf 0 ~ d---~xD 2 _ f 0 ~ d~y2D-vd Y ( 22v + L2v)-d/2 8 ( 2 y L - I - L ) O ( L x
< 2).
y
(4.31) The second one y
> y,
(4.32)
y--~2-1L
yields --z-x x
which using .~ =
L
~ y
22v
L2 v
-d/2 8 @ -
L)|
< 2),
(4.33)
8(.~- L)|
< 2).
(4.34)
finally gives
Ll+efoCX~d'- ~xD-efo~X~dyYD(22V+y2V) d / 2--y - x
2
313
Polymerized membranes, a review
Replacing the second integral on the r.h.s, of (4.28) by (4.34) gives
LO--LO( ! i ~ )
I~"
~')L =
Ll+efo c~ dXxDx fo ~
dyyDy
(x2v']- Y2V)-d/2
x max(x, y)-E6(y - L).
(4.35)
Now one distance (here y) is fixed, whereas the integral over the other distance (here x) runs from 0 to c~. The former constraint max(x, y) = L has been transformed into the factor of max(x, y)-e times the constraint y = L. Before generalizing this formula, we shall show how it can be used in practice. The residue in 1/e (which determines the corresponding one-loop counterterm) is given by the simple formula with dc(D) = 4 D / ( 2 - D):
f0
,
(4.36) The subleading term can analogously be calculated by expanding (x 2v + 1) -(a-a'(~ and m a x ( x , y ) -~ in e. We obtain the convergent integral representation
o -fo
'1
Xx~ x
x
('
2-D
~
ln(x 2v + 1 ) - ln(max(x 1))
)
. (4.37)
This method extends to the integrals which appear in the counterterms associated to the contraction of any number of points. In general, we have to compute integrals over N (N - 1) distances x, y . . . . . of the form f
I (e) = I
Jm a x ( x , y
f(x, y .... )
(4.38)
.... ) < L
with a homogeneous function f such that the integral has a conformal weight (dimension in L) x: l(e) ~ L K. For the integrals which appear in n-loop diagrams, this weight is simply x = ne. (4.39) The integral over the distances is defined by the D-dimensional measure (4.21). The residue is extracted from the dimensionless integral
J (e) = x L -x I (e) = L -K L 0OL l ( e ) = L f m
a x ( x , y .... ) = L
f ( x y,
x max(x, y . . . . )-K.
) (4.40)
314
K.J. Wiese
The domain of integration can be decomposed into 'sectors', for instance {...
(4.41)
{...<x
and we can map these different sectors onto each other by global conformal transformations. For instance we can rewrite the integral (4.40) as J (e) - L
f=L:y ....
f
f (x, y . . . . ) max(x, y . . . . )-,c _ L I Jr - - L ; x
f (x, y . . . . ) ....
x max(x, y . . . . )-'r The constraint on the maximum of the distances is replaced by the constraint on an arbitrarily chosen distance. This mapping of sectors is one of the basic tools used in calculating more complicated diagrams, e.g. at the two-loop order or for the disorder dynamics. Also note that it implies the universality of the leading pole in l/e: fixing the longest, the shortest or one of the intermediate distances always give the same leading pole. Finally, starting from these regularizations, any other one can be constructed.
4.4
Factorization for D = 1, the Laplace-De Gennes transformation
The methods described so far also apply to polymers. For polymers however, some simplifications are valid. Let us first give a simple (perturbative) example before stating the general result, namely the equivalence of polymers with the limit of zero components for a q~4-model. In perturbation theory, we may consider expectation values with two dipoles inserted. The simplification for a polymer is that its one-dimensional nature enforces an ordering of the intervening points, resulting in topologically different diagrams, as exemplified by the equation below:
(4.43) It turns out that topologically different diagrams are also different analytically, whereas for membranes, a single diagram contains all these contributions. This is due to the additive nature of the polymer correlation function, which reads for
2
ordered points
Polymerized membranes, a review
x3:
Xl < x2 <
C(xl
315
x2) +
-
C(x3
x2) ~-
-
C(x3
-
(4.44)
Xl).
This property is used to simplify the perturbation expansion. On a more formal level, there exists an integral relation between two-point functions for polymers and a local scalar field theory, which was first discovered by De Gennes (1972) and which we discuss now, following the derivation in Sch~ifer (1999). Consider the discretized version of the Edwards model (suppressing all indices '0' for bare quantities)
Lf
G (r, ro, L) = 1--I
ddrs
(47r ~.)a/2
s=l
• exp
( - ~ (rt--rt+l)2 b)~2~ ~ t=0
4~.
4
)
6d (rt - ru) , (4.45)
t=0 u=0
where the first and last monomers are fixed by r(0) = r0, r(L) = r. L is the number of momonmers, ~. their length, and the integration measure is normalized such that
f
ddrs
(4n.~.)d/2 exp(--r2/4~.)= 1.
With this normalization and setting b = 0, G is the probability conserving diffusion propagator at time t = JkL. The self-avoidance interaction can be disentangled through an auxiliary field qJ(r)
G(r, r0, L) =
L ) f LD[qJ] d d rI-Is ((rt-rt+l)2(47r)~)d/2 ~ Jr~ i)~qJ(rt) exp s=l
•
t=0
4Z
t=0
q'(r)2) b '
(4.46)
where a suitable normalization factor is absorbed into the integration measure D[qJ]. One then sees that the first part, namely the partition function of the polymer in the potential qJ(r)
'fd"rs
G(r, ro, L; qJ)"= H
s=l
(4rr~.)d/2 exp
(~(rt--rt+l) 2+ ZL i)~qJ(rt)) t=o
4~.
t=O
(4.47)
316
K.J. Wiese
satisfies the equation G(r, r0, L + 1" tp) --
f
ddr' -(r'-r)Z/4)~ei~qJ(r ) t (4rrz)d/-.......-------~e G ( r , ro, L; tp).
(4.48)
In the limit of a continuous chain, ~ becomes small and the r.h.s, can be expanded in ~., with the result G(r, ro, L + 1; tp) = (1 + i~.qJ(r) + ),Ar + O(~.2)) G(r, ro, L; qJ).
(4.49)
This can also be written as (e := ~.L) 0 r a G ( r , ro, s qJ) = (itP(r) + Ar)G(r, r0, s tp). Of.
(4.50)
Using a notation inspired from quantum mechanics, the solution to this equation is G(r, ro, e; ~ ) = l r l e - e ( - i q " r ) - A r ) l r o } . (4.51) The usual method, to solve this equation in quantum mechanics, consists in going from the 'time-dependent' SchrSdinger equation to the 'time-independent' one. In statistical mechanics one equivalently writes down the Laplace transform O~
(~(r, ro, t; qJ) "-- / dte -et G(r, ro, g; ~),
(4.52)
0 which gives
(
(3(r, ro, t" qJ) -- r
,
t - iqJ(r) - Ar
)
ro .
(4.53)
This is, up to a factor of Z -n, the correlation function of an n-component scalar field theory, (7,(r, ro, t; qJ) = Z -n
f
7)[r162 (r)r (ro)e- 89f~ r162
(4.54)
where Z is the partition function of the one-component version, Z = f D[Ole- 89fr r
(4.55)
Formally, the factor of Z -n is easily eliminated by setting n -- O. Combining (4.46), (4.52), (4.53) and the latter statement, one obtains that the Laplacetransformed polymer correlation function (~(r, r0, t) "- / dte -el G(r, r0, (), 0
(4.56)
2
where
G(r, r0, s
(7(r, r0, t) --
317
Polymerized membranes, a review
is the continuum version of
G(r, r0, L), equals
n-~olimf D[qJ]D[ck]qbl(r)cpl(ro)e- 89fr 4(r)(t-iqJ(r)-Ar)~(r)e-
fr qJ(r)2/b. (4.57)
The path integral over ~ can still be performed to obtain the final result
f D[,,,,(r)~l (ro)e -~.4[4~1
(3(r, ro, t) = n--+olim
~4[~]
fddr( ~q~ t-'2( r ) +
~1 [Vq~(r)] 2
+ ~b [~(r'] ~)
(4.58)
This is the path-integral representation of a correlation function in the ncomponent q~4-model, after taking the limit of n ~ 0. This remarkable result, first discovered by De Gennes (1972), gives two seemingly unrelated methods to calculate the same physical quantities. The derivation given above allows for some straightforward generalizations. Consider as in (4.45)
G(r, ro, L)
~Lf .
=
d dry.
(47rx)d/2
(
with
L-I ,].l,gen , 1 polymerlr] = Z t=0
p(r')
gen
exp -7-Lpolymer[r]
)
(4.59)
s--I
(rt -- r t + l ) 4k
2
+
f
d d r'.T'[p (r') ].
(4.60)
is the polymer density L
p(r') = X Z
6d (rt -
r'),
(4.61)
t=O
and .T'[p(r')] any functional of
p(r').
Then, following the same lines as above,
G(r, r0, t) - n~01imf D[q~lq~l (r)~bl (r0)e -~[4~1 t
"]-L2r[qb]=fddr(-~2(r)
1
q--~[g~(r)]
2
-k ~- [ ~ ~ 2 ( r ) ] )
9 (4.62)
Some examples are (in continuous notation)
f dx ~d(r(x)) e, > f ddr.~
1 q~2(r)
fdxfdySd(r(x)-r(y))
< > fddr
(4.63)
I2 ]2 ~2(r)
(4.64)
318
K.J. Wiese
f dx f dy f dz ~d(r(x) -- r(y))3d(r(x) -- r(z)) < > f dx f dy(_Ar),a(r(x)_r(y))~.
> f dar[l-'2 ~q~
ddr
(r) ] ( - A r )
q~2(r) (4.65)
[~q~2] (r)
.
(4.66) These relations are exploited when studying the generalization to membranes of a N-component 4~4-model in Section 13. The derivation also shows, and this is the reason why we have discussed it in detail, that a similar procedure is not suitable for membranes. Trying to generalize the above methods to membranes leads to the problem of solving the diffusion equation of a (D - l)-dimensional membrane in a random potential, which is a formidable task. It seems that no local field theory which is equivalent to selfavoiding membranes can be constructed, even though this is hard to prove. Another hint comes from the following observation: suppose one wants to construct a scalar field theory with upper critical dimension dc = 4 D / ( 2 - D), i.e. the same upper critical dimension as for D-dimensional polymerized tethered membranes. In addition, this theory shall have two nontrivial renormalizations, as otherwise the critical exponent v* is a simple algebraic function in D and d which certainly is wrong. Now suppose that the Hamiltonian has the form 7-/= ~ 0 4- 7"~int, where 7-/0 is the free Hamiltonian and 7"~int the interacting part. 7-[0 is 7"/0 = [ 4) (r) K (r - r')q~ (r'), dr, r t
(4.67)
with some kernel K (r - r'). First, in order to have a nontrivial renormalization, K (r - r') has to be local, for reasons similar to what we discussed in Section 3.9. The first possibility, K (r - r') = 3 d (r - r') gives a trivial theory, so the simplest nontrivial choice is K (r - r') -- 3d (r -- r ' ) ( - A ) n, with n > 1 integer, leading for the field 4) to a canonical dimension of [4~]r = (2n - d)/2. (There may of course be an additional massive term.) The interaction can also be chosen local or nonlocal, where we do not consider 6-1ike interactions, by assumption. Nonlocal interactions of the form fr, r' ~)m(r) K' (r - r')q~m(r t) with a suitable kernel can be rendered local through an auxiliary field. However, since only local interactions renormalize, this fixes the interaction to have the form 7"~int "=
fr t~ m (r),
with
m =
dc2dc - 2"--~"
(4.68)
Only for special values of dc will m be integer and a nontrivial renormalization be possible. Thus it seems already impossible to construct a nontrivial scalar field theory with the correct upper critical dimension.
2 Polymerizedmembranes, a review 5
5.1
319
Proof of perturbative renormalizability
Introduction
In this section, we discuss the proof of perturbative renormalizability to all orders in perturbation theory as given by David et al. (1993a,b, 1994, 1997). (A pedagogic presentation of some of the ideas can also be found in a course by Guitter (1997), in French.) The proof itself is lengthy and rather involved. It is impossible to give more than a condensed version here. Since the proof stands in the context of methods developed for local field theory, as the 4~4-model, we start our discussion there. The first to construct a general theory of renormalization were Bogoliubov and Parasiuk (1957). They introduced what since then is called an R-operation, which subtracts the divergences from a given Feynman diagram. However, it still took more than 10 years before Hepp (1966) could prove rigorously that this operation indeed renders Feynman integrals not only finite, but even absolutely convergent. (This means that the integrals involved have no more nonintegrable divergences and can, e.g. be performed by numerical integration, assuming that one is looking at an Euclidean theory.) This was done by considering each ordering of the distances in the Feynman integrals, the since then so-called Hepp sectors, separately. Renormalization, however, demands more: a theory is said to be renormalizable if and only if divergences can be absorbed into a finite number of 'renormalized' (i.e. redefined) quantities, which are in general proportional to the original quantities, with a proportionality factor of Z, which may also be a matrix Zij to allow for mixing of the operators. It remains to show that the R-operation can indeed be interpreted as a multiplicative renormalization, i.e. to introducing Z-factors. This was most clearly demonstrated by Zimmermann (1969), who reformulated the R-operation in terms of forests, i.e. mutually disjoint or included sets. A lot of material can be found in Callan (1975); Collins (1984); Smimov (1991) and Rivasseau (1991). An equivalent formulation, which in some respects is technically more convenient uses nests. It is this formulation of the proof of perturbative renormalizability, introduced by Berg~re and Lam (1976), which finally has been generalized by David et al. (1993a,b, 1994, 1997) to polymerized tethered membranes. In order to do so, they had to overcome three major difficulties: first, the proof of Berg~re and Lam (1976) works in the Schwinger-proper-time formulation (a-parameter representation) after elimination of the momentum integrals. In the language of membranes, this is the special case of polymers, for which a general membrane diagram splits into topologically disjoint polymer diagrams. This allows the simplification that a Hepp sector is just an ordering of nearest-neighbour distances. Clearly, such an ordering is no longer possible for tethered membranes, leading to a more involved definition of Hepp sectors. Bergbre and Lam (1976) then construct what they call a 'tableau' in order to show the absolute convergence of
320
K.J. Wiese
the subtracted Feynman integrals in each sector separately. Also this construction gets more involved in the case of membranes as will be discussed below. The second difficulty was that membrane integrals over relative distances are distributions instead of simple integrals as in the case considered in Berg~re and Lam (1976). These two problems were first clarified in the context of a simplified model, namely the interaction of a phantom membrane with an impurity, i.e. a single 6-interaction (David et al., 1993a,b). Finally, for self-avoiding membranes, also bilocal (or more generally multilocal) counterterms as, e.g. the interaction ~d (r(x)--r(y)) had to be incorporated into the R-operation. This was achieved by introducing the multilocal operator product expansion (David et al., 1994, 1997), discussed in the preceding sections. Let us now proceed to the proof. In order to do so, we have to introduce some notations and definitions.
5.2
Proof
1. Definition: ,~', .A, I, I c~ [. Consider a general term in the perturbation expansion of the observable (_9(~1. . . . . Zm). The interaction is 6-like and [ is the number of points on which it depends, e.g. [ = 2 for the usual self-avoidance. Perturbation theory consists of N-fold integrals over the positions .,4 -- {s . . . . . s } of all of the N involved vertices, xl . . . . . xN, denoted by ,-Y := {xl . . . . . XN}. Note that we carefully distinguish the vertex xi from its position s Explicitly, the term has the form Io(-~l
-~,N) - - ( O ( Z l . . . . .
.....
Zm)(1)(.~l . . . .
)...
(1)( . . . .
-~N))o"
We also consider connected expectation values i conn,,--' (.9 l,X l , . .
.,
-~N)-
(O(Zl,..
.,
Zm)(1)(.~l . . . .
)..
.
(1)(..
.,
\conn -~N)/o 9
may be the ([ = 2)-point ~d-interaction, but the same formulation applies to ([ = 1)-, ([ = 3)- or higher point interactions and is unnecessary to be specified for the following.
2. Definition: Diagram, connected components. A diagram is a partition of any subset of 2' into disjoint subsets Ci, which we denote by 79 = {CI . . . . . Cn }. The elements of ? are called connected components. 3. Definition: Inclusion, intersection, and union of diagrams. A diagram ~1 is included in a diagram 792, 792 -< 792, if every connected component of ~1 is a subset of a connected component of ?2. Note that inclusion defines a halfordering on the set of diagrams.
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321
Fig. 13 Connected components (dashed lines) and f-interactions, solid lines. The resulting connected molecules are shaded in grey.
The intersection "Pl /~ "792 of two diagrams ~ l and 792 is the unique maximal -< Pl and 7~ -< 792 =~ 79 -< diagram 79 included in both Pl and 792 (i.e. u "jD1 A ~D2). The union 7'1 v ~2 of two diagrams/)l and 7:'2 is the unique minimal diagram which contains both ~1 and ~2 (i.e. u with 7:'1 -< P and 7:'2 -< 79 => 7:'1v ~ 2 -< P). 4. Definition: Root. A set C of points is rooted by choosing one of its elements o0 as root, and is denoted Ca, = {C, w}. Similar, a rooted diagram ~a, is a diagram where each connected components Ci is rooted ~a, = Ui{Ci, o)i }. 5. Definition: Connected Molecule. Consider an arbitrary diagram ~ with connected components Ci. Grouping together those connected components, which are linked together by a-interactions (see Fig. 13) yields a partition of the diagram into subdiagrams ~i. These subdiagrams 7:'/are called connected molecules.
6. Definition: Forest. A forest is a set ;~ = {Pl . . . . . Pro} of pairwise distinct connected molecules which each consist of at least one diagram with at least two points I~i[ >__ 2, and which are either disjoint or included, i.e. for all i g= j: 7)//x ~ j = {}, or 7)/ -< ~ j or 7:'/ >- ~ j . This inclusion defines a half-ordering for the diagrams in ~'. The empty set {} is also considered a forest. It is important to realize that this definition depends on the theory considered via the a-interactions; and may turn the analysis difficult. This is the reason why, later, a different formulation in terms of nests will be introduced which is independent of the underlying interaction (cf. example 62). 7. Definition: (Compatibly) rooted forest. A forest ~" is rooted, if its elements are rooted. It is called compatibly rooted and denoted ~'~, if for any two elements {'jOi, O)i }, {'Pj, O.)j } which are included, i.e. ~i -< ~ j , either wj -- o) i or o)j ~ "Pi. Note that any forest can be compatibly rooted.
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K.J. Wiese
8. Definition: Weight of a compatibly rooted forest. The weight of a compatibly rooted forest is defined as the inverse of the product over all roots of the number of points in the largest connected component, containing the root: W(~3) "= I-]`0i~i~e [max(lCjl, O.)i E Cj)] -1. This definition is chosen such that summing over all compatible rootings of a given (unrooted) forest equals unity, ~-~compatible rootings ~ W (~'~) -- I.
9. Definition: Characteristic function of a diagram. The characteristic function X (C) of the connected component C ~ 79 is defined as X (C) := I-Ixl,xmEC {~)(L Is - s I). The characteristic function of a diagram 7:' is defined as the product of the characteristic functions of its connected components: X (7~) := I-Ici~7:' X (Ci). 10. Definition: Dilation-and Taylor operator The Taylor operator TT:',o of a diagram 79,0 is defined as the sum over all divergent contributions in the shortdistance expansion, when contracting the connected components in 79,0 towards their roots, as given by the MOPE. If the divergence is logarithmic at the upper critical dimension, the latter is multiplied by the characteristic function X (T'oj). (Recall that only for logarithmic divergences, a cut-off is needed.) More specifically, we define a dilation operator DT:'~o, which contracts points in the same connected component towards their common root by
DT:,,o "~'m ~ ~'m (~-) :=
{ "~'`0i "F- X (Xm -- "~,0i) -~m
if Xm (F_ Ci,`0i if not 3i, Xm E Ci.
The MOPE then yields the short-distance behaviour .....
\ ~'~
......
a
which defines T ~ ), as discussed in Section 3.5", see also the examples in Section 3.6. All terms T ~ ) I which are superficially UV-divergent when integrating over the points in Ci,`0i but O.)i have to be subtracted. To this end, define O'max as - D times the number of integrations O'max : = - D ~ i (ICil - 1). The Taylor operator T7% of a rooted diagram 7',0 applied to 1 is then defined as (setting e =0) ~7< O'max
The last term gives rise to a logarithmic divergence at e = 0, thus demanding the introduction of an IR cut-off (regularization scale). Note that it may, as well as the other terms, be absent from TT:,,o. We also use the rule that a Taylor operator does not act on the IR cut-off. This is important when applying several Taylor operators successively. There is now the following important lemma.
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11. L e m m a : Factorization of Taylor operators in a forest. Taylor operators for diagrams that belong to the same forest factorize. 12. Proof: Let 791,792 ~ ~ be elements of the same forest. Then either T'I/x 792 = {} or without loss of generality Y'l -< 792. In the first case, this is a consequence of the MOPE. In the second case, this is a consequence of the MOPE and the fact that a Taylor operator does not act on the IR cut-off. We are now in a position to define the subtraction operation. 13. Definition: R-operation. The subtraction operator R is defined as the sum over all compatibly rooted forests ~ of the Taylor operator defined by this forest:
Since the empty set is also considered a forest, it contributes the identity. R could also be defined as the sum over all forests, then choosing an arbitrary compatible rooting. Also note that Taylor operators of forests that contain only superficially UV-convergent diagrams vanish identically, and can be excluded from the sum. We now state the central theorem of David et al. (1997). 14. Theorem: Renormalizability. (i) The renormalized integral f~
Rlo(.xl
.X'N)
I ..... -XN
is UV-finite at e = 0. (ii) The renormalized integral, which contributes to the connected expectation value of the observable O at nth order, o ( n ) (Zl . . . . .
Zm) := f Rl~~ ds ! ..... -~'N
.....
-~N),
with N = In, is UV-finite and IR-finite at e = 0.
(iii) In perturbation theory, the renormalized expectation value of an observable is given by OR(Zl .....
-b)n " t o ( n ) (Zl 9
Zm) " =
n!
"JR
.....
~m)
n=O
(iv) The subtraction operation R is equivalent to multiplicative renormalization, i.e. to introducing Z-factors in the standard way.
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K.J. Wiese
15. Proof: Multiplicative Renormalizability. Part (iv) of the above theorem is an immediate consequence of the forest structure of the diagrams. 16. To prove parts (i) to (iii) of the theorem, we first rewrite the subtraction operation R in terms of nests instead of forests. Then, by analysing the reformulated subtraction operation in 'Hepp sectors', i.e. within a given ordering of the distances, finiteness of the integrals will be proven. To do so, we need some more definitions. 17. Definition: Complete diagram, completion of a diagram. A complete diagram is a diagram that contains all points of ,t". In other words, it is a partition of k'. An arbitrary diagram 79 can be completed by adding for any of the not already included points xi the set {xi }. We shall sometimes use the same notation 7~ for this completed diagram, whenever confusion is impossible. The completion of the empty set is the union of all sets containing one of the points xi, and is (in the sloppy notation mentioned above) denoted 7'1} = {{xl}. . . . . {XN}}. We also note 7'2' = {{xl . . . . . XN}} = {X}. 18. Definition: Nest. A nest 92 is a set of l + 1 (with l < N) complete diagrams, that are strictly ordered by inclusion, and that contain 791}: 92 = { T o - 7'1} -< 791 -< 792 -< -.. -< 7~t}. The smallest possible nest thus is {7'1}} and not {}. Note that sometimes (David et al., 1997) nests are defined as strictly ordered sets of diagrams instead of complete diagrams as in David et al. (1993b), necessitating the notion of complete nests for our definition 18. In our treatment all elements of nests will be thought of as completed by definition 17. 19. Definition: Compatibly rooted nest, its weight. A compatibly rooted nest is a nest that is compatibly rooted by the definition (7) as for forests. Also the weight of a compatibly rooted nest is the same as for a compatibly rooted forest. Summing the weight over all compatible rootings as in definition 8 yields unity. We now reformulate the subtraction operation R as sum over nests instead of a sum over forests. 20. Lemma: R-operation in terms of nests. The R-operation defined in (5.2) can equivalently be written as the sum over all compatibly rooted nests,
R:=-~--~_W(92~) H (-TT:,,o). 91,
(5.3)
~0~9~
21. Proof: For every nest 92, one can define the associated forest ~'(92) as the union of all connected molecules with at least two elements, that build up the elements of 92. If the nest is compatibly rooted, also the forest will be compatibly rooted, and is denoted ~', (92,). This defines equivalence classes on the set of all
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Polymerized membranes, a review
325
nests. (An explicit example is given in example 62.) One then shows that for a given forest ~ e
E
I-I
9le, ~e(91e)=~e 7:'o~~91e
FI
7:',,,~~e
and that the weight factors inside the class ~e(92e) coincide. The final step is to remark that the sum over nests also contains the trivial diagram 7911, which accounts for the global minus sign in (5.3). 22. The analysis and proof of finiteness of the subtracted integrand RI is performed separately for each ordering of the distances in I, the 'Hepp sectors'. This construction avoids problems with 'overlapping divergences' known from standard scalar field theories and is a generalization of the construction in terms of 'tableaus' in Berg~re and Lam (1976), which reduces to the latter in the case of D = 1. We have to introduce some more definitions. 23. Definition: Hepp-Sector, spanning tree. The spanning tree of 2' = {xl . . . . . XN} is defined as follows: (i) Link the two vertices xi, xj 6 2" together that have the smallest mutual distance. This yields Xl ---- -t-(.~i - .~j), and fuses xi and xj into a compound. (ii) Repeat this procedure recursively, without forming a closed loop, i.e. link together points xi, xj ~ 2" that belong to different compounds, and that have the shortest mutual difference. This yields ~.k,k-- I . . . N - 1 . Further denote by T the incidence matrix of the above construction, i.e. the information about which vertices are linked together by which ~.k. The Hepp sector 7-{r associated with T is the part of the domain of integration .A for which the above construction leads to the same incidence matrix T. 24. Definition: Saturated nest. A saturated nest is a maximal nest of 2", i.e. a nest | that consists of N ordered complete diagrams and contains T'I} and 79A,, |
-- { sO --'P{} -< s l -< "'" -< S N-2 -< s N - I
='PA-'}.
(The upper indices are introduced for later convenience.) 25. Definition" Extended Hepp sector. To each saturated nest G is associated an extended Hepp sector ~ . Define for any diagram S the distance between its connected components Ci and C j as dij " : min (I.~ - .vI, x E Ci, y ~ C j). The minimal distance in a diagram S, dmin(,S) is then defined as the minimal distance between its connected components, dmin(,S) : : min(dij, with Ci, Cj S and Ci g: Cj). If S has only one connected component, we define dmin(S) : : ~ . The extended Hepp sector (or sector for short), 7-/~5, is finally defined as the set of coordinates s such that dmin(T~{}) < dmin(8 I) < ... < dmin(S I) <
326
K.J. Wiese
I
Fig. 14 Rooted union T vo., R (thick lines) of T (dashed lines) and Ro~ (thin solid lines) with roots to (squares).
9"" < dmin(PR') = cx:~. It is the disjoint union of Hepp sectors 7-/h spanned by trees Ti. Trees/~, that yield the same extended Hepp sector | = | form an equivalence class. Furthermore, the extended Hepp sectors themselves form a partition of the domain of integration .,4: If | # | then 7-/6 A 7-/6' = {}, and
U e ~ 6 = A. 26. Definition: nest
92+
=
Tree associated to a saturated rooted nest. A saturated rooted = P{} -< A/'~, -< -.. -< A/'~, -< -Al- O"1+1 -< .." -< NON_, 91+ I
naturally defines a spanning tree T as follows" when going from N ' ~ to M I+1 " "O91+1
'
two rooted connected components C~t)l., and Co~,. 2''2 of A/'~t, are fused into a single connected component. For the spanning tree choose ~./ := +((51,1 - ~5t,2). 27. Definition" Rooted union and subtraction of diagrams. The rooted union of a diagram T and a rooted diagram R,,, with roots w (Fig. 14) is defined as the unrooted diagram T vo~ ~ := [ T \ ( ~ \ w ) ] v 7~. The most intuitive way to understand this definition is to consider the connected components of T, and to modify them by moving the points of each connected component 7"r of 7~ into the connected component of T, where its root wi lies, or leave it unchanged, if its root is not in any connected component of T. This gives the connected components of T vo~ R. (Empty sets are eventually to be disregarded.) We also define the rooted subtraction T / , , , R as T/ojTZ := T \ ( R \ w ) . It is an unrooted diagram, which is obtained by replacing in T all points of R by their corresponding root. (This can be seen as the set-theoretical action of the subtraction operator TTzo~.) 28. We now introduce 'tableaus'. This is a general construction to group terms together, which cancel in a given Hepp sector. The result of this construction will
2
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Polymerized membranes, a review
sector I
~1 9
']i'l1
"'"
.
']i'I
.
.
.
.
.
,~N-I
.
.
9
9
"'"
9
9
.
.
.
9
9
9
9
Fig. 15 Tableau.
thus crucially depend on the Hepp sector 9 Some illustrating examples are given in Section 5.3, see examples 63-65. 29. Definition: Tableau 9Let |
= {S O = 791j -< S l < ... -< S N-I = T'x} be a
saturated nest and 7-(~ the associated generalized Hepp sector. Let further 92~ {790 = 7:'11 -< Pl -< ".. -< P r } a compatibly rooted nest with roots {w0. . . . . wr} (defining the Taylor operator T g ~ in the subtraction operation R) to which for convenience we add (the unrooted set) 79r+1 - 79x (even if it might already be present in 92~; for T'{] and P,t' see definition 17). We then define the tableau T~
as ~ I "-- ( S / V o ) j P j ) A T~j+ l .
(5.4)
We think of the tableau as a large matrix made out of diagrams, where the columns are numbered by the subtracting diagram 79j, J - 0 . . . . . T and the rows by the sector element S t, I = 0 . . . . . N - 1 (see Fig. 15). 30. L e m m a : Ordering of the tableau. The tableau defined in (5.4) is ordered, when reading it like this article (from left to fight and top to bottom). Formally, qF~ ~ ~ + 1 and T~ - l -< q[~3j+1. 31. Proof: The first of these inclusions follows directly from the definition. For the second one, we have ~ - 1 = (79,t, vo)j 79j)A79j+] = Px/xPj+l = P J + l = 79{1vo)j+l PJ+I -- qi'~ l , proving not only the inclusion, but even the equivalence. 32. This shows that the elements of the tableau form an (unrooted) nest 9l(| 92~) which contains 92 and which has many identical elements. We now look for all nests 92, that under the above construction yield the same ~ ( | 92~). The key idea for that construction is that complete lines may be removed from the tableau, if two vertically adjacent diagrams are equivalent; since then all elements of the
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K.J. Wiese
tableau in between them are also equivalent, and thus redundant. One then shows that this is equivalent to eliminating one element from 92~. We need the following lemma. 33. Lemma: Reduction of the tableau. If T~ - T~+ l, then eliminating all elements between these two members of the tableau (in the above discussed natural reading ordering) including one of them, is equivalent to constructing the tableau from 92~ := {T0 = 7:'{} -< T'] - < . . . ~ D j - I -< " P j + I -< " ' " -< '~DT}, which is obtained from 92. by omitting the element T'j. 34. Proof: This is proven in appendix E of David et al. (1993b). 35. This procedure can be repeated, until no longer do two vertically adjacent elements of the tableau coincide. There is another important lemma. 36. Lemma: Commutativity of the reduction operation. The reduction procedure as defined above is commutative, i.e. independent of the ordering. It therefore defines a unique rooted minimal nest 920, such that ~ ( | 9lo) = ~ ( O , 92.). Note that 920 never contains T'x, since the latter is added as element 7:'r+l to 92.. 37. Proof: This is also proven in David et al. (1993b, p. 617 ff). We can now define equivalence classes of nests. 38. Definition: Equivalence class of nests. The reduction procedure prescribed above defines equivalence classes on the set of all compatibly rooted nests (i.e. those appearing in the R-operation in (5.3)) by Co(92.) :=
39. Lemma: Characterization of Co(92~). Given a compatibly rooted nest 92~, then its equivalence class Co(92~) is the set of all compatibly rooted nests 92~ such that 92o C 92~ C ~ ( | 92o). 40. Proof:
(i) 92~ ~ Co(92~) ~ 920 C 92~ C ~ ( | 92o), is immediate: the second inclusion, 92~ C ~1(| 920) has been shown above, and the first one follows from the uniqueness of the reduction procedure to the minimal nest
(ii) 920 C 92~ C ~ ( 0 , 92~ :=~ 92~ ~ C o ( ~ ) . of David et al. (1993b). We now need the following combinatorial.
This is proven in appendix F
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Polymerized membranes, a review
41. L e m m a : Sum over weights. Let 92~ a compatibly rooted nest and 93I an unrooted nest which contains 92~. Then denote by 9Jl:~ D 92~ all rootings of 93t that are compatible with 92~. For the sum over these roots, we have w(~)
= w(9/~).
(5.5)
9J~ D9~,93t fixed
42. Proof: The idea of the proof is to construct all possible rootings of a nest 92 as follows: enumerate the vertices. In a given diagram, the root is always assigned to the vertex with the smallest number. Then check that the weight of such a configuration is given by the number of possible enumerations of the vertices that yields the same rooting, divided by 1921!, where 1921is the number of vertices in 92. The proof of the above lemma is then straightforward; applying the construction to 9Y~. (Details are given in appendix G of David et al., 1993b.) 43. As stated above, this allows us to rewrite the R-operation as sum over equivalence classes, and to reduce the analysis of the finiteness to that of the subtracted integrand within each equivalence class. To this aim start with l
RC~(9~)' 920~minimal w.r.t.
w(~r
FI (-T~)
To rewrite Rce(,y0~ ) in a factorized form, we first insert (5.5) with 931: = ~(|
cYSt), and then use the characterization of C@(92~) to g e t
R~(~,-:-
~
~ ~
w
w )
~
)H
(-T~)
FI (-T~).
This can still be rewritten as RC~(9~ ) = -
• 1-I
y~
w(~)
l-I
where we used the fact that the subtraction-operation for diagrams in 92~ is always performed, whereas a diagram in ~)t~\92 ~ can either be included or not, leading to the factor of (1 - T-p,).
K.J. Wiese
330
Since the equivalence class Tt(G, 9"~) can equivalently be characterized by a maximal nest 99~ with 9J~ - ~ ( | 93t~) - ~ ( | 920), we obtain the final result. 44. Theorem: Formulation of the R-operation as sum over equivalence-classes of nests. The R-operation can be written as
R =
~
W(ffYt~)R~.
9J~maximai w.r.t. G :---
where r
I-I
I-I
and 9J~ are minimal and maximal elements from the same equivalence
class ~ ( ~ , 9Ji:~) = r
910 ).
45. Since any term in the perturbative expansion is a sum over all sectors, using the above theorem, it is sufficient to show that R ~ I (Xl . . . . . XN), is finite in the sector | for all 93t~ maximal with respect to | This is done in three steps: first, one applies l ] p o e ~ (-TT:~. ) to I. This leads to a factorization of I into subdiagrams. One then shows that the remaining factors of (1 - Tp~) act independently on these subdiagrams. This is, as in iemma 11, a consequence of the MOPE. (For details, see David et al., 1993b.) Finally, one introduces tree variables which are appropriate for the sector; finiteness is proven in terms of these variables. 46. Definition" Construction of a spanning tree. First of all, note that in general a tree associated with the sector ~ does not take into account the factorization introduced through l-IT~eg~ ( - T . / ~ ) . To construct such a tree, we proceed as follows: consider the tableau, which is constructed from the sector G and the minimal nest 91~, and thus can no further be reduced. Each of its elements is contained in 9Y~, thus the rooting of 99/7e can be transferred to the tableau. We have also seen above in proof 31 that the first element of each line, ~ 92~. Each line thus encodes what happens inside one of the 19201 factorized (rooted) subdiagrams 7)j "-- qI'~ qI'~ A spanning tree for the connected components Pj,j of 7:3j is constructed as follows: consider PJ, j "= S I A Pj,j. The s e t ~ J , j
:=
UI, #!j.j##I.-~l #lj,j
is a saturated nest for the points in Pj,j,
with a rooting induced by ~Y~,, and thus defines a spanning tree on Pj,j, using definition 26. Its elements are denoted by ~.1J,j . the tree is ordered with increasing indices I. Also denote the union of the trees for 75j as G j "= Uj ~J,j. The spanning tree defined by ~.tJ,j will be used for the integration in the following. Note that this tree contains exactly N - 1 elements, i.e. not all combinations of I
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Polymerized membranes, a review
331
and J, j contribute; there may be elements ~.$,j with the same sector index I, but then, necessarily the diagram J or the connected component j are different. 47. L e m m a : Inequality for the distances in the spanning tree. The following inequalities hold for the length of the elements of the spanning tree 1 - <
I~Iy,j I
~j,j I'1
and
-
t < I
Vl < I .
~J',J' I -
48. Proof: Consider a tree T, and the corresponding generalized sector | = {,SO. . . . . S m-I }. Then the distance between the I elements of S I-1 is smaller than the shortest distance between any element of S I - 1 and the additional element of S / . Denote by ~.i, i = 1 . . . I - l the elements of a spanning tree for ,31-1 and by ~./the shortest junction between a point in S / - l and the additional point of S / . Then a vector connecting an arbitrarily chosen point in S 1-1 with the additional element of S 1 can be written as ~. = ~.1 + Y~.i ~.i, where the sum goes over at least 1 - l elements. Using the triangle inequality, ~. is bounded by I,~1 _< I,~#1 + Zi I~.il _< ll~,ll. -'I Since any ~,j,j can be realized as restriction of a spanning tree for the sector G, the first couple of inequalities is proven. The second one then follows immediately from the nested structure of G. I 49. Definition" Index sets. The index set Ind := {I, 3J, j with )<j,j ~ Gj,j} = {ll < 12 < ... < IE} is defined as the set of all indices I, such that there is a j,j in the spanning tree. We also define the index set of the subdiagram 7:'j as 1 ~ | I n d ( 7 5 j ) - = {I, =lj with )~j,j
50. Definition: Integration variables. The tree variables defined in 46 are now used to perform the integration. To give explicit bounds, they are still reformulated as follows" first, choose for all I ~ Ind a representative 71 := 7<1 Jo,jo"
1 Second, for all ~.j,j I with (J , j ) # (Jo, jo) define Xj,j
.=
i~Ij,jl/I ~.11. According
I -< I 9Third, d e f i n e ~ I := I~1 I/ 1~1+1 I, I is bounded" 1/I _< Xj,j to lemma 47, Xj,j
and set ~.#+ l . _ ~.I, if ~.I+ 1 would be undefined otherwise. (This happens since Ind does not contain all indices I < N.) fll thus measures the ratio of successive representatives ~t with I ~ Ind. According to lemma 47, fit is bounded by 0_
I-I
/~Ind
1-I
J=0
I-I
j,s.t.#j.jEf~j
9
K.J. Wiese
332
This will be rewritten in terms of reduced variables fl 1, Xj,jl and angles still need the following notation.
~'2IJ,j.
We
52. Definition: Cumulated level. The cumulated level of a subdiagram 75j, I' E(l, T~j) is defined as the number of ~.j,j with I t < l, and J fixed. It counts the number of integrations over variables with an index smaller than or equal to I in the subdiagram 7~j. Replacing recursively the integration variables ~.I by/51, the cumulated level E(l, ~ j ) yields the power of/~1 from the integration measure as
(~I)DE(I,~j). We also define the completed cumulated level of a diagram 75j, E(l, 75j) as the number of integration variables ~.~', j, with I' < I, where J ' is either J or 75j, is a proper subdiagram of 75j. 53. Lemma: Reformulation of the measure. The measure in 51 can equivalently be written as (with A (I, J, j) to be discussed later)
d/~t I
d
I
-~
I J=O
I-I
(i~l)DE(l,fgJ)
/Elnd(75j)
I •
I-I f dSI,J
~.Ij,jC-~J
A(I,J,j)
FI ~.tj,j#xt
t and/~I are only bounds, and not Note that the limits for the integration over Xj,j all of it contributes. Due to these bounds, the integral over Xj,t j is always finite. The same remark applies to the integration over/~ t, with the important difference that we have to check the limit of small/~ t. This is the purpose of the following. 54. Integration over angular and relative variables.
The angular integration
f d~/j
does not induce any divergence, as long as D > N - 1. Since short-distance divergences (which are physical) and small-angle divergences (which come from the measure) are disentangled, we do not expect the latter to destroy the proof of perturbative renormalizability. A proof sensu stricto has not been given in the case of multilocal theories. It has, however, been checked at second order (Wiese and David, 1997). To avoid unnecessary complications, we will not specify the angular integrations any further and simply denote it for each t is always finite, and the power subdiagram by Also the integration o v e r Xj,j A (I, J, j) need not be specified. Analogously to the angular integration, we shall
f~.
denote this finite integral, including the factors of
( ! ) A(I'J'j) X j,j , as fxJ"
2
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Polymerized membranes, a review
We can now give bounds on each subdiagram separately. 55. Lemma: Bound for a subdiagram. Using theorem 44, we can write the renormalized manifold integral as a product over the subdiagrams 79j in the form t
19~,l-I
1-I 71-I f
t~lnd
d/3 /
ic
79j
(~/I '
filE),
9
(5.7)
J=O
0
where the subdiagram associated with 75j and C is i c (~1, . . , filE) . : 7~j ,"
H
(ill)DE(I,75j )+g' 6-Y]i[Bi ]
I (51n d ( ~ j )
,pj
j,j(fl ' , Xj,j, '
.'j,j) t] 9
(5.8)
i c (~I)is the MOPE coefficient, when contracting the points inside 75j and PJ J'J extracting from the MOPE the term proportional to C. Eventually TT:,j yields more than one counterterm proportional to a relevant or marginal operator C. In that case, we have to sum over these contributions. Since the following arguments work for each term separately, we do not explicitly write down this summation. The operators Bi are insertions into 79j from subdiagrams. [Bi] is their naive scaling dimension; relevant operators have a positive dimension, marginal operators dimension zero, and only such terms are included in the subtraction prescription. As for C, we only consider the insertion of a single operator from each subdiagram, g' is a topological number, the number of 'loops' or equivalently the number of interactions contracted in the subdiagrams of P j , and thus always positive. If we denote Imax(75j) := max ( I 6 Ind(75j))
'
then I 79j c (/3" ' . . . . /3rE) is
bounded by (/3'm"xOSJ)) ee-[cl, g is the total number of 'loops' in x
r
and its
/
subdiagrams, so ~ > g'. Since this term is positive and vanishes in the limit of e ~ O, we shall not specify ~. For I
< Imax(~J),
Ic
T~j
(fill
'
....
ill[:) is regular and bounded
by
(~I) 6 with
3>0. 56. Proof:
First consider a subdiagram 7~j with no proper subdiagrams, and
334
K. J. Wiese
I
]c (~t l s2t pj , XJ,j, J,j)
:--
H
(~l)DX(l'~J)+e'e-~-'ilBil
/ elnd(~j)
(,
x 1C ~ j , j ( ~ l 7:'j
J,J'
j,j)
.
One first checks that the limit of small/~t is equivalent to applying the dilation operator defined in 10 to all points in S t f3 73j, with a root given by the nest 9Yt~. The Taylor operator Ts/n75j with the same root uses exactly the same dilation operator to determine the terms that have to be subtracted. The limit of small /~t will thus be finite, if one of the factors (1 - TT:,~) in (5.8), when applied to the subdiagram 75j, reduces to (1 - Tsln75j) with the roots induced by 9Y~. However, the latter is a consequence of the construction of the maximal nest ffYr Therefore, in the limit of small/~t ' [ T'j c (/~t , Xj,j, t f2tj,j) will behave as 0((~1) ~) with 8 > 0. (One can furthermore show that for self-avoiding tethered membranes in the limit of e ~ 0, 3 = min(2 - D, D).) Note that with the above rescaling, the argument is independently applied to all variables/~t, with I
First, lemma 55 is applied to each factor I c (~1, 79j
,'"
., fltL.). We
only have to check that the product of all powers of fit is positive. First of all, the p o w e r s (/~Imax(~J))-[C] are cancelled at each level, since the operator C is at the same level I = Imax(73j) inserted into another diagram, thus there appears a factor (i~lmax(~J))+[Bil with Bi -- C. The remaining factors have the form (fit)& or even (fit)~ (with e and ~ as defined in 55 and 56), which are finite for e > 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
Polymerized membranes, a review
(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~
~
-<
o..o. | o. 9 , o. -o..
I
9 9 O. e
,
'" "'1 :ei:e o. ..o.
with a relative minus sign with respect to r to 923 is the same as the contribution of ~.
63. E x a m p l e : Tableau construction, convergent subdiagrams. As an example consider the sector O ie:'
..o;ie,' o.: o . : ' io o.:
"
The subtraction shall be given by the nest
+ :xe
which we have chosen maximal.
~
'e: '
"
O.:
'
Roots are marked by crosses.
The tableau
2
Polymerized
membranes,
337
a review
defined in definition 5.4 is sector ),
:0..o. ....
....
~,i~.~
.'..'.~
.~,~o.
:o-o:
:of-,:
:01e
i~
o o"
~01~ol
.o. ~e:
0 ~01:
e9 o -
o..
"~O....o.
,O.. o :
o9. . . o l
o.
qi9 'e..
.-o ' o .
o
:~e. o.,
...9 o ./
: ~o.
;o
.o. .
'~
:o .o
o~ 9 o-
.
,1
:o: .:,: : e - 0:
reduction :~ 0. .
:o..o.
:.ol?o..
:j io-:
-o 0 . :o ..o,
oo~ ~.o.o.
:~ .o.-
:e
0:
e 0. I.O
O:
The only reducible element is ",x.." O, ~ corresponding to the global subtraction. Note that this element always has to be reducible, since in any sector the global divergence has to be subtracted. 91(G, 91e) - 91 and the equivalence class CG(91e) consists of three elements, namely
c~~-ii~
....
'
'
~
:x 4: ~ :,~0: 9 ~x~ x~'l I ....... :~x 0 . '
:X-o:
'~ :x ' o .
~ " " 1o. I
~ :,~-0: ~
'* . o . o . !
Its minimal generating nest is 9 1 o = [ ::x:,::~: :........ ::~i-.:} ,, o..' :'x $: '
9
Concerning the subtraction operator R, this is the situation, where the counterterms (given by 91.) belong to sectors not present in | and are therefore finite in | This leads in theorem 44 to two terms
"1, ......
"1..
, . . I - ( ' - ' ) ' '
,,,j-(l-,.-),.,~..
which are just distinguished by their roots.
....,
338
K.J. Wiese
64.Example:
Tableau construction, (maximally) divergent subdiagrams. In this example, we study the opposite case, where all (but the trivial diagram) are reducible. Let (~5 : ~
and
. o . . e ; ' .e. i~i:' e;!e. ,:' 9 .o. 1'~'~ ''~ ' ' ' ' 1
92e:=[x~' ,~ ' ,,, ' ,x ,O..}
"
Then the tableau is sector ~0 ,
reduction
o:,i
),
,~o
0:01
e-0:
-o-e
.o.. .iol
.e' .e.
-o-..o. iO.;-0,
.o-. -'0: .O 0:
.0"..'0. .O. :'O.
'0"..'0". .O.~.O..'
09 0. -.o.- -o.
.0..'0. -o.. 9
-0. O .o.. .o..
o
e o .o.ol
.o o". o.o.
oo o
9
o 0". O.O
,., o O:
,..o.: -o ~0
,.,. ) o...o..
. - ," .o 0.:
(
o" 9 o.
It is (maximally) reducible, such that the minimal generating nest is
and C ~ ( ~ )
is given by (see classification lemma 39):
I xx}
{ o", e , ' ,0,,-el' '" "" o,,o,-, o"'1 e
C
"
It has 23 = 8 elements, as long as the roots are not specified. Concerning the subtraction operator R, this is the situation, where all subdiagrams in ~ belong to a subdivergence of ~5. In theorem 44 the contributions for different rootings belong to different maximal nests. In the case of 99~E ) _
xi~x
.<
9o,.
x ,o:
"<
O..e ....
o.
"< x
}
'
the subtraction operator R ~ e is
Note that the global minus sign in (5.6) has been cancelled against the factor (-T..) = - 1 , and that there are seven other possible maximal nests, and factorized subtraction operations.
2
339
Polymerized membranes, a review
65. Example: Tableau construction, single subdivergence. In this example, we study the case of a single subdivergence. Let as before
(~
+o+,.:e=' ,o.. O:'
"--
.~ .o.: ' ".O.o.:
'
but .e.
[ ' '"
"
:e.e:-
:e+,: |
Then the tableau is sector >
.,,..:0: ,. :,::
.,...:,:
:.+.:,::
..,.. +::
o.. :,::
..o... :,:,.
o~ :,:.
:+0. 0: o ,.
. . + o :. . : ,:;:.
...o:. ~..:. ::,:.
i 0. o e/
o o.. o o
e 0: o..0. i
+o o.. ,o. o..
. o o.
.
r
.
.
,.. :,.:.
reduction
.
+++.g:
!
oo: 9 +,
,...:. .:. , .
:.....
....
9O
O .. . :
o + 0.:
.....: :,:
69 o
: 0 .0: o.: :O.:
.O
O:
) \
The minimal generating nest is
~o and 9l(|
i~:+ .. I ,
'
..
..
,
91~) - 9l. Let us explicitly write down the six elements of C~(91~)"
I I
..... .~"~. ""1 . . { ~x~~+,,'"x~ "" '
.
'
.~
,
.
l,
.... 9 '
.
~ .
,
.
'
x ... ",x ' ~. ' .
'
.
' :.•
.
. ' .
"'"1,' .x
9
o...y.
'
o
The simple subdivergence occurs, when in the subdiagram x~::i(:. the distances 0 .
,
~.:: are contracted. The last four elements of C~(91~) subtract the global divergence, the first and second two of them are distinguished by their rooting. In
340
K.J. Wiese
theorem 44, the corresponding subtraction operators are written as
.i
o
.I
o
(,-'r..)(,-,,,..),,-.. (,-,r..)(,-'r..)+..
i: :o I :
66. Example: Tableau construction, a small change. In this example, we give a variant of example 65. We will finally obtain a maximal nest, of which some elements contain the minimal nest of example 65, but whose minimal nest is different. Let as before (~i :----
, ,i.
o:
o.?
9
..
9o'
O o'
..
l
9 o.!J '
and 9~
.=[~x~x~,~x-,}.
Then the tableau is sector .......
I (::~.
reduction
o oi~
:o e: :.,e:
oi li:
iO
.. o
:07-?
:o e o, 0
e 0 .: .i
e.e:-
. i
.? .?
. :el,
:O 0
9 el
..+ .:o+
. : .ie~
o9 0O:
o~i.
0!)
:o:~ie:
.o
o:
9
O..
The minimal generating nest of the tableau is
and -
92(|
[...i~...~ ..,
..,~.,+~,
..I
. .
.
/
341
2 Polymerized membranes, a review
Let us explicitly write down the seven elements of C~ (92~):
~ x- I I ~84 ,-, . . }
co~~-I{,
{ ~~~~~" ..... " ' " / 9 '
,,
'
..'
'
~
O.
x
iO.,x'
I ~,~, ' . , x,'-,, "}.
I ~~" ........... ~'1
'
'
'
,.,
'
,.
~
'
:O
XI
.o.
'
'
:e
"
In theorem 44, there are four contributions to the subtraction operator, corresponding to four different maximal nests 93l~:
"1 .
6
.....i('-~)('-~) ~ . . . . . . ~= ( , - ~ . . ) ( , - ~ . . ) ~ .
Calculations at t w o - l o o p order
In the introduction, we had mentioned that the first calculations using some criterion to fix the expansion point at one-loop order were performed in Hwa (1990). The result of df ~, 3.5 for membranes in three dimensions even violated the geometric bound of 3 discussed above. It became therefore necessary to perform two-loop calculations, not only to test the proof of perturbative renormalizability, but also to obtain more reliable values for the fractal dimension. This is the aim of this section.
6.1
The two-loop counterterms in the MS scheme
In this section we apply the formalism explained in Sections 3 and 4 to determine the counterterms which renormalize the theory at second order (Wiese and David, 1997; David and Wiese, 1996). If we consider the bare theory, given by the Hamiltonian (3.1) when setting Z = Z b = 1, power counting gives three UVdivergent diagrams (together with their weights): 1 (b) = -
::~
:.;
342
K.J. Wiese
which give short distance singularities when the points inside the subsets are contracted to a single point. These singularities give double and single poles at e = 0. There are two other potentially dangerous diagrams:
~i~i
'i~~,
9
'~;.
--.
(6.2)
These diagrams do not give n e w poles at e = 0 for reasons similar to what happens with diagram (3.92). Now one has to remember that the model is already renormalized at one-loop order, i.e. that we use the renormalized Hamiltonian (3.93), with the counterterms (3.112) and (3.113). As a consequence there are five additional divergent diagrams, which come from the insertion of the one-loop counterterms:
@/E._I -....
1 - 2 ~ ,:4~;, (h)= ~u
+
2
(6.3)
There are other potentially divergent diagrams, analogous to those depicted in (6.2), which factorize into convergent diagrams. The first four terms in (6.3) are a combination of a diagram divergent at oneloop order (giving a single pole) times a divergent one-loop counterterm (which gives another single pole). The fifth term is more peculiar: it is the combination of a convergent diagram (which corresponds to a contact term) times two one-loop counterterms (thus giving also a double pole). Owing to the MOPE, diagrams (a), (e), ( f ) and (h) give a divergence proportional to the insertion of the local operator +. They can be subtracted by adding a counterterm proportional to the divergent part of the integral of the corresponding MOPE coefficients .
2
343
Polymerized membranes, a review
-~t--)e_i ) 2 9 (6.4) Since we use the minimal subtraction scheme, we want to subtract only the double and single poles in e at e - 0. To isolate these poles, we have to perform a -2 Laurent expansion of the various terms in (6.4) and to keep the terms of order e and e - l but to drop the analytical part. Setting the renormalization momentum scale/z = L -l , we obtain the final expression for the renormalization factor Z at two-loop order Z-I
- b(2-D)
+~
1 ( : ~ +)[-'F)e~ .....
....
~,
+
_ + b2 ( 2 - D ) I
(("~")e-') ....
~
+
-2,
E-I
+ O(b3)"
(6.5)
Here (I)e"~ .....e"p denotes the sum of the terms of order 8 nl 8 np in the Laurent expansion of (I)L, taken at L = 1. Similarly, the diagrams (b), (c), (e) and ( f ) give a divergence proportional to the bilocal operator - -. An analogous analysis leads to the following expression for the coupling-constant renormalization factor Zb at two-loop order . . . . .
-I
-2~-!
(6.6)
6.2 Leading divergences and constraint from renormalizability Renormalizability, once established, completely determines the terms of order b2/e 2 in the renormalization-factors, since the RG-functions have to be finite. In
344
K.J. Wiese
our case the following modified Z-factors fulfil this requirement up to order b2: zr
-b(2 -D)(:~,
+}e-,
-.}e-v(d, + 2 ) ( . ~ .
x[l+~b(({~}l,
x [ l + ~ (2({i:~ii,1..
vd(~
")e-'-
+)e-,)]
(6.7)
+)e-')] "
(6.8)
This can be verified by calculating the renormalization-group functions/3(b) and at order 2. Moreover, these functions are identical to their one-loop counterpart. It is therefore useful to factorize the Z-factors loopwise (with respect to the RG-functions) through the ansatz:
v(b)
Z = Z (1) • Z (2) x . . .
Zb = Z~') x Z(o2) • ....
(6.9) (6.10)
If our considerations are correct, Z (2) and Z~2) do not contain terms of order b 2/~,2, but only terms of order b 2/~'. Moreover, by construction they just contain the two-loop contributions to the RG-functions. Explicitly, from (6.5) and (6.7), we obtain for Z C2) the expression Z (2) = 1 + b 2 ( 2 - D)
~
+
-2,e-~
-(Q +)_, ...
+2
.....
+1],
2
(6.11)
where we already used the fact that/%,)]+~0 = 0 (see (3.104)). Finally, one has to remember that (... I...)ep are the terms of order of the Laurent series of the integral over distances of the corresponding MOPE coefficient
eP
<.......,t
= distances< f L
(...I...).
(6.12)
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Polymerized membranes, a review
345
Using this fact one obtains the decomposition Z (2)
(6.13)
= 1 + b 2 (2 - D) [f~ + f 2 + 73 + 0 ( ~ ~
where in the next subsection, each term will be shown to be expressible as 1/e times a convergent integral. These terms are:
(6.14)
~ ~(Q+I. 1
x
(/o:~ i+/, + .2_ 0.., + 2.
(o +l ...
(6.15) 1
.3 ~(ol+)~.t<~:~:l:
:/,-{~~~.l-
:/~.t
.6.6.
Even if not written explicitly, we will only calculate the residue of .~l . . . . . Z'3 at L -- 1. Similarly, dividing (6.6) by (6.8) we obtain for Z (2)
4
3
..,,
]
-2~-I
(6.17)
that we decompose as Z ( 2 ) = 1-+- b 2 [CI 4-C2 4-C3 4- O(6~
(6.18)
346
K. J. Wiese
with ,--
--,
2
(6.19)
o),.+ (0
o>,.
(6.20)
(6.21) The coefficients C1, C2 and C3 as well as .T'l, .T'2 and .T'3 will be shown to be 1/e times a convergent integral in the next section.
6.3 Absence of double poles in the two-loop diagrams The simplest case is the diagram CI in (6.19). A subdivergence occurs when two dipoles are contracted to a single dipole. When this contraction is performed first, the MOPE coefficient factorizes as _-
-_)~
({~}1=
-_)C~},I~.
--).
(6.22)
There are three different subdivergences and one finally obtains that the double pole associated with this diagram is given by l
1
The factor of 1/2 comes from the nested integration (Duplantier et al., 1990): the double pole results from the integration over a 'sector' where the distances inside the subdiagram are smaller than all other distances. Note that by subtracting all subdivergences in each individual sector, the only remaining divergence is the integral over the global scale. Using the procedure of Section 4.3 it is expressed as l/e times a convergent integral. Similarly, let us consider the diagram C2 in (6.20). A subdivergence occurs when the single dipole to the fight of the diagram is contracted to a point. The MOPE coefficient factorizes as
o)
624,
2
347
Polymerized membranes, a review
Consequently, the double pole for this diagram is
({~i~1
=
=)~-2=2
.... + _,({~i~:i,1=
--)~_,, (6.25)
where the factor of 1/2 again comes from the nested integration. Finally, let us consider the term .F-1 in (6.14). Four sectors contribute to the double pole of its first term, which correspond to the subcontractions depicted here:
(~i~]+)
~ ({~}t=
=)(~
+)"
(6.26)
Each of the contractions appears with a combinatorial factor of two. The double pole for this diagram is therefore
1
Thus, .~'1 is also finite. For the remaining terms .~'2 and ~'3, we have to show, that the singular contributions in the bracket cancel. To this aim, we make use of the equation of motion to compute the effect of the insertion of the operator + in the counterterms. In Exercise 5 (Appendix H), the reader can show that (Wiese and David, 1997) 2
+O(e ~
i
(6.29)
Using these identities implies that also ~2 and C3 only contain a single pole.
6.4 Evaluation of the two-loop diagrams The evaluation of ~1, ~e, ~3, CI, C2, C3 is only for D = 1 analytically amenable. For all other dimensions, the convergent integrals have to be calculated numerically. These calculations are challenged by a couple of difficulties: first of all integrable subdivergences have to be eliminated by suitable variable transformations.
348
K.J. Wiese
Two types of subdivergences exist: divergences for small distances (as predicted by the MOPE) and divergences for small volume of the span of the integration points, arising from the D-dimensional measure. They require separate, while dependent variable transformations. The result of these procedures are convergent integrals over up to five variables, which have to be integrated numerically. Since in addition, these integrals are strongly peaked (up to about 104 times the average) only specialized integration routines can handle them. We implemented an adaptive Monte Carlo integration. For details of the numerical integrations, which took about 103 h CPU time on a workstation, we refer the interested reader to Wiese and David (1997).
6.5
RG functions at two-loop order
For completeness, we still give the explicit form of the renormalization group functions at two-loop order:
,6(b) = - e b +
2-D +2e v(b) =
2
[
F ( 22___~O)o +
(C~ + C2 + C3) +
1
2
4D
]
d (f'l + f'2 + .F3) b 3 + O(b 4) (6.30)
1 + ~--~b + 2e (.~'l + .~2 + .~'3) b 2 + O(b 3)
]
9
(6.31)
Solving fl(b*) = 0 for b* and insertion of b* into (6.31) then yields v*(D, e). We shall discuss this function in the next section.
7
7.1
Extracting the physical informations: extrapolations
The problem
In Section 3.1, we have seen that the coupling b for self-avoidance has dimension e = 2 D - ((2 - D ) / 2 ) d . We have then performed a perturbative expansion in b to order 1 (Section 3.8) and 2 (Section 6). As in standard q~n-theory, the/3-function has a nontrivial fixed point b* with/3(b*) = 0 and positive slope, which thus is IR-attractive and governs the behaviour of the membrane at large distance. We have seen in (3.1 17) that b* is of order e, such that the perturbative expansion of the scaling function v(b) can be written as a perturbative expansion in e of the form v*(D, e) -- 2 - D + v l ( D ) e + v2(D)e 2 + O(e3). (7.1) 2
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Polymerized
membranes,
349
a review
0.14 0.12
r '----
~-
-
i
-'--.
0.08 ~ 0.06
-
"-.
i
,. , .,,
". " " \...
0.02 t
-~----- - - , . , , . . . . . . , . . , ~ , . . ~ - , - , . ~ _ ~
L
0.00
.....
1.0
"
1.1
'
1.2
9
1.3
..
- '""
1.4
1.5
1.6
1.7
~ ''~
1.8
\
~
-"~t_.~__-"-~
1.9
2.0
D
Fig. 16 The functions v I (D) (dashed line)and v2(D) (solid line). The latter is given with the statistical error from the numerical integration.
Let us mention here that similar expansions are valid for other exponents and other models, but let us focus on v*(D, e), whose coefficients are plotted in Fig. 16. Our goal is to obtain informations about self-avoiding polymers and membranes, and to this aim, we can expand about any point on the critical curve e(D, d) = 0 (see Fig. 12, Section 3.1). For polymers, one usually expands about the point (D = 1, d = 4), i.e. for D -- 1 fixed. For membranes, naively setting D = 2 and any finite d as, e.g. d = 3, i.e. e = 4 in (7.1), yields the wrong result 0. To obtain the correct result, the general idea is of course to expand about some point (Do, do) on the critical curve e(Do, do) = 0. If the function v*(D, e) were known exactly, the result of this expansion is expected to converge towards v*(D, e), but eventually only in some part of the possible range of D in 0 < D < 2, and possibly only after resummation. However, only one or two terms of the series in e are available, such that the result will depend on the expansion point and on the expansion parameters. The simplest scheme is to extrapolate towards the physical theories for D = 1, 2 and d -- 2, 3 . . . . . using the expansion parameters D - Do and d - do. However, this set of expansion parameters is not optimal, and better results are obtained by using combinations involving De(d) = 2 d / ( 4 + d) and e(D, d) = 2 D - ( ( 2 - D)/2)d. Furthermore, it is advantageous to make expansions for quantities such as v*d or v*(d + 2) rather than v*, as we will show below. The latter are systematic expansions about the Flory and variational (mean-field) results. Historically, the first such extrapolation was introduced in Hwa (1990). A systematic treatment was developed and tested in Wiese and David (1997), where also the expansions about the mean-field and Flory results were first introduced.
350
K.J. Wiese
In the meantime, these methods have been proven useful in Bowick and Guitter (1997); Bowick and Travesset (1999a,b); Le Doussal and Wiese (1998); Wiese and Le Doussal (1999) and Wiese and Kardar (1998a,b). Let us now turn to the general analysis.
7.2
General remarks about extrapolations and the choice of variables
As already stressed, the t-expansion given by (7.1) cannot be used directly for membranes, as it can for polymers, since directly setting D -- 2 in the t-expansion for v* gives a trivial, but absurd, result, since all the terms Vn(2) of the t-expansion vanish! Moreover, when D = 2, e = 4 irrespective of the value of d. This simply means that the point D = 2, e = 0 (which corresponds to dc = oo) is a singular point and that it is not possible to perform a direct e-expansion about it. To define an expansion about any point on the critical curve (D0, do), note that the expansion (7.1) is exact in D and of order n in e. For the example of a two-loop calculation, on which we will focus here, n = 2. Thus v*(D, e) can be expanded up to order 2 both in D - Do and e. Then one can change the extrapolation path through any invertible transformation {x, y} = {x(D, e), y(D, e)} and re-express D and e as functions of x and y up to order n in x and y around the point (x0, y0) on the critical curve, yielding
v*(D, e) = fi*(x, y) = v0.0(x0,-* Y0) + Axv~,0(x0, Y0) + Aye*0,1(x0, Yo)
"+-I(Ax)2~,0(X0,Y0) + AxAy~)~,I(xO, YO) ~-I(Ay)2p;,z(X0, Y0)q-... Ax = x -- x0,
Ay = y -- Y0.
(7.2)
The goal is to find an optimal choice of variables {x, y }. The guidelines for such a choice are the following: (i) the estimate for v* should depend as little as possible on the choice of the expansion point on the critical curve; (ii) it should reproduce well the known result for polymers (D = 1); (iii) for membranes (D = 2) the limit d ~ c~ should be nonsingular and in agreement with large-d results. This demand turns out to be quite stringent. Finally, one must choose a resummation procedure to extrapolate v* from the knowledge of the series (7.2) up to order n. Since only a few terms are available and since insufficient knowledge is available on the large-order behaviour of these series or on the analytical structure of the resummed series, sophisticated resummation methods as Borel transforms cannot be used. The series therefore has to be summed and boldly truncated at order 2. Let us now discuss possible extrapolation variables. Statements about their goodness always apply to the only available two-loop calculations (David and
2
Polymerized membranes, a review
351
Wiese, 1996; Wiese and David, 1997), but are partially transferable to other cases (e.g. Le Doussal and Wiese (1998); Wiese and Le Doussal (1999); Wiese and Kardar (1998a,b); Bowick and Guitter (1997); Bowick and Travesset (1999a,b)). {D, e}: this works well for polymers (D = 1), since both at one- and twoloop order we get results which are quite stable with respect to Do, but no prediction is possible for membranes. {D,d}: similar to {D,e}. {D, Dc(d)}: recall that one of the problems of the expansion in d is, that for D ~ 2, dc (D) ~ oo and therefore d - dc (D) becomes large. (In general dc(D) ~" 1 / ( 2 - D).) However, by using the above variables, one maps the region of possible values of (D, d) onto the square [0, 2] 2, and the critical curve Do, dc(Do) becomes a straight line. In general, these variables work well, both for polymers and membranes. {e, Dc(d)}: another promising method is the expansion in e and Dc(d). This expansion is also regular for D ~ 2 and is perhaps more in the spirit of an e-expansion. Some examples are given for polymers in Fig. 17 and for membranes in Fig. 18 (using one more reformulation of the perturbation expansion as discussed below). In general, one observes a plateau for D ~ 1.5, and the exponent is evaluated at this plateau, since there, the result depends the least on the expansion point ('minimal sensitivity method'). The broadness of the plateau gives a measure of the goodness of the extrapolation variables, and of the result. Another criterion, which evidently can only be used at two- or higher loop order, is to demand that the results of the one- and two-loop extrapolations (or higher if available) agree. In general, this methods yields comparable results to the minimal sensitivity method. It has also been checked in Wiese and David (1997) that these schemes work well for a toy model, i.e. when expanding the result obtained by a variational approach War = 2 D / d in x and y, by going to higher orders. In that case, the series becomes convergent,* but only in a part of the interval 0 < D < 2, depending on the dimension d. Let us conclude by remarking that apart from the above general rules, for any new model and exponent, one has to experiment in order to find the best way *It is interesting to expand Vvar = 2 D / d in D and e. The result is War-
2-D 2
1 2-D 1-(e/(2D)) = 2
(
E 1+ ~
~2 + ~
) + ....
Thus boldly truncating the series at a given order in e becomes increasingly bad when approaching D=2.
352
K. J. Wiese
(D,d)
=
(I,3)
(D,
d)
=
(1,3)
o t~
o6
.....................................
.......
.................
v o4
o2
oo
o~
i Do
,3
0
05
~
1.5
2
o
Fig. 17 Extrapolation for polymers in d = 3. The left plot is an extrapolation of v* using as extrapolation variables D and e. For the right plot, De(d) and e have been used. The first-order results are given by the dashed line, the second-order results by the solid line.
to extract the physical information. What may seem to be a disadvantage of the above method, namely its ambiguity, upon careful thinking turns out to be one of its main advantages, since it allows us to estimate the confidence of the result, i.e. its systematic error.
7.3
Expansion about an approximation
In Section 2.3, we discussed approximations for the exponent v*, namely the Flory and the variational approximation. It is an intriguing idea, to use these approximations as a starting point for a systematic expansion. To build such a systematic expansion on the level of diagrams is a rather hopeless task. However, as first observed in Wiese and David (1997) and David and Wiese (1996), this is possible in the perturbative expansion of critical exponents, as we show now. The general idea is that the approximations can be written as a nontrivial relation among the renormalization Z-factors of the form
(7.3)
Z b .~ Z a.
This relation can be formalized by setting o/
Zb = Z ~ Z p
Z = ZoZp,
(7.4)
where we think of Zo as a large renormalization factor and Z p as a small perturbation. We have also introduced an exponent ~, but it will drop out from the final
2
353
Polymerized membranes, a review
(D, d) = (2, 3)
(D, d) = (2, 3) ......... .....'"'"
..
12
14
Do
16
1.8
2
0
1.2
1.4
Do
1.6
1.8
Fig. 18 Extrapolation for membranes in d = 3. Both plots stem from extrapolations of v* (d + 2). The extrapolation variables are (left) D and d; (fight), d and e. The first-order results are given by the dashed line, second-order by the solid line. result. In order to eliminate Z0, one first solves the defining equation (3.115) for the fl-function, expressed in terms of Z0 and Zp, for Z0: 0 lnZo=
O---b
(~ + d / 2
,6 ( b )
1-
~+
2
b
~
lnZp
"
(7.5)
Inserting this into the definition of the anomalous dimension v(b), and using the definition e = 2D - ((2 - D)/2)d, yields
a
/~(b)
v(b)(d + 2(~) - 2D + (2 - D)~ + fl(b)-~ ln(ZbZ -C') + - - ~ - .
(7.5)
The last term vanishes at the fixed point b*, given by fl(b*) = 0. We therefore obtain a nontrivial e-expansion for the critical exponent v* = v(b*), reading 0
v*(d + 2(~) = 2D + (2 - n ) ~ + fl(b)-~ ln(ZbZ-~')lb=b..
(7.7)
This expansion is formally obtained, when expanding v* (d + 2c~) instead of v* in e. In the following sections, we will explain and specify the two cases of interest, (~ = 0 (variational approximation) and ~ = l (Flory approximation).
7.4 Variational method and perturbation expansion With the help of a Gaussian variational ansatz, which becomes exact when d --, cx), Goulian (1991), Le Doussal (1992) and Guitter and Palmeri (1992) have shown that in the latter limit I)*
-- War =
2D d
9
(7.8)
354
K.J. Wiese
Another way to obtain the same result is to suppose, that the dipole does not necessitate any renormalization, thus Zb = 1. Using (7.7), and there setting Zb = 1 and r = 0 according to (7.4), one recovers (7.8). Let us now analyse whether there is a limit in perturbation theory in which Zb = 1. The counterterm for the renormalization of the wave function at oneloop order is given by the residue -+-)e ~ 1 as D ~ 2,
(7.9)
whereas the counterterm for the renormalization of the coupling constant is
o
o),
o, as o - ,
which is exponentially smaller than (7.9) when D ---> 2. A similar exponential factor appears for the renormalization of the coupling constant at two-loop order, but not for the renormalization of the wave function. There is no rigorous proof that this persists to all orders in perturbation theory, but a convincing heuristic argument, which we present now. In Fig. 19 we have depicted an example of a MOPE coefficient f ( s , t . . . . ) from the contraction towards a dipole. The important observation is that f has always the structure
f (s, t . . . . ) - - f (s 2-D h-t 2-D
~ o o
For e ~ O, this can be re-expressed as 2-d/Zf
(l
~(s 2-D + t z-D) . . . . .
.)
)
.
(7.11)
(7.12)
(Recall from Section 3.2 that typical integrals over the dual of the embedding space read fp e -pQp ~ det(Q)- 89 leading, for a d x d-matrix Q, to the exponent of - d / 2 . ) For D --~ 2, the expression 89 2-D + t z-D) is always of order 1. Supposing that all subdivergences are subtracted, the expression to be integrated is finite and the integral of order 1. The argument is concluded upon remarking that 2 -a/2 = 2 -2D/(z-D) + O(e). (7.13) If this exponential bound In Zb << In Z for D ~ 2 is correct, then v* - War ~" 2 -a/2 when d ~ e~. For smaller, but still large d, it is useful to use (7.7) with ~ = 0, i.e. (Wiese and David, 1997) 0
v*d -- 2D + f ( b ) - ~ ln(Zb)]b=b.
(7.14)
2
Polymerized membranes, a review
355
Fig. 19 Example of a contraction towards a dipole.
for extrapolations. Let us also mention that in the context of the q~4-theory, the variational approximation corresponds to the mean-field approximation, for which we know that the specific heat exponent a, which for membranes has to be defined as (Wiese and Kardar, 1998a) a := 2 -
v*d
~, D
(7.15)
vanishes.
7.5 Expansion about Flory's estimate In the introduction, we have shown that a simple heuristic argument supposing that the elastic energy scales like the contribution from self-avoidance, yields the Flory estimate 2+D ~ . (7.16) V~ory = 2 + d In perturbation theory, a similar ansatz would be to set Z = of tr = 1 in (7.7), which results in (Wiese and David, 1997)
Zb. This is the case
(7.17)
v* (d + 2) = O + 2 +/~(b)-ff-s In - b=b*
This makes it clear that if the wave-function and coupling-constant renormalizations are the same, or more precisely if Zb/Z stays finite at the IR fixed point b*, ,8(b)(O/Ob) vanishes and the Flory result becomes exact.
ln(Zb/Z)lb=b,
Moreover, the e-expansion of v* (d + 2) is clearly an e-expansion about Vnory. This expansion seems to be the most satisfying numerically. In particular, the method of minimal sensitivity and that of minimizing the second order term give
K. J. Wiese
356
IT'
"
1.0 !,
0.8
x~-
',
....
i '~\~ 9
\
0.6 r
\\t_; \
o.4
", ,
i
"~.. .....
1
9
i
0.2" !
__n.ot 2
4
6
8
J
10
12
14
16
,
18
i
20
d
Fig. 20 Extrapolation of the two-loop results in d and e for membranes D = 2 in d dimensions, using the expansion of v*(d + 2) (squares). The solid line is the prediction made by Flory's approximation, the dashed line by the variational ansatz.
generally close results. Let us also stress that good expansion-parameters for v* are not necessarily good for v* (d + 2) and vice versa. For instance, the expansion in D and d is bad for v* but works quite well (although not optimal) for v* (d + 2). Examples of a two-loop extrapolation, using the expansion (7.17), are presented in Fig. 18.
7.6
Results for self-avoiding membranes
Results of a two-loop extrapolation for v* are given in Fig. 20 for membranes (D - 2) in d dimensions (2 _< d < 20). We see that for d ~ oo the prediction of the Gaussian variational method becomes exact, as argued in Section 7.4. For small d, the prediction made by Flory's argument is close to our results. This is a nontrivial result, since the membrane case corresponds to e = 4 and in comparison with polymers in d = 3, where e = 1/2, the two-loop corrections were expected to be large. In fact, they are small when one expands around the critical curve e = 0 for an adequate range of D ~ 1.5 (depending slightly on d and on the choice of variables) and a suitable choice of extrapolation variables. In this case the two-loop corrections are even smaller than the one-loop corrections and allow for more reliable extrapolations to e = 4. This can be understood from the large-order behaviour, as will be discussed in Section 14. Let us now turn to the physically relevant case of membranes in three dimensions (D = 2, d = 3). Our calculations predict an exponent v* ~ 0.85 or
2
Polymerized membranes, a review
357
equivalently a fractal dimension of df ~ 2.4, which is in agreement with those experiments and simulations which find a fractal phase. As we have discussed in Sections 2.5 and 2.6, this is still under debate. Let us therefore give some heuristic argument why eventually no fractal phase exists in d = 3 (Wiese and David, 1997; Wiese unpublished, 1997), i.e. why self-avoiding membranes might always be fiat, as suggested by the numerical simulations discussed in Section 2.6. Let us start from phantom membranes (without self-avoidance). The fractal exponent v* is then 0 in the crumpled phase, 1 in the fiat phase, and equal to
vc =
-~+O
~
(7.18)
at the crumpling transition (David and Guitter, 1988; Paczuski and Kardar, 1989). This last estimate is the result of a large d expansion. Its applicability to low dimensions is thus not clear a priori, but numerical simulations (Kardar and Nelson, 1987) show that even in three dimensions this approximation is reasonable (vc = 2/3). Let us now ask whether self-avoidance is relevant at the crumpling transition. By naive power counting we find that this is the case if c* d > 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
Polymerized membranes, a review
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
Polymerized membranes, a review
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 <0
D>
for
1,
D = 1, D
364
K.J. Wiese
This implies that for 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
Polymerized membranes, a review
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
Polymerized membranes, a review
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)
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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
2 Polymerized membranes, a review
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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 ~ Ix<~ and since the mean distance R to the binding centre scales as R -~ Ix-v, we have (10.6)
R ~. Ib - b*l -v/
Let us now turn to the case of polymers, which has been extensively treated in the literature. It is also interesting, since there a striking simplification occurs. For polymers, and polymers only, the result (10.5) stays true to all orders in perturbation theory. This is proven as follows. Consider the contraction of (n -4-1) g-interactions. We claim that (D = 1) (10.7) n+l
where the n arguments on the r.h.s, are the n nearest-neighbour distances of the l.h.s.. This is proven as follows: first, by changing the integration variables enforcing the g-interactions and ordering the distances, i.e. xi < X i + l , we can write .:i"i"
Silt1
i"~i}, : ~n+l
e ipr(xl)
...
"
P kl
kn
e ikilr(xi)-r(xi+l)l .
(10.8)
i=1
The MOPE coefficient
n+l
is obtained from (10.8) by dropping the integration over p (which finally gives 9), setting p = 0 and taking the expectation value, leading to
. . . . . . . . . .
n+l
(10.9)
,
kl
kn
i=l
374
K.J. Wiese
For polymers, this is easily verified to factorize as
(..,--;, ;,-,:,[o)_f ...f i-I(ei~tr(x~,-r(xi+,,l) n;l
kl
kn
0
f e -k?t Ixi-xi+ll
n = I-I i=1
i=1
(10.10)
ki
n
= H Ixi - X i + l l - d / 2 i=1
as should be demonstrated. However, when performing perturbation theory in x-space, at each order new terms appear, and it is extremely difficult to show that the el-function can still be recast in the simple form (10.4). The simplest remedy is to introduce as in Section 4.4 a chemical potential t conjugate to the length L of the polymer, and then to integrate over the polymer length. Since L = Ein__l Ixi+l - xil + e., where e is the length of the free ends, this leads in (10.10) to
~11 f e-(k2Wt)lxi-xi+ll ki
(10.11)
and integration over all positions Xi gives
" f
l
(10.12)
"-- ki which factorizes trivially. Finally, one has to sum the perturbation expansion; this is easily done, since the perturbation expansion is a geometric series. Thus (10.4) is exact, if one there sets
:"*""
,,........ ,
9 = s
lWk 2
= F
2-
9
(10.13)
(Note that the equivalent problem arises in the presence of nonlinear growth (Kardar et al., 1986). A pedagogical treatment is given in Wiese (1998c).) Let us now turn back to the general case of membranes. One should note that the model (10.1) is equivalent to the model of a directed polymer (or membrane) in a (d + D)-dimensional space binding to a D-dimensional plane (or line) 7-/pin[r] = 2 -
, D /12 (Vr(x))2 + bZblze / x
x
~d(h(x))
2
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Polymerized membranes, a review
2- 1 D
fl
2 (Vh(x))2 -+ bZb
~f~d
(h(x)) + const, (10.14)
x
x
with r(x)=
x )
h(x)
(10.15)
"
h(x) are the orthogonal fluctuations. (For a review see L~issig, 1998.) If the membrane is rigid, the model is further modified to ~pin[h] = - 2 ( 4 -
D ) ( 2 - D)
f,
-~(Ah(x)) + bZb# ~
x
(10.16)
x
where now
e = D - vd,
(h(x)),
4-D
(10.17)
v= ~ , 2
which can be renormalized along the same lines as (10.1) (David et al., 1993a,b), using now the correlator C(x - y) = Ix - yl 4-~ A mathematically specific intriguing case is the limit of D --+ 2, since in that limit Vh(x) is dimensionless, such that an infinity of couplings appears, of the form fx f(Vh(x)) ~d (h(x)), with an arbitrary function f . This problem was considered in Wiese (1996b) and it was concluded that the fixed point of (10.16) is still relevant, but is unlikely to be reached for an arbitrary microscopic model.
10.2 Tubular phase In (2.10), Section 2.3, we have introduced an effective free energy for isotropically polymerized tethered membranes. An interesting question is, what happens in the case of anisotropy (Tokuyasu and Toner, 1992; Radzihovsky and Toner, 1995). Anisotropy should be relevant physically for tubules, which are synthesized by polymerizing a fluid membrane with parallel-oriented embedded stiff objects, e.g. DNA molecules. This leads in generalization of (2.10) to a model with different elastic constants in the parallel (Xll) and (D - 1) orthogonal (x• components
~r~--
fx y(~,~r) xz ~+
x~l( ~(0~r) ~ + V
O2r)
(0~r) + yt•
(oo rO~r) + (.,,+o,~(0, r0,r;+~
(O•
+ ~ tll
(0,,r)~
0,r
+ot(O:rO:r) ~ + o~(O~rO,,r)~ b
+g f f , 6"(r(x) - r(x')).
(10.18)
K. J. Wiese
376
~h2
hI Fig. 24 A membrane in the tubular phase. Note that the membrane has not necessarily to be a tube. In Section 2.3, we have seen that for t < 0 rotational symmetry is spontaneously broken and the membrane becomes flat. In the case of two harmonic elastic constants/11 and t• one may well have tll < 0 and t_L > 0, leading to a tubular phase with length proportional to the microscopic length L and diameter Rg "~ L v* (see Fig. 24). In contrast to isotropic membranes, where the flat phase is stable at all scales, the tubular phase of tethered membranes behaves at large distances as a stiff polymer, which is known to resemble a flexible polymer for even larger scales. The tubular phase has indeed been found in simulations (Bowick et al., 1997a). Since it is difficult to treat the full model (10.18), to catch the effects of selfavoidance, it was proposed (Radzihovsky and Toner, 1998) to study a reduced model by expanding about the mean-field solution
r(x) = ( ~xll + u(x) ) h(x)
(
1 V/ Itlll II it ' ~- u lf + o fl
(10.19)
where u (x) and h (x) are the fluctuations. (Note the difference from the definition (2.18).) Keeping only the leading harmonic terms in u and h, we arrive at 7-(' [r] -- ~
Xll Ollh
+ t• (0_Lh)2 + 4 Itill (01iu) 2 +
t• (Oq_l_U)2
+ -~ b fx fx,'d-I (h(x) - h(x')) 3(((Xll - Xil) + u(x) - u(x')).(10.20) The last term ~(((Xll - xll ) + u(x) - u(x')) can be simplified, if fluctuations in u are subdominant with respect to the leading term ((xll - xl). Let us suppose
2
Polymerized membranes, a review
377
that this is the case; it was a posteriori verified in Radzihovsky and Toner (1998). Then
~(r
1
- xll) + u(x) - u(x')) = -~(xll - xil)
and the degree of freedom u (x) decouples, leading to
7-f" [h] = ~
~Ctl Oiih
+ t_L (19_Lh)2 + ~ -
,
(10.21) This model can be studied using the methods of Sections 3 and 4. The dimensions in units of x• ~ 1//z are [Xll]x• = 1 and g := [ b / ( ] u - 2D - 3 _ v ( d - 1) with u "= [h Ix• = -~ 5 - D / 2 . Note that since v ( D = 2) > 0, direct calculations at D = 2 are possible in contrast to the isotropic case discussed in the rest of this review. An important simplification arises from the fact that the interaction in (10.21) does not depend on the distance in the parallel direction Xll - Xil, thus cannot give a renormalization of KII. The argument is identical to what happens in the dynamics of self-avoiding membranes and is discussed in more detail in Section 11.3. The calculations at D = 2 to first order in ~ were performed in Bowick and Guitter (1997), and later generalized to arbitrary D in Bowick and Travesset (1999a,b). As discussed in Section 7, the advantage of calculating at arbitrary D and then extrapolating to D = 2 is a better numerical precision and control of the error. The results of Bowick and Travesset (1999a,b) ( v * ( D = 2, d = 3) ~ 0.62) for v* are found to be comparable with the Flory result
I)* -~-tubule 1+ D vFl~ = 1 + d '
(10.22)
obtained by remarking that the dimensional arguments that led to (2.15) apply to the self-avoidance in (d - 1) dimensions in the (D - 1) transversal directions; thus leading to the replacements of d ~ d - 1 and D --~ D - 1 in (2.15). However, for larger g, i.e. when lowering d further, one can no longer expect that the anharmonic elastic terms in (10.18) can be neglected. In Radzihovsky and Toner (1998) it is argued that this happens at d < dt -~ 6. Since we know of no systematic treatment of both self-avoidance and anharmonicity, we refrain from further discussions. Let us only note that, in the case of the non-self-avoiding but anharmonic model, anomalous exponents have been calculated (Tokuyasu and Toner, 1992; Radzihovsky and Toner, 1998), also for the case of unidirectionally polymerized, and otherwise fluid membranes (Toner, 1995).
378
11
K.J. Wiese
Dynamics
11.1 Langevin dynamics, effective field theory In this section, we want to study the dynamics of polymerized tethered membranes, including polymers. To this aim, we add to the membrane position r(x) a time argument t, and study the evolution of r(x, t). The simplest diffusive dynamics which can be constructed, is given by the Langevin equation
d- - t - r 0i ( x , t ) = 1 . o
- ( 2 - D ) ~ r ~ ( x- ~ , t )
+
(x , t)
.
(11 . 1)
7Y is the static Hamiltonian as given in (3.1). In order to keep the derivation transparent, we give it in terms of bare quantities, and shall introduce renormalized ones later. The Gaussian noise ((x, t) has correlations
((i(x,t)(J(x',t')) = 2(2-- D)SD~D(x -- x')~iJ~(t
--
t').
(1 1.2)
ko The factors of (2 - D) and So in (11.1) and (11.2) are introduced in the same spirit as those of the static Hamiltonian in (3.1) in order to obtain simple expressions for the response and correlation functions and to compare with the static case. Readers only interested in the general procedure can safely ignore them. Since in our arguments and presentation we closely follow Wiese (1998a), let us also mention that there these factors were hidden in the normalization of the integration measures, but are equivalent to the above. In order to define the time derivative in (11.1), a discretization has to be introduced. We use prepoint (It6) discretization (Janssen, 1992)
r;(x,t + r)r
r;(x, t) = ~.0 - ( 2 - D) 3r;(x, t-----~+ ~ (x, t) .
(11.3)
In the terminology of Hohenberg and Halperin (1977), this is model A. In the polymer community, this is called the Rouse model (Rouse, 1953; Doi and Edwards, 1986) and computer scientists will recognize the definition of a molecular dynamics algorithm. The construction ensures that the static correlation functions obtained from the Boltzmann weight e - ~ are correctly reproduced, see the discussion in Section 12.3, Zinn-Justin (1989) and Janssen (1992). This model has been studied using scaling arguments for polymers (De Gennes, 1976a,b) and membranes (Kantor et al., 1987). For polymers, a renormalization group analysis has been performed at one-(AI-Noaimi et al., 1978; Oono and Freed, 1981; Oono, 1985; Purl et al., 1986; Wang and Freed, 1986; Schaub et al., 1988) and two-loop (L. Sch~ifer, personal communication)
2
379
Polymerized membranes, a review
order. The result to all orders was given in Wiese (1998a), and we follow the procedure outlined there. The Langevin equation (11.1 ) can be transformed into an effective field theory (Martin et al., 1973; Janssen, 1992) with action
l/f[
,7" [r0, ;0, (] -- 2 - D
x
;0(x, t)
(
•0(x, t) + •o(2 - D) ~ro(x, t"------~- ((x, t)
)
t
+_4_~o ( (X, t )
(11.4)
.
Expectation values of an observable O are calculated by integrating over all fields ro and ?o and the noise
{O>bo=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) :=
'<'
-~(r(x, t) - r(O, 0)) 2
)o x2
_
Ixl 2-D
2
( 4 k i l l ) (2-D)/2
-
+ f
-ds- s (D-2)/2 (e - s -
1)
S
o x2
"-
r
-7
e- ~zTT+
f
-ds- s
D/2e_s
(11.11)
S
o
1
R(x, t) "= -~ (r(x, t)F(0, 0))0 = (9(t) (4zrXltl) -D/2 e -x2/4xltl SD(2 -- D),
(ll.12)
where as usual So is the volume of the unit sphere in D dimensions (see (A.2)). The normalization of the integration measure fx is the same as in the static case (cf. Appendix A). This ensures that in the static limit (x 2 >> kltl)
C(x, t)
~ Ixl 2-o.
(11.13)
If X2 is much smaller than Xltl, the correlator approaches the finite value
C(x, t) - (4Jtlk)v + O(x2). F(D/2)
(11.14)
2
381
Polymerized membranes, a review
It is useful to note that the propagator is simply related to the time derivative of the correlator by 1. R(x, t) = (O(t)-~C(x, t). (1 1.15) This is the perturbative version of the fluctuation dissipation theorem, further discussed in the context of nonconserved forces in Section 12.3.
11.2 Locality of divergences In analogy to the static case, we can now construct perturbation theory. The perturbative expansion of an observable (9 is in analogy with (3.12) (O) b = Norm Z
n
(Xblz~ ZbZx)n n!
f (o
~i'1)0
, (11.16)
where the normalization Norm has to be chosen such that ( 1)b = 1 and the integral is taken over all arguments of the interaction. We claim that, as in the static case, divergences only occur at short distances and short times. To prove this, look at a typical expectation value
e'n)o=~/..,
f fg(xl--Xm,tl--tm, kl,km) e- 89 k2n (11.17)
where each contribution consists of a function fe, which is a product of propagators, correlators and k's and an exponential factor, with
Qij -- -C(xi - xj, ti - tj).
(11.18)
fi is a regular function of the distances. As we have seen in Section 3.2 in the static case, divergences at finite distances can only occur if Qij is not a positive form. We shall show that aij is a positive form for all ki which satisfy the constraint Z k i --0. (11.19) i This constraint, implicitly inforced by f,, always holds (see (11.9)). For equal times, positivity of Qij is just the statement that the Coulomb energy of a globally neutral assembly of charges is positive, as derived in Section 3.2. One simply
382
K.J. Wiese
identifies C with the Coulomb propagator and case, write
f ao, f 2---s
Qij = ( 2 - D)SD
(27r)D
ki with the charges.
In the dynamic
e ip(xi-xj)+iw(ti-tj) (11.20) 2k 092 -t-- (Xp2) 2
The exponential in (11.17) now is
Z kikjQij = ( 2 i,j XZ
f D)SD
dDp
(2n.)O
law
2k
2--7 092 _+_ ()~p2) 2
kikjeip(xi-xj)+iw(ti-tj)
ij 2
= (2-
D)So
f
dDp (~-~O
~
2X Z w2 _F(~p212
ki e i (pxi -I-wti) i (11.21/
Due to equation (11.19), the integral is ultraviolet convergent and thus positive. It vanishes if and only if the charge density, regarded as a function of space and time, vanishes. This is possible if and only if endpoints of the dipoles (which form the interaction) are at the same point in space and time. No divergence occurs at finite distances. 11.3 Renormalization It is easy to see that, as in the static case, there exists a multilocal operator product expansion (MOPE), as introduced in Section 3.5. The general criterion for renormalizability, theorem 59 (Section 5.2) then ensures renormalizability. We show now that the counterterms which render the static theory (3.1) finite are also sufficient for the dynamic case (11.6). As an illustration we first calculate the one-loop counterterms for the renormalization of the field and of the coupling constant. The first singular configuration appears when both ends of the interaction (11.9) are contracted towards a single point. The leading term (MOPE coefficient) of this expansion is (a more detailed demonstration of the derivation of the MOPE coefficients in dynamic theories is given in Sections 12.4 ft. in the case of disorder dynamics) .~.-. . . .
2 f [7(x,
t)(ik)] e -k21x-yl2-~ [(ik)(r(x, t) - r(y, t))]
+ ....
.
k (1 1.22)
2
Polymerized membranes, a review
383
We now expand r (y, t) as 1
r(y, t) = r(x, t) + (y - x)Vr(x, t) + -~ [(y - x)V]2" r(x, t) + O(Ix - y[3). (11.23) The leading term in equation (11.22) is
f r(x , t ) ( ~ A r ( x
( x - y ) 2 - k~ e 2 _k21x_yl2u D d
, t)) ~
k
= ~(x, t ) ( - A ) r ( x , t)
-~
Ix - Yl
Denoting with ( ~ : 1,~+)the MOPE coefficient of ~
9
(11.24)
proportional to
,4- = ?(x, t ) ( - A ) r ( x , t), this can be written in the form =
-+-),
(1 1.25)
where
( ' ~.... , -+-) -- --2-'-DI [x -- y[D-vd
(11.26)
is the static MOPE coefficient (see (3.96)). This implies that the one-loop counterterm for the wave function renormalization is the same as in the static case. Let us now consider the counterterm for the coupling-constant renormalization. Using the techniques explained in Section 3.6 we obtain for the contraction of two interactions towards a single one
(~:l:~
-) : ~ <
x and y are the distances between the contracted endpoints of the dipoles, and t is their time difference. The trick is now to write this expression with the help of (l 1.15) as
1| - 2--~
~d
[C(x, t) + C(y , t)] -d/2
.
(11.28)
To evaluate the diagram, we have to integrate over all times. If we use no cut-off in the time direction, the time integral will simply give the value of the function at its lower bound: OO
(11.29) o
384
K.J. Wiese
The r.h.s, is the MOPE coefficient of the static theory (see (3.79)). We easily convince ourselves that this relation implies the same counterterm as in the static case, if we take care of the additional combinatorial factor of 2 for the time ordering of the interactions. One knows from general arguments that the divergences associated with short distances in space are removed by the static counterterms (Zinn-Justin, 1989). This implies that new divergences can only appear for short times. Using the general theorem 59 (Section 5.2) about renormalizability, we have to subtract all divergences proportional to marginal and relevant operators. Thus, there may be new divergences proportional to = ~(x, t)/'(x, t),
(11.30)
which have to be subtracted in the MOPE. We now consider a general contraction of n dipoles towards ..-.-- : ~____~n
~ ~,,,,,.-.
(11.31)
In order to obtain the operator -~,-, one has to contract all fields r and ? except the field ?(z, t) with the largest time argument (all other contractions give 0). One also has to leave uncontracted one arbitrarily chosen field r. Due to the structure of the interaction (11.9), the field r always appears in the form r ( x , t - r) - r ( y , t - r). So the contraction yields r(z, t) [r(x, t - r) -
r(y,
t -
r)] M(distances),
(11.32)
where M denotes the MOPE coefficient which depends on the distances in space and time. Now, r ( x , t - r) - r ( y , t - r) has to be expanded about (z, t). The leading term has at least o n e spatial gradient. No term of the form (11.30) can be constructed. Therefore, there is no singular contribution of this type in any order in perturbation theory and no renormalization of ~ is needed. The last marginal operator at e = 0 is = ?(x, t) 2.
(11.33)
Its renormalization can either be obtained from the fluctuation-dissipation theorem (further discussed in Section 12.3)
O(t)~-~
~ (r(x, t) - r(0, 0)) 2
-- ~.
Z~. (r(x, t)~(0, 0))b,
(11.34)
which relates the full correlation and response functions. This was done in Wiese (1998a). Let us give a direct derivation here: first note that the interaction can be
2
385
Polymerized membranes, a review
written as =
(11.35)
= f t ,x t) -- 7(y, t)] (ik)e ik[r(x't)-r(y't)]. k
Study now the contraction of n interactions ,~ )'
9
- towards , ~ (11.36)
~,,,~.
Using the same arguments as those leading to (11.32), the contraction gives [7(x, t) - 7(y, t)] [7(x', t') - 7(y', t')] M(distances).
(11.37)
The leading term is proportional to (VT) 2, and no term proportional to ~ = 72 is generated. Let us note that this is sufficient to prove renormalizability of polymer dynamics without using the general theorem of renormalizability (theorem 59, Section 5.2). Two classes of diagrams exist: (i) 'static' diagrams (i.e. those correcting and ,~ -), which using the fluctuation dissipation theorem are reduced to diagrams of the static theory, and for which in the case of polymers one can use the equivalence to 4~4-theory; (ii) 'dynamic' diagrams, proportional to , ~ and ,~-,~, which vanish identically. In (11.6), we have introduced renormalization group Z-factors for the fields r, 7 and for ~. and b. The absence of counterterms proportional to ~ and in (11.7) implies that Z2, = 1 and 2Z)~ = 1, i.e. 1
= -Z
and
Z~ = Z.
(11.38)
Equation (11.7) takes the simple form
'iS
,7" [r, 7] = 2 - D
~
+ ~ Z ~ , - - ~. ~
+ ~blz ~ Zb
xt
SiS
~
--.
xyt
(11.39) Solving the renormalization group equations for ~. and 7, one obtains the new dynamic exponent z* (Wiese, 1998a). Let us give a more intuitive derivation here, using the technique of exact exponent identities developed in Section 3.9. The absence of a counterterm proportional to fx ft 7 (x, t)/" (x, t) gives the identity for the full dimensions [r]f -k- [r]f -k- D [x] -- 0. (11.40) Analogously, the absence of a counterterm proportional to fx ft 7(x, t) 2 yields 2 [7]f + D [x] + [~.t]f = 0.
(11.41)
386
K.J. Wiese
Eliminating [r]f from these equations, gives
[~.t]f = 2 [r]f -t- D [x].
(11.42)
The autocorrelation function therefore scales as 1 (r(x t) - r ( x t')) 2} ~ It - t'l 2/z*
(11.43)
Ib
with D z* = 2 + -!)* -.
(11.44)
For polymers (D = 1), this relation was given using scaling arguments in De Gennes (1976a), for membranes (D = 2) in Kantor et al. (1987). This result was followed by perturbative calculations for polymers in one-loop (AI-Noaimi et al., 1978; Oono and Freed, 1981; Oono, 1985; Puri et al., 1986; Wang and Freed, 1986; Schaub et al., 1988) and two-loop (L. Schiifer, personal communication) order. The result to all orders is due to Wiese (1998a). Let us also mention that non-self-avoiding membranes at the crumpling transition or in the fiat phase have been treated in Niel (1989) and Frey and Nelson (1991). It is interesting to note that, for polymers and membranes, (11.44) can be written in the form (11.45)
z* - 2 + d f ,
where d f is the fractal dimension of the membrane or the polymer.
11.4 Inclusion of hydrodynamic interaction (Zimm Model) While the Rouse model defined in (11.1) and (11.2) is particularly simple, it cannot be realized in experiments. Experiments on polymers and membranes are always carried out in solution. The question is, how the dynamic exponent z*, defined in (11.43), will change in the presence of additional hydrodynamic degrees of freedom, z* should experimentally be observable via dynamic lightor neutron-scattering methods. To our knowledge, no such experiment has been performed. Hydrodynamic interactions for polymers were first introduced by Zimm (1956). He wrote down the following Langevin equation, which will also be used for membranes: -d-~ro(x, t l = D .
-(2-D)~r
~ +((x,t).
(11.46)
2
387
Polymerized membranes, a review
Fig. 25 The different phases for a D-dimensional self-avoiding membrane embedded in d dimensions including hydrodynamic interactions. The region with 3 = 2 - vd < 0 and e < 0 is the Gaussian phase. Self-avoidance and hydrodynamic interaction are irrelevant. Hydrodynamics is naively relevant for 3 > 0, but renormalization reduces the range of relevance to the grey domain. It becomes irrelevant if d > df + 2, where df = D/v* is the fractal dimension of the membrane (Rouse dynamics). We have drawn two different estimates for the cross-over line from the hydrodynamic to the Rouse domain (see main text). Self-avoidance is always relevant when e > 0. Here, '.' denotes the scalar product of the matrix operator D and the vector is defined by
~7-[/~ro,which
f.
g
f fi (x)gi (x).
(11.47)
qo,
x
The hydrodynamic interaction is
~)iJ(x,
y,r,r')
= ~.o~ij~O(x
--
y) + ~.0r/0
\k 2
k4
. (11.48)
k We will not repeat the derivation (Zimm, 1956) of equation (11.48) here. However, let us note that one supposes that the hydrodynamic degrees of freedom are fast enough, so that their dynamics can be neglected and that screening effects are irrelevant. This might be wrong for membranes and in this case our results would only apply to membranes with large holes. (For a discussion of screening effects
388
K.J. Wiese
for fluid membranes see, e.g. Seifert, 1997.) For 17 = 0, (11.46) reduces to purely diffusive motion (Rouse model). The noise correlation is 2(2- D)SDSiJ~(t--t')~D(x--
((i(x,t)(79.()J(y,t'))=
(11.49)
y).
This ensures that the static behavior is correctly reproduced, see the discussion in Section 12.3, Zinn-Justin (1989) and Janssen (1992). In analogy to (11.6), we obtain the dynamic functional in It6 (prepoint) discretization
ii[
ff [r0, ~0] = 2 -- D
] f
~0" ~-r0 - r0" 79. P0 +
t
,-
~0" 79. tSr-----o" (1 1.50)
t
This model has to be renormalized. Again divergences only occur at small distances. They can be analysed via a multilocal operator product expansion (MOPE). Renormalizability is ensured if counterterms for all possible marginal and relevant operators are included into the action. It has been shown in Wiese (1998b) that the model is renormalized by introducing one additional renormalization for 0, leading to a dynamic renormalization different from the Rouse model. The two regularization parameters e and ~ are the canonical dimensions of the coupling constants e := [b0]g = 2D - ud 3 " = [r/0]u - 2 - yd. As in Section 9, this leads to a double-E expansion, which has to be performed about the point (3 = 0, e = 0), i.e. (D = 1, d = 4). The dynamic exponent z* is derived along the same lines as for the Rouse model. In contrast to the latter, now a counterterm proportional to p2 is needed, such that ( 11.41 ) no longer holds. At one-loop order, this is due to the contraction of the hydrodynamic interaction. Denoting •
~
_~. ~i ( x ,
t)rj (y, t) f(,ij~:~ kikJ)eik[r(x,t)_r(y,t,] k4
(11.51)
k
the diagram .....
~
-
2 d ( d - 2)
(11.52)
is thus positive and ensures the stability of a fixed point 1/* > 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-" /
"'\ .
-\
",..~,,,; ~ ;,
--"~"-'<::/'--- ...... "i
" . ' - ' ~ "
.~',.~X_. V.-":/-'-.'--
/"~/
:.~,. :; \,
.... .
.........
.....
",x
,,
Fig. 26 Longitudinal (potential) disorder, transversal disorder and isotropic disorder (from left to fight). The latter one is obtained through superposition of 7 x 7 Fourier modes, the first two by applying the longitudinal and transversal projector respectively. the disorder correlator, decorated with two response fields. Short-range, i.e. 8correlated disorder thus leads to 8-interactions between different points on the membrane, very much in the spirit of the ~-interaction enforcing self-avoidance. Physically, static disorder is relevant in most of the accessible cases. Potential disorder normally leads to trapping, ultraslow dynamics, anomalously small response to external perturbations and pinning which cannot be studied using standard RG-techniques (Fisher, 1985, 1986; M6zard and Parisi, 1991; Nattermann et al., 1992; Balents and Fisher, 1993; Kinzelbach and Homer, 1993a,b; Narayan and Fisher, 1993; Blatter et al., 1994; Cugliandolo and Le Doussal, 1996; Kardar, 1998; Leschhorn et al., 1997). Nonpotential disorder has been studied much less. Whereas physical realizations of frozen nonpotential disorder are rare, similar problems appear for driven dynamics in the presence of quenched disorder (Balents and Fisher, 1995; Krug, 1995; Giamarchi and Le Doussal, 1996; Schmittmann and Bassler, 1996) or domain growth in the presence of shear (Onuki, 1986). The problem itself is intriguing: does the nonpotential part of disorder still lead to glassy dynamics? We will show below that this is not the case, but that physical observables are described by a new perturbative fixed point, which is universal with a large domain of attraction. Moreover, the fluctuation--dissipation theorem is broken, which can be interpreted as a rise in temperature under renormalization. The origin of this phenomenon is easy to understand: nonpotential disorder can only be kept up by constantly pumping energy into the system, thus raising temperature. This leads to a dynamic exponent ;3 not substantially larger than 2. It also prevents possible computer simulations from the usual problems of ultraslow dynamics. This section heavily relies on Wiese and Le Doussal (1999), to which the reader is referred for further details and discussion. Our main aim here is to explain the underlying principles and to show how the techniques developed in this review serve to analyse disorder situations. This is also the reason why we
2
r(x)
Polymerized membranes, a review
391
=-/ (b
Fig. 27 elastic manifolds (polymers D = 1) in random flows: (a) directed polymer; (b) isotropic chain. will restrain our attention to one of the situations studied in Le Doussal and Wiese (1998) and Wiese and Le Doussal (1999), namely an isotropic membrane in shortrange correlated disorder, visualized in Fig. 27.
12.1 The model We again consider a D-dimensional manifold imbedded into d dimensions, including particles (D = 0), polymers (D = 1) and membranes (D = 2). In generalization of (11.1), we study the Langevin dynamics
d
d-t- r~)(x, t) -- ~-0 ( Ar 0i(x, t)
+ (i (x , t) + (2 - D)Fi[ro(x, t)])
(12.1)
where the term proportional to Ar~ (x, t) is the derivative of the internal (entropic) elasticity. The Gaussian thermal noise (i (X, t) is denoted by angular brackets and is the same as in (11.2). F i [r] is a Gaussian quenched random force field with correlations:
Fi[r]FJ[r '] -- A; j (r -- r').
(12.2)
Disorder averages (over F) are denoted by overbars. We consider a statistical rotationally invariant force field with both a potential (L) ('longitudinal') and a divergence-free (T) ('transversal') part whose correlations depend only on the distance r - r'. Both parts contribute separately to the disorder correlator:
A~ (r) =
O
0 A~(r)_ ( r
Ori Or-----f
OrI OrI
0 O)AT(r) Or i OrJ "
(12.3)
392
K.J. Wiese
In Fourier space
A ~ (r) -- f
A;j
(k)e ikr,
(12.4)
k
where the normalization of the k-integral is as usual (see Appendix A). Several cases for the correlation of the random force are of interest. We will mainly focus on forces with short-range correlations, for which the force correlator scales like a g-distribution A ij (r--r') ~ ~d (r--r'), with, however, a nontrivial index structure. In Fourier space, the correlator then reads at small k A0j
=
+ 809
where PLJ(k) = k i k J / k 2 is the longitudinal and P;J (k) = ~ij _ pLJ(k) the transversal projector, already encountered in (11.48); see also Appendix C. As we shall show below, this is the genetic situation, relevant for all short-ranged correlations, even those for which one starts with a force correlator which is formally shorter ranged than the g-distribution (e.g. decaying faster than l/rd). Long-range correlations can also be studied (Wiese and Le Doussal, 1999), but for simplicity of presentation are omitted here. Note also that if the forces can be derived from a potential V (r) in the Hamiltonian ~ [rl = 2 -
lD f
+ + V[r(x)l,
x
and denoting by V (k) the Fourier transform of V (r), then AiJ(k) = k i k J V ( k )
(12.6)
AL(r)---- V(r).
(12.7)
or
(This suggested the terminology 'potential disorder'.) Before developing a field-theoretical description for the above model, let us state the (systematically wrong, but numerically often satisfying) predictions made by a Flory-type ansatz; supposing that (12.1) imposes the scaling for the radius of gyration
RG t leads to
RG x2 4
/ 1 ~/ Rd 2
R G ~" x T+-~ ~ t T~-~.
(12.8)
(12.9)
However these estimates are certainly much too crude to catch the physics of the problem, especially the difference between potential and transversal disorder.
2
12.2
393
Polymerized membranes, a review
Field-theoretical treatment of the renormalization group equations
We start from the equation of motion (12.1) and convert it into an effective field theory analogous to what we did in (11.6). The effective dynamic action in prepoint (It6) discretization reads J [r0, r0] = 2 - D
, f
r~(x,t)
X,I
x [/'~)(x, t)+ LO (--Ar~(x,t)+ ( i ( x , t ) + ( 2 - D)Fi[ro(x,t)])]. (12.10) The noise ((x, t) and the disorder force F[r] are Gaussian, see (11.2) and (12.2), and can thus be integrated over. This gives the dynamic functional J [r0, ~0] -- 2 - l f ~- ~ 1~7 6~1 7 '611 7 76 6'
'
X,t
2~2.1
f
rO~i(x,t)A~[ro(x,t)-ro(y,t')]~Jo(Y,t').
(12.1 1)
x,y,t,t ~
Here and in the following, contraction over indices is implied, whereever confusion is impossible. Note also that the factor of 89disappears, if we use the time-ordered disorder interaction. The disorder correlator is
AoJ[ro(x,t)-ro(y,
t)] =
f
+
,'2 '2,
k
In order to simplify notations, we introduce the following graphical symbols:
~4- = ~(x, t)(-Ax)r(x, t) ~.- = ~(x, t)i(x, t) = ~(x, t) 2
i,~v~
_-v~J = f ~i(x,t)eik[r(x't)-r(y't')]~J(y,t t) k
~
-- f r(x, t)eik[r(x't)-r(y't')lr(y, t') k
L ~
--_
fk
eLJ (k)r i (x, t)eik[r(x't)-r(y't')]r j (y, t')
"
(12.13)
394
K.J. Wiese
T ~ _ f pTj (k)~ i (x, t)eik[r(x't)-r(y't')]r j (y, t'). k The action takes the symbolic form
1 /
J Jr0, r0] -- 2 - O
(''~o-4-~.0""~o- ~.o''~'~o)
x,t ~2f f( 2 x,t y,t'
~'~
L
--vw~g ~ + ~
T
--~0 gT ) ,
(12.14)
where the index '0' denotes bare quantities. The dimension of the coupling is e'--
g
u
=
Ix] g
u
= 2 + D - ~ d . 2
(12.15)
Disorder is relevant for e > 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
Polymerized membranes, a review
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
Polymerized membranes, a review
+ 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
Polymerized membranes, a review
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
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Polymerized membranes, a review
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
Polymerized membranes, a review
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
Polymerized membranes, a review
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)
Polymerized membranes, a review
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
2 Polymerized membranes, a review
0 (N)-field theory
<
D-,l
~ N---~O
self-avoiding polymers <
O(N, D)-manifold model ~ N--~O
D--~l
self-avoiding D-dimensional tethered membranes
Fig. 31 Schematic description of the new model, and its limits. 13
N-coloured membranes
Field theories have strong connections to geometrical problems involving fluctuating lines. For example, summing over all world-lines representing the motion of particles in space-time, is the Feynman path integral approach to calculating transition probabilities, which can also be obtained from a quantum field theory. Another example is the high-temperature expansion of the Ising model, where the energy-energy correlation function is a sum over all self-avoiding closed loops which pass through two given points. Generalizing from the Ising model to N component spins, the partition function of a corresponding O(N) 'loop model' is obtained by summing over all configurations of a gas of closed loops, where each loop comes in N colours, or has a fugacity of N. In the limit N ~ 0, only a single loop contributes, giving the partition function of a closed self-avoiding polymer (De Gennes, 1972). There are several approaches to generalizing fluctuating lines to entities of other internal dimensions D; it is important to note that such extensions are not unique. The most prominent generalizations are string theories and lattice gauge theories, both describing D = 2 world sheets (David et al., 1996). The low-temperature expansion of the Ising model in d dimensions also results in a sum over surfaces that are (d - 1)-dimensional. Each of these extensions has its own strengths, and offers new insights into field theory. Here we introduce a generalization based on tethered membranes as defined in Section 3.1, which have fixed internal connectivity, and thus are the simplest generalization of linear polymers. The resulting manifold theory depends on two parameters N and D, with limiting behaviours related to well-known models as depicted in Fig. 3 I. The model is defined by its perturbation series, and as in string theory not obviously derivable from a local Hamiltonian. The work discussed here has first been published in Wiese and Kardar
410
K.J. Wiese
(1998a,b). We give a shortened presentation of the main ideas, focusing on the construction of the generalized manifold model. This is important from a conceptual point of view, especially since an alternative subtraction scheme is used, in which no divergences proportional to (Vr) 2 appear at the one-loop level. We therefore felt the necessity to present in detail the derivation as well as the discussion of the relation between the two models. On the other hand, the original article shall not be rewritten, and therefore most of the applications are only sketched; for more details, the interested reader is referred to the original publications. This section is organized as follows: we first review the high-temperature expansion, to motivate the formal constructions done later. In a second step, we derive results of the O(N)-model in the polymer language. This is well known, but is done in a way, which can be generalized to membranes. Finally, we discuss some applications: more precise estimations of exponents in the O(N)-model, the limit of N ~ oo, cubic anisotropy, the random temperature Ising model, and finally a conjecture for the nature of droplets, which govern the Ising model at criticality.
13.1 The O(N)-model in the high-temperature expansion In this section, we briefly review the high-temperature expansion of the O ( N ) model. (For more extensive reviews, see Domb and Green, 1974; Savit, 1980). The Hamiltonian is 7"[ = - J N ~"~ Si . S j , (13.1) (i,j)
where the sum runs over all nearest neighbours of a d-dimensional cubic lattice. To obtain the partition function, we have to integrate over all Si subject to the constraint that Iail = 1, resulting in (K = 13J) Z =
e-#~ i}
=
f, i} Iq(i,j)
e NKSi.Sj.
"
(13.2)
The high-temperature expansion is obtained by expanding the exponential factors in (13.2) as e N K S i S j - 1 + N K S i . Sj + . . . . (13.3) Typically, only the first two terms in the Taylor expansion are retained. This is justified as we are only interested in universal quantities, for which the weight is already not unique and may be modified [exp(N K Si 9Sj ) ~ 1 + N K Si 9Sj ] in order to cancel subsequent terms in the Taylor expansion. We can represent the various terms in the perturbation expansionin the following manner (see Savit, 1980; Kardar, 1996): for each term N K S i 9 S j , we
2
9
9
9
9
9
9
9
9
9
.
o
9
2'~3'
9
9
(a) 9
9
411
Polymerized membranes, a review
9
9
9
9
.4'
9 o
(b) 9
9
(c) 9
9
9
9
Fig. 32 Some terms in the high-temperature expansion of the O(N)-model. draw a line connecting sites i andj. At any given site i, up to 2d such lines may join. The integral over the spin Si is nonzero, if and only if an even number of bonds end at site i. For calculational convenience, we normalize the integrals by the corresponding solid angle such that
f
dS/ = 1.
(13.4)
Let us now study the first few terms in the perturbation expansion (see Fig. 32). The diagram (a) is (a) = (K N) 4 / d S , . . , dS4 S~ S~ S~ S3~ S~' S~' S~,S~.
(13.5)
To do the integrations, note that
f dSi ?S2i=
f d~Si 1 = 1,
(13.6)
and therefore
f dSi S~Si~ = 1
(13.7)
Performing all but the last integration in (13.5), we obtain
(a)
g 4N f d,~l ,~2 = g 4N.
(13.8)
For any nonintersecting loop, this result is easily generalized to g numberoflinksN,
(13.9)
412
K.d. Wiese
i.e. every closed loop contributes a factor of N. Let us now analyse what happens when loops intersect and to this aim calculate configuration (b). Doing all but the integration over Sl, we obtain (b) = K8N 2
f
(~2)2 = K8N 2 _-- (a)2.
(13.10)
Two graphs which have one common site thus give the same contribution in the high-temperature expansion as if they were disjoint. This is not the case if they have one bond in common, see (c). The integral contains an odd power of the field Sl, and therefore (c) = 0 . (13.11) This high-temperature series can thus be reinterpreted as the sum over all selfavoiding (nonintersecting) loops. Bonds are totally self-avoiding (see, e.g. configuration (c)), while vertices are also partially self-avoiding, as can be seen from the following argument. There are three possible ways to build up configuration (b): one may take two small loops, but there are also two possibilities to use one loop only (note that these configurations come with a different power of N). The latter have to be excluded from the partition function. (There are additional constraints associated with multiple intersections.) On the other hand, as we are only interested in universal quantities, taking precise account of these configurations should be irrelevant as long as bond-self-avoidance is present. In the direct polymer approach of Edwards and Des Cloizeaux (Edwards, 1965; des Cloizeaux, 1981; des Cloizeaux and Jannink, 1990) discussed below, this corresponds to taking a smaller initial (bare) coupling constant. A single loop can now be viewed as a random walk, i.e. as the trace of a particle moving under Brownian motion. The corresponding Hamiltonian is 1 2 + Lt, 7-/0 = f0 L dx ~(Vr(x))
(13.12)
where r(x) ~ I~a is the trajectory of the particle at time x (equivalently, x is the polymer arc-length). The total length of the loop is L = f ~ . In addition, one has to demand that the panicle returns to its starting point, i.e. that the polymer is closed. To make it self-avoiding, Edwards and Des Cloizeaux (Edwards, 1965" des Cloizeaux, 1981" des Cloizeaux and Jannink, 1990) added an explicit repulsive interaction upon contact, leading to 7-/=
l
dx ~(Vr(x)) 2 + - - T
dx
f0
d y g a ( r ( x ) - r ( y ) ) + Lt.(13.13)
The factors of 1/4, as well as the normalization hidden in 8, are the same as for membranes in (3.1) (see also Appendix A). # sets the renormalization scale. In
2
Polymerized membranes, a review
413
the high-temperature expansion, there appear loops of all sizes. We thus have to sum over all different lengths of the polymer, weighted by a chemical potential t conjugate to the length, mimicking the constant K in (13.2). To avoid possible confusion, let us stress that although closely related, in K and t are not identical. While K is defined as the fugacity for the length of the lattice walk, the chemical potential t is conjugate to the coarse-grained length. In principle, the same lattice walk can be represented by curves r ( x ) of different length L. However, as far as universal quantities are concerned, this is unimportant. Both parameters have to be tuned to reach the critical point, and only their deviations from the critical value, but not the critical value itself, have some physical correspondence.
13.2 Renormalization group for polymers We now discuss the perturbation expansion of the Hamiltonian in (13.13). Let us start with the correlation functions of the free (non-self-avoiding) polymer. One has to be careful in distinguishing between open and closed polymers, which will be denoted by subscripts 'o' and 'c' respectively. For open (or closed, but infinitely long) polymers, the correlation function C o ( x ) = -~
)o
( r ( x ) - r(0)) 2
,
(13.14)
is the solution of the Laplace equation 1 -ACo(x)
2
-- 6(x),
(13.15)
which is easily found to be
Co(x) = Ixl,
(13.16)
and is the same as (A.10) for D = I. For closed polymers, (13.16) has to be modified. The reason is that the information has two equivalent ways to travel around a polymer loop of size L, leading to Cc(x)-
Ixl(L -Ixl) , L
V Ix[ < L.
(13.17)
We next calculate the weight of a polymer of length L. For open polymers this is simply e -Lt, (13.18) where t is the chemical potential. For closed polymers, an additional factor of
(~d
(r(L) - r(O))) ~
(13.19)
414
K.J. Wiese
has to be added, which measures the probability of finding a closed polymer among all open polymers. The expectation value therefore is taken with respect to the weight for an open polymer, and calculated as follows:
(~d (r(L) -- r(O)))o = fk (eik(r(L)-r(O)))o : fk e-k2C~ = fk e-k2L = L -d/2.
(13.20)
The normalizations of gd and fk are chosen as usual (see Appendix A). To get the quantities obtained in the high-temperature expansion of the loop model introduced above, we still have to integrate over all possible lengths of the polymer. We define the free density of a single polymer as (see also footnote on p. 421) zlO)=
0
l fdL TLL
:= 2
_d/2e_Lt
1 = ~F
( 1 - ~ d)td/2_ 1.
(13.21)
We have chosen to integrate over a logarithmic scale ( f d L / L ) in order to make the integration measure dimensionless. The factor L counts the number of points which may be taken as origin, and the factor of 89has been introduced to reproduce the results of the free (Gaussian) field theory (see below). Additional insight is obtained from a different way of calculating ZI ~ If we do not perform the last integral in (13.20), ( 13.21 ) becomes
zlO) _ 1
e - Lt
f
L
1f 1 -- -2 Jk k2-+ t"
(13.22)
This term of the polymer-perturbation theory is equivalent to a term in the perturbation theory of the field-theoretical description of the O(N)-model. If not explicitly noted, the diagram is regularized by the chemical potential t. In the usual treatment of the O ( N ) field theory, the hard constraint of ISI = 1 is replaced in favour of a soft constraint, implemented by the Hamiltonian 7-/r
t "2
~ (Vq~(r)) 2 + ~r (r)
f [' ddr
b# e + -~(t~Z(r)) 2].
(13.23)
In this description, one has to take the limit N - , 0 in order to allow for only one connected piece. (Remember that every closed loop counts as a factor of N.)
2
415
Polymerized membranes, a review
This equivalence, first pointed out by De Gennes (1972), is not accidental and was demonstrated in Section 4.4. It reflects the fact that both the field-theoretical formulation of the O(N)-model, as well as its lattice equivalent, belong to the same universality class. We can now also comment on the factor of 89introduced in (13.21 ). According /
to (4.63), O
= ..(l(~(r))2) 0 is the Laplace transform of
I
t
f dx __(~d(r(x)))O =
L x L D/2, such that f dL/L has to be accompanied by a factor of 89 Also note that a factor of (4rr)d/2 has implicitly been incorporated into the measure, since in (4.63) the 8-interaction appears as 8d (r(x)) and not as gd (r(x)) (for the definition of ~d (r(x)) see Appendix A). We now perform the perturbation expansion of the polymer Hamiltonian in (13.13). The first term is the expectation value of one 8-interaction with respect to the free theory of a closed polymer, integrated over all positions of the interaction on the polymer of length L, and then over all polymer lengths. This is explicitly 1
Io dL L-d/2e-Lt YoL dx foL dy
I eik(r(x)-r (Y)),c
l fo~ dL e -Lt ~0 L dx ~0 L dy [ Ix - yI(L f_,- Ix - yl) ] -a/2 L -d/2 -- -~
--
f0
dL e -Lt
f0' dx y0xdz [z(L - Z)] -d/2
- f o ~ d Z fo~176 fo~dy'e-t(x'+>"+Z) fpe-p2z f e -k2(x'+y') 1
l
(k 2 + t)2 p2 + t =2~--=4
0
x O.
(13.24)
(Remember the factors of 1/2 for each connected component, introduced in (13.21).) The relation to 4,a-theory is again apparent: the integrals in (13.24) are ultraviolet divergent. The leading divergence is subtracted via a finite part prescription, the subleading term is treated via dimensional regularization as a pole in e = 2 - d/2. (13.25) (Note the factor of 2 difference from the more usual definition of e = 4 - d.) Let us now introduce a renormalized Hamiltonian. Since in contrast to the model introduced in Section 3. l, an additional chemical potential appears, three renormalizations may be required: a renormalization of the field r, of the coupling constant b, and of the chemical potential t. Denoting the bare quantities with a
416
K.J. Wiese
subscript 'o', we set ro - - ~ r Z r ,
to = Ztt, bo = lz EZ d/2 zbb.
(13.26)
This yields the renormalized Hamiltonian =
z f dx 24 (Vr _ 1(x))
+ ~bUeZbfdxfdySd(r(x)-r(Y))+zttfdx' (13.27)
where # sets the renormalization scale. It is possible to subtract at the scale of the renormalized chemical potential t, but this turns out to be rather confusing when deriving the renormalization group equations. We can now eliminate the divergence in (13.24) by setting (the index e means as in (3.105) just the pole term in 1/e of the diagram)
Zt -- 1
. E
(13.28)
e
This is seen by expanding e -'~ with H given in (13.27). From (13.21) and (13.22), we read off the numerical value of 0 ' yielding b Zt = l + ~ . 2e
(13.29)
The next step is to study the renormalization of the interaction, to which the following two diagrams contribute: )[('
~"~"C"
(13.30)
To calculate the first diagram, change coordinates to x0 and Y0, which indicate the points midway between the contacts on each polymer. The shorter relative distance between these points on each polymer is denoted by x (or y), while the longer one is indicated by f2x (or f2y). The arbitrariness in this choice leads to a combinatorial factor of 2 per polymer loop, for an overall coefficient of 4. For each contribution of )[[(=
f
f
e-t(~x+f2y+x+y)
ko,xo,Yo k,x,y x(ei(?+k)[r(x~176176176
(13.31)
2
Polymerized membranes, a review
417
short-distance singularities appear in the integration over x and y. The leading term in the short-distance expansion is in analogy to the MOPE (see the derivation following (3.79))
f
(eik~176176
fk e-k2(C,.(x)+Cc(Y))e-t(x+Y).(13.32)
O,xO, YO
,x, y
For small arguments, the correlation function can be approximated by its infinite volume limit, leading up to subleading terms to
3(/, ,x,y ..
x
e -(k2+t)(x+y)
=
..
x
(13.33)
( k2 4- t) 2"
The final result is
)::(
=)..(
+ subleading terms.
x 20
(13.34)
The second diagram in (13.30) has already appeared in (13.24), and we can symbolically write
1
( d) t 2-d/2 .
=fk (ka_7t._t)a=F 2 - ~
----20
(13.35)
This diagram appears with a combinatorial factor of 2 for its left-right asymmetry, and another factor of 2 for the u p ~ o w n symmetry of the leftmost interaction. Adding these contributions yields the following renormalization factor at oneloop order (note that the combinatorial factors of 4 cancel with that of the b / 4 in the Hamiltonian (13.27)),
Zb= l - F -g
4-
--
~
= I-F~.
g
(13.36)
g
No field renormalization is necessary (Z = 1). We will discuss the apparent difference from the renormalization factors in (3.112) and (3.113) later. The next step is to calculate the renormalization group functions, which measure the dependence of the renormalized quantities upon a change of the renormalization scale #, while keeping the bare values fixed. The derivation of these functions is given in Appendix D, and results in a/%function -eb ~6(b) -
u
b -
0
1 -+- b~b In
Zb+ ab~ In Z
(13.37)
418
K.J. Wiese
and a scaling function for the field r
1
1
0
v(b) = ~ - ~/3(b)~-~ In ( Z Z t ) .
(13.38)
We are now in a position to calculate the exponent v* in one-loop order. The /3-function is at this order
= - e b + 2b 2 + O(b 3) + O(b2e),
(13.39)
and the scaling function v(b*) becomes v(b*) = 21
210
~b* + O(e 2)
=
e
d- O(e 2)
+
1
e
= g k- ~" -+- 0(62).
(13.40)
This renormalization scheme is also used in 4~4-theory. At one-loop order, no renormalization of the wave function is necessary. Only the reduced 'temperature' t is renormalized. There is another scheme, equally useful, to perform the renormalization of polymers, which is also used in the broader context of polymerized membranes. This scheme also works for infinite membranes. Naturally, for infinite membranes, no renormalization of t can occur as it is identically zero. It is also known for the renormalization of standard field theories that one has the choice to work either in a massive (t ~ 0) or a massless (t = 0) scheme. For the polymer model, let us find a renormalization scheme where t is not renormalized, and therefore the limit t --+ 0 can be taken without problem. The key observation is that only the combinations Z Z t and Z b Z d/2 enter the renormalization group calculations, and these combinations are left invariant by changing the Z-factors to !
Zt=l, (13.41)
Z' = Z Zt , t Z b =
ZbZ
e-2 t
9
For a derivation of this property as a consequence of the rescaling invariance of the underlying Hamiltonian, see Appendix E. In terms of the modified Z-factors,
2
Polymerized membranes, a review
419
we obtain f l ( b ) --
-eb
l 4- b ~ In Z~, -t- d b ~ In Z'
v ( b ) --- .l . 1. f , (. b ) 0 In Z ' . 2 2 Ob
(13.42)
This is the scheme used in the rest of this review, and it is the only suitable one for higher loop calculations. On the other hand, it may lead to some confusion as it necessitates a renormalization of the field, even in the case of polymers. This may not have been expected from the one-to-one correspondence on the level of diagrams for the N --+ 0 limit of 4~4-theory, and polymers. As shown above, the two schemes are completely equivalent and one may use the one better suited to the problem at hand. Let us stress another important difference between the two approaches. When using the MOPE, the divergence of a dipole on approaching its ends was obtained in (3.75). Specializing to polymers gives the result
x ..... y
= Ix - y l - d / 2 1
l lx -- yl l-d~2 -+- +
2
_
.
.
.
o
(13.43)
The divergence proportional to the operator 1 can be subtracted by analytical continuation. In the absence of any boundary and for infinite membranes, this term has no effect on the renormalization functions. The second term is more serious and has to be subtracted. This is done by renormalization of the field, thus introducing the renormalization factor b Z' = 1 -~ 2e'
(13.44)
equivalent to Z' defined in ( 13.41). Upon expanding the Hamiltonian, this yields a counterterm proportional to + , which cancels the divergence. Let us now study the renormalization of the coupling constant in this scheme. Using the MOPE, we can write down the following two UV-divergent configurations: *~'--~~
:~~,
(13.45)
from which we shall extract terms proportional to the interaction -denote as
(:i~)
1-"
*)t_i,
(...~...~. e.
~)t_l ,
-, which we
(13.46)
420
K.J. Wiese
where L = t - l plays the role of the IR cut-off. The first is written in the notation of polymer theory as
({-~) l= =)t-! = ) i i ( "
(13.47)
Indeed this diagram was subtracted when we renormalized the interaction in (13.36), where we also subtracted the term
(:
:~-~-~'=~ =)t-' = '"~"(
"
(13.48)
The MOPE (3.92) now tells us that _~=-.
.
.
.
-- x ~ . , . . . .
(13.49) .
This result implies that having introduced a counterterm for @.., i.e. a renormalization of the field, no counterterm for the diagram in (13.48) is needed. We can check for consistency by comparing the fl-functions from the two schemes at one-loop order. In the massive scheme, we had ,...
fl(b)=-eb+b2 ( ) i i ( e +~" " C e) +O(b3)+O(b2e)'(13"50) In the massless scheme, we obtain
fl(b)
-+-)~) + O(b3) + O(b2e) (13.51)
It is now easy to see that expressions (13.50) and (13.51) are equivalent up to order O(b 3) and since
O(b2e),
and
d(~.2 . +)~ = -~d= l + O(e).
(13.53)
Another observation is that in the massless scheme, vertex operators like e ik[r(x)-r(y)] a r e finite, whereas in the massive scheme they require an additional renormalization. Let us also note that these relations can be understood within the concept of redundant operators, discussed in Section 4.1.
2
13.3
421
Polymerized membranes, a review
Generalization to N colours
Having performed a careful analysis of the different renormalization schemes, we are now in a position to generalize to the case N > 0, i.e. to an arbitrary number of self-avoiding polymer loops. To this aim, we introduce polymers of N different colours, and for the time-being, work in the massive scheme. In addition to ~
,
which renormalizes the chemical potential t, there is now a second
contribution, namely
OO This diagram is easily factorized as
0 - - 0 - 0 xO,
(13.55)
and is therefore equivalent to the digram already encountered in (13.24) and absorbed in Zt (for N = 0 in (13.28)). Let us now determine the combinatorial factor: a configuration (13.56) can be made out of one polymer or out of two polymers. The latter comes with an additional factor of N, accounting for the N different colours introduced above as well as with a relative factor of 1/2 for the additional connected component as introduced in (13.21) t. Zt is thus modified to Zt = 1
1+ E
=l+~-~e
+ O ( b 2)
e 1+-~-
+O(b2).
(13.57)
This is indeed the same combinatorial factor as derived from N-component theory. For the renormalization of the coupling constant, in addition to
)ii(
and
.~..~..(
q~4_
(~3.58)
*Note that in the original article (Wiese and Kardar, 1998a), no factor of 1/2 was introduced in the definition of (13.21); in compensation, it was argued that ( [ ~ could be made out of one polymer in two different ways rather than in one way. The same additional factor of 2 was associated to the configurations in (13.58), leading to the same final result for v* as given here. For membranes, the normalizations are accordingly modified (see footnote on p. 423).
422
K.J. Wiese
there is the possibility that an additional loop mediates the interaction between two given polymers, described by a configuration ) " O " ( "
(13.59)
The configurations in (13.58) are realized in two different ways each in the high-temperature expansion, while for (13.59) there is only one realization which comes with a factor of N for the N different colours. Zb is therefore modified to Zb = 1 - t - -
-t-
--
4":-
E'
....
e
8
b(8 + N) = 1 + ~ . 4e
(13.60)
Evaluating the critical exponent v* as before now yields 1 e2+N v* = - -t ~ . 2 28+N
(13.61)
It is again possible to switch to the massless scheme. At this stage this is not very enlightening, as for polymers all diagrams are essentially equivalent. We will therefore discuss this scheme in the context of membranes, which are introduced in the next section.
13.4
Generalization to membranes
We shall now apply the above construction to polymerized tethered membranes, as introduced in Section 3. Including a chemical potential to -- tZt, the Hamiltonian reads 7-/= 2 - D
~(Vr(x)
+ blzeZb
(r(x) - r(y)) + tZt~, (13.62)
with the same conventions as before (see Appendix A), and
:= f oax = so f .
(13.63)
In (3.15), we had obtained the free correlation function for infinite membranes as Co(x) = Ixl 2-o. For finite membranes, the latter is modified. The reason is that any function Co(x) defined on a closed compact manifold fulfils
f ACo(x)
-
O.
(13.64)
2
Polymerized membranes, a review
423
For the correlator to satisfy the above condition, the usual Laplace equation, AC0(x) ~ gO(x), has to be modified to 1
1
AC0(x) = gO(x)
(13.65) ( 2 - D)SD f2' where f2 is the volume of the compact manifold. The numerical prefactors come from our choice of normalizations in (13.62) (see Appendix A). In the infinitevolume limit, the correction term disappears, and the usual equation is regained. It is then easy to deduce that
1(1
)0
CO(X) - -~ -~(r(x) - r(y)) 2 -- Ix - yl 2 - ~
vS-----~~lx - y12+ subleading terms. Dr2
(13.66)
The coefficient of the correction term clearly agrees for D = 1 with the exact result for closed polymers in (13.17). The considerations of Section 13.2 can now be generalized to the case of membranes 9 The free density of a single polymer, i.e. the sum over all sizes of a noninteracting polymer, (13.21), is generalized to the membrane-density*
z
:= =
c(D) o 8
f
d~2 f2 ~-vd/De-t~2
-fi-
E. c ( O ) $ 2 F (_~ D 8
1 ) t e/D-1.
_
(13.67)
We have chosen to integrate over a logarithmic scale,
dx x
1 dr2 Dr2
To emphasize the arbitrariness of this choice, we have included an additional factor of c(D), which is further discussed in the next section, and a factor of $2/8. The latter factor is chosen in order to render the final result as simple as possible 9 These factors are important, as they also appear in the ratio of divergences due to self-interactions of one membrane, and those of interactions with other membranes 9 The factor f2 in the integrand of the above equation originates from the possible choices of a point x0 on the membrane, while the factor
~2-vd/O
)
~d (r(xo)) 0'
,368,
*As in definition (13.21) (see footnote on p. 421), an additional factor of 1/2 has to be introduced in (13.67) at variance with the original article (Wiese and Kardar, 1998a). Here, we furthermore introduce a factor of with which we reproduce the final result of Wiese and Kardar (1998a). The reason is a conceptual improvement: whereas in Wiese and Kardar (1998a), the combinatorial factor for an additional connected piece was taken in analogy to the polymer case, here we succeed in its explicit calculation (see (13.88)and (13.92)).
$2D/4,
424
K.J. Wiese
is the probability that at this point the membrane is attached to a given point in space. As usual, we have introduced a chemical potential proportional to the size of the membrane. Let us now generalize (13.24) for the effect of one Sa-insertion from the expansion of the interaction. For the time being, we fix the size of the membrane to ~, and evaluate
fxfy{~a(r(x)-r(y))}o .
(13.69)
This integral is (see (13.20))
KaSof Co(x)-d/2'
(13.70)
and we have to remove all UV-divergent contributions. To do so, we expand for small x. Up to UV-convergent terms, this is (using (13.66))
Co(x)-a/2
~-2fdxx D(-vd --x
~O
x
dvSo x O-vd + ...). DF2
+-5
(13.71)
The first term is strongly UV-divergent and has to be subtracted by a finite part prescription, while the second is (up to terms of order c ~ equal to
I faE/o
(13.72)
Note that we have cut off the integral at the upper bound X m a x = ~'21/D. This procedure may appear rather crude, but the residue of the pole in 1/e is not affected (see Section 4.3). Upon integrating over all scales, the membrane density (to first order) reads
z l" = ( ~ )
b"" Z " T ~ - -g-o V - a - a
+ ~162
b(fat)e/o + D
8
. . . ] , (13.73)
~
which upon integration over fa results in
E b
1
z l ' ) = zl ~ 1+ ~ + . . . .
(13.74)
Note the difference in factor of 2 between (13.72) and (13.74), which is due to the nested integrations as discussed in Section 6.3. This factor of 2 can be interpreted
2
Polymerized membranes, a review
425
as being geometric. The counterterm is only needed in the half-sector x < ~"2lID and not in the half-sector x > fll/O. Introducing now a counterterm for t yields b
Zt = 1 + - - .
(13.75)
2e
The bare and renormalized quantities are then related by generalizing (13.26) to r0 = x//Zr, to = Zt t,
(13.76)
bo = I,Ze Z d/2 Zbb,
leading to the renormalization group functions (compare with (13.37) and (13.38)) fl(b) = #
-eb
+1
b=
1 + b ~ lnZb + d b ~ lnZ
0
v(b) =
2-D
l
2
2
O In( zz}Z_D)/D ). fl (b, ~-~
(13.77) (13.78)
(The derivation is given in Appendix D.) The combinations Z Z } 2-D)/D and Zb zd/2, which enter the renormalization group calculations, are left invariant by changing the Z-factors to t
Z t - Zt/Z~,
Z t-
ZZ (2-o)/o,
(13.79)
Z bt = Zb Zeot/D-2 For a derivation of this property as a consequence of the rescaling invariance of the underlying Hamiltonian, see Appendix E. In order to eliminate the renormalization of t, we chose Zot = Zt ,
(13.80)
resulting in !
Zt=l, Z t - - Z Z ~2-D)/D
(13.81)
Z ot = Z b Z t / D - 2 ,
and the renormalization group functions fl(b) =
v(b) =
-eb
1 + b ~ In Z; + ~-b~ In Z' 2- D 2
1 0 In Z ' 2 $(b)- O---b "
(13.82)
426
K.J. Wiese
With this change of variables, (13.75) is replaced by Z~ = 1, and
2-Db
Z' = 1 -~
2D
e
.
(13.83)
The above result is precisely that obtained by using the multilocal operator product expansion technique for infinite membranes, see (3.105) and (3.112). The interpretation of this formula is simple, as l +
-1 -- 2D
(13.84)
is just the residue of the diverging contribution from the MOPE, of one gd_ insertion. For N = 0, the renormalization of the coupling constant in the massless scheme is analogously (see (3.113)) (13.85) with
(!:~)l-
~')c - 2
- D F(22~DD) "
(13.86)
Alternatively, in the massive scheme (Z = 1), z~ =
(
1+-~:~i: g
- ' + ;b
i l) "---"
• z,~
+ l)
Let us now study the generalization to N components in the massive scheme. The membrane density to first order in b is
-
~ ~ , _ z _ J
-
2 N bS----ffD t Z e ~ C --
+ O (b2).
(13.88)
The factor of 2 in front of (-~--{'-~ is due to the fact that the ends of the Sinteraction may be interchanged between the two membranes, whereas the factor of N counts the number of different colours of the additional membrane. We now calculate the diagram ~ for membranes, a s
= c(D) S2Df dr2 ~2 ~2-vde-tf2 D 8 f2
(13.89)
2
427
Polymerized membranes, a review
We already have given the derivation of a similar integral in (13.67). The only difference is that now a second factor of ft appears to take into account the additional point which moves on the membrane. Integration over f2 yields
0
= c(D)D$28F(e)-~ t-e/D=c(D)-ff-S2-e l te/D +O(e~
(13.90)
Equation (13.75) is therefore modified in the same manner as (13.57) to
Zt-
1 +~e
1+
2
"
(13.91)
There are several possibilities to derive the modification to the renormalization factor of the interaction. For a direct derivation generalizing (13.60) we note that the effective interaction is modified by a new term proportional to N,
)(
sg
) " 0 " ( ( 1 3 . 9 2 )
The renormalization factor Z b therefore becomes Z b =
1+-
!i~)--
-- + 1 +
4
"
(13.93)
E
It is now easy to derive the renormalization group functions --)+1+
c(D)N) 4 +
O(b2e)'
O(b3) +
(13.94) and
v(b)-
2~ 2
1+ ~
1 -~- -------~-
+ O(b2).
(13.95)
At the nontrivial (IR-stable) fixed point, this yields the critical exponent to order e
e V :r
2- D 2
1+ c(D)N
1 -I
2 2D
1
I-' ( 2_--~DD)2
2 - D F(22_-~DD)
+1+
,
c(D)N 4
representing our central result for the generalized O(N)-model.
(13.96)
428
13.5
K.J. Wiese
The arbitrary factor c(D)
In calculating the free partition function in (13.67), we introduced an arbitrary factor of c(D). In principle, any function of D which satisfies c(l) = 1,
(13.97)
reproduces the correct result for linear objects. The additional freedom (or ambiguity) is apparently a reflection of the nonuniqueness of the generalization to manifolds. Even after restricting to the class of hyperspheres, there is a remaining ambiguity in the choice of the measure for the size of these manifolds. This arbitrariness carries over to our generalization of the O (N)-model to N-coloured membranes. (Note also that c ( D ) is independent of the introduction of factors like the 1/(2 - D) in (13.62).) Two choices have been studied in Wiese and Kardar (1998a), namely c(D) = 1 and c ( D ) = D. The second one seems to be the best suited for numerical extrapolations, and we shall in the remainder focus on it. 13.6
The limit N - , cx~ and other approximations
As in the O(N)-model, it is possible to derive the dominant behaviour for large N exactly. In the standard q~4-theory, one starting point is the observation that (($2)2(r)) = (S2(r)) 2 ,
(13.98)
since in the limit N --, oo, spin-components of different colours decouple (Amit, 1984; Ma, 1986; Zinn-Justin, 1989). Here, we pursue a slightly different approach, based on the diagrammatic expansion. Note that for N --, oo, only simply connected configurations survive. (The vertices are made out of membranes, the links out of ~d-interactions.) For example, the diagram 0 : : ( ) w h i c h
~
"O
is doubly connected, and the diagram
which includes a self-interaction, each have one factor of N less than
the simply connected graph O " O " O " The leading diagrams for the membrane density at the origin are then given by (we suppress a couple of geometric factors which are unimportant for the derivation)
Q ,-- o
+o . . o
+o-.o--o
+o--o
+(13.99) ....
2
Polymerized membranes, a review
429
The above sum can be converted into a self-consistent equation for f by noting the following: successive diagrams can be obtained from the first (bare) diagram by adding to each point of a manifold a structure that is equivalent to f itself. This is equivalent to working with a single noninteracting manifold for which the chemical potential to is replaced by an effective value of to + b0 f . Calculation of f for this manifold proceeds exactly as in (13.67), and results in the integral f--
f
d~21-vd/Oe-f2(to+b~ f2
(13.100)
The above integral is strongly UV-divergent, and leads to a form f = B(to + boY) - ~ a - I
+ A,
(13.101)
where B is a constant. The strong UV-divergence, controlled with an explicit UV cut-off, is absorbed in the constant A. It is dropped in a dimensional regularization scheme, as in (13.67). The radius of gyration R is now related to f as follows: from (13.100) we note that to + b o f is the physical chemical potential conjugate to f2, thus leading to a typical volume of f2 ~ 1/(t0 + b o f ) . Since there are no self-interactions in the effective manifold introduced above, its radius can be related to the volume by R ~, f2 v/~ Thus, up to a numerical factor which is absorbed into the definition of R, we obtain 2D R - rZ-r5 = to + bo f . (13.102) Eliminating f in (13.101) with the help of (13.102) yields 2D
20 -d
R-rZ-t~ - (to + boA) + boB Rr:-~
.
(13.103)
Identifying the difference in temperature to the critical theory as [ = to + boA,
(13.104)
the critical theory is approached upon taking [ ~ 0 and R ~ oo. This occurs if and only if d is larger than the lower critical dimension d>dl=2_
2D
D 9
(13.105)
If d is in addition smaller than the upper critical dimension, i.e. 4D d < du = 2----Z-~ ,
(13.106)
the left-hand side of (13.103) vanishes faster than the R-dependent term on the fight-hand side, and we obtain the scaling relation R "~ [ - l / ( a - r
(13.107)
430
K.J. Wiese
In the large N limit, the exponent v* is therefore given by *
I)N._.,c ~
D
--
w
2D
d
"
(13.108)
2-0
We can verify that the standard result (Zinn-Justin, 1989) is correctly reproduced for D = 1 as 1 * (13.109) VN_.~(D = 1) = ~d - . 2 Note that for d > du, the leading behaviour from (13.103) is 2-D
R ~/---rb--,
(13.110)
implying the free theory result 2-D
v- ~ . 2 13.7
(13.111)
Some more applications
In this section, we briefly discuss some possible applications. As explained in Section 7, to extract the physically relevant O(N) exponent for D = 1, one has the freedom to expand (13.96) about any point (D0, do) on the critical curve e(Do, do) = 0, see Fig. 12. As discussed in Section 7 and depicted in Fig. 33, the resulting extrapolation for v* varies with the extrapolation point. (We have found the extrapolation for v*d (see Section 7) to be the best converging one, and have chosen it for the example.) Guided by previous results for polymers and membranes (Wiese and David, 1997), the criterion for selecting a particular value from such curves is that of minimal sensitivity to the expansion point, and we thus evaluate v* at the extrema. The broadness of the extremum then provides a measure for the quality of the result, and the expansion scheme. Although we examined several such curves, only a selection is reproduced in Fig. 33. Our results are clearly better than the standard one-loop expansion of v* = 1/2 + (N + 2)/[4(N -t- 8)]. In analogy to tethered membranes (David and Wiese, 1998), we expect the above expansion scheme to be better controlled than the traditional e-expansion. (The e-expansion should become quasi-convergent for D ---, 2.) However, since the exponents of the O(N)-model are already known to high accuracy, the generalization to O(N, D) is valuable if it offers insights beyond the standard field theory. Furthermore, the scheme will have limited appeal if it cannot be extended to other types of field theories. In the rest of this section we shall demonstrate that: (a) the model provides insights about the boundaries of droplets at criticality
2
Polymerized membranes, a review
431
0.8
V0.7
3
......
0.6~ 0
0.5
N
0
1
2
0.601
0.646 0.676 0.697
v*, from Zinn-Justin (1989) 0.589 0.631
0
"0
3
, .
v*, our result
4
~
0.5
1 Do
0.676 0.713
1.5
2
Fig. 33 Extrapolations for the exponent v* of the O(N)-model in d = 3, using the expansion of v*d with c(D) = D. The dashed lines represent the best known values from Zinn-Justin (1989).
in Ising models; (b) a generalized manifold model is constructed with cubic anisotropy, which exhibits a reverse Coleman-Weinberg mechanism not present in standard field theory. Furthermore, it provides us with a one-loop fixed point for the random bond Ising model. For the Ising model (N = 1), a different geometrical description is obtained from a low temperature expansion: excitations to the uniform ground state are droplets of spins of opposite sign. The energy cost of each droplet is proportional to its boundary, i.e. again weighted by a Boltzmann factor of e -tfl. Thus, a lowtemperature series for the d-dimensional Ising partition function is obtained by summing over closed surfaces of dimension D = d - 1. For d = 2, the highand low-temperature series are similar, due to self-duality. For d = 3, the lowtemperature description is a sum over surfaces. What types of surfaces dominate the above sum? Since there is no constraint on the internal metric, it may be appropriate to examine f l u i d membranes. However, there is no practical scheme for treating interacting fluid membranes, and the excluded volume interactions are certainly essential in this case. Configurations of a single surface for N = 0, selfavoiding or not, are dominated by tubular shapes (spikes) which have very large entropy (Cates, 1988). Such 'branched polymer' configurations are very different
K. J. Wiese
432 I-
0.8"
/
0.6~ v", 0.4'.
J
08:
08:
o.6i. . . . . / v'. jz 04~---.............
06: v" 0.4:
JJ
0.2i
0.2:
a=2.75
'
0.2'.
d=2.25
,t=2.5 ,,~
....
o,8
/'
i
i ~ o , , ..... ~ ~
....
o:~ ....
0.6t..................... / "'1 /~ 0.4
"/
/
~o0
.... /
0.2! ,:~ ....
~"17-
a=.~o ,,,
,o,,,~
Fig. 34 Test of (13.112) for Ising models in d = 2.25, 2.5, 2.75 and 3. The dashed curves are from the standard extrapolation (D = 1), while the solid curves are from the dual description (D = d - 1). The exponent v* is estimated from the maximum of each curve, which are obtained by extrapolating v*d with c(D) = D, linearizing in N, and dividing the result by d D. from tethered surfaces. However, for N # 0, it may be entropically advantageous to break up a singular spike into a string of many bubbles. If so, describing the collection of bubbles by fluctuating hyperspherical (tethered) manifolds may not be too off the mark. To test this conjecture, we compare the predictions of the dual high- and low-temperature descriptions. Singularities of the partition function are characterized by the specificheat critical exponent a ( D , d , N), or (using hyperscaling) through In Zsingular "~" It - tc] v*d/D. The equality of the singularities on approaching the critical point from low- or high-temperature sides, leads to a putative identity v*(l, d, 1 ) -
v*(d - 1, d, 1)
d-I
.
(13.112)
Numerical tests of the conjecture in (13.112) are presented in Fig. 34. The extrapolated exponents (the maxima of the curves) from the dual expansions are in excellent agreement. Nevertheless, higher-loop calculations would be useful to check this surprising hypothesis. The simplest extension of the O(N)-model breaks the rotational symmetry by inclusion of cubic anisotropy (Amit, 1984). In the field-theory language, cubic anisotropy is represented by a term u ~-~i ~)4, in addition to the usual interaction o f b ~~ij ~ 2 ~ ] .2 In the geometric prescription of high-temperature expansions, the anisotroplc coupling u acts only between membranes of the same colour, while the interaction b acts irrespective of colour. Stability of the system of coloured membranes places constraints on possible values of b and u. To avoid collapse of the system, energetic considerations imply that if u < 0, the condition u + b > 0 must hold, while if u > 0, we must have u + N b > 0 (Amit, 1984; Wiese and Kardar, 1998a). These stability arguments may be modified upon the inclusion of fluctuations: In the well-known Coleman-Weinberg mechanism (Amit, 1984), the RG-flows take an apparently stable combination of b and u into an unstable regime, indicating that fluctuations destabilize the system. In
2
Polymerized membranes, a review
433
Fig. 35 Regions with different RG-flow patterns in the (N, D)-plane (top), and the corresponding RG-flows (bottom); shaded regions are unstable. the flow diagrams described below, we also find the reverse behaviour in which an apparently unstable combination of b and u flows to a stable fixed point. We interpret this as indicating that fluctuations stabilize the model, a reverse Coleman-Weinberg effect, which to our knowledge is new. We have shaded in grey, the unphysical regions in the flow diagrams of Fig. 35. As in their O ( N ) counterpart, the RG-equations admit four fixed points: the Gaussian fixed point with bb = UG * = 0; the Heisenberg fixed point located at b*n ~ O, u *n = 0 ; the Ising fixed point with b t = O, u I ~ 0; and the cubic fixed point at bc* ~ 0, u* ~ 0. Furthermore, as depicted in Fig. 35, there are six different possible flow patterns. In the O(N)-model, the flows in (i) and (ii) occur for N < 4 and N > 4,
434
K. J. Wiese .7
. . . . .
"cubic II , ', Ising
-
i
0.65
, ...
'~\\
-~,. . . .
~
"~i
~/
\\..
0.64
0.55
,
\\\\
.
\
\
0
i i ,
\ \\
0.5
i
t
,cubic \
\
0.5
.
_
i i A
1
.
.
Do
.
.
1.5
2
Fig. 36 Extrapolations of v* from the expansion of v*d with c(D) = D, for the O(N)model in d = 3. Exponents at the Heisenberg fixed point for N = 0 are compared to those of the Ising and cubic fixed points. The crossing of the latter curves yields an estimate of v* = 0.6315 for the 3d lsing model.
respectively. The other patterns do not appear in the standard field theory, as is apparent from their domain of applicability in the (N, D)-plane in Fig. 35. Note that there are two stable fixed points in three out of these four cases. The N --~ 0 limit of the above models is interesting, not only because of its relevance to self-avoiding polymers and membranes, but also for its relation to the lsing model with bond disorder. The latter connection can be shown by starting with the field theory description of the random bond Ising model, replicating it N times, and averaging over disorder (Harris and Lubensky, 1974). The replicated system is controlled by a Hamiltonian with positive cubic anisotropy u, but negative b = -or (~r is related to the variance of bond disorder). From the 'Harris criterion' (Harris, 1974), new critical behaviour is expected for the random bond Ising system. But in the usual field theory treatments (Harris and Lubensky, 1974), there is no fixed point at the one-loop order. In our generalized model, this is just the borderline between cases (i) and (iii). However, we now have the option of searching for a stable fixed point by expanding about any D ~ 1. Indeed, for N = 0 and 1 < D < 1.29, the cubic fixed point lies in the upper left sector (u > 0 and b < 0) and is completely stable, as in flow pattern (iii). The extrapolation for v* at the cubic fixed point is plotted in Fig. 36, where it is compared to the results for the Heisenberg and Ising fixed points. The divergence of v* on approaching D -- 1 from above is due to the cubic fixed point going
2 Polymerized membranes, a review
435
to infinity as mentioned earlier. Upon increasing D, the Ising and cubic fixed points approach, and merge for D = 1.29. For larger values of D, the cubic fixed point is to the right of the Ising one (b~ > 0), and only the latter is stable. Given this structure, there is no plateau for a numerical estimate of the random bond exponent v~x~, and we can only posit the inequality vDO*> Vising*, which is also the consequence of an exact argument (Chayes et al., 1986). While this is derived at one-loop order, it should also hold at higher orders since it merely depends on the general structure of the RG-flows. One may compare this to four-loop calculations of the random bond Ising model (Mayer, 1989), which are consistent with t~ = 0, i.e. at the borderline of the Harris criterion (Harris, 1974), with v* = 2/3.
14
Large orders
If perturbative calculations are simple at first order, as discussed in Section 6, they present considerable difficulties at second order, and require a lot of analytical and numerical work. An important issue is to understand if these calculations make sense beyond perturbation theory, or if nonperturbative effects destroy the consistency of the approach. A first step is to understand the large order behaviour of perturbation theory. In this section we shall formulate the problem of the large order behaviour for the Edwards model in a way which is directly applicable both to polymers and to membranes. Using the formulation of the self-avoiding membrane (SAM) model as a model of a 'phantom' membrane (without self-avoidance) in a random imaginary external potential V, we show that the large orders are controlled by a real classical configuration for this potential V, which is the analogue for SAM of the instanton for ~4-theory (Lipatov, 1977a,b; Le Guillou and Zinn-Justin, 1990). This 'SAM instanton' potential V is the extremum of a nonlocal functional S[V], which cannot be calculated exactly. We obtain the general form for the asymptotics of the term of order n, which is
n d12 ( - C ) n (n!) 1-e/D,
(14.1)
where as usual D is the internal dimension of the membrane, d the dimension of bulk space, e = 2D - d(2 - D)/2 the engineering dimension for self-avoidance and C a positive constant depending on D and e (or d). This behaviour is universal: the constant C obtained from the instanton does not depend on the internal shape or topology of the membrane. This section closely follows David and Wiese (1998), where the problem was first addressed.
436
K.J. Wiese
14.1 Large orders and instantons for the SAM model We consider a D-dimensional manifold .At with size L and volume V = L D (typically the D-dimensional torus TD = [0, L] D) in d-dimensional Euclidean bulk space. The partition function is Z(b; L) = f D[r] e -7"tIr;b'L1
(14.2)
with the Hamiltonian
7-t[r" b, L] =
dDx ~(Vr(x)) 2 + ~
dDx
dDy ~d(r(x) -- r(y)).
(14.3)
b > 0 is the repulsive two-point interaction coupling which describes selfavoidance, and r throughout this section the bare field. Also note the difference in normalizations from (3.1); rescaling the field r(x) -~ r ( x ) ~ / ( 2 - D)SD in (3. l) and not absorbing any geometrical factor in the 8d-distribution, b and b0 are related by 2( 47r ) d/2 b -- b0 S2D
( 2 - D)SD
.
(14.4)
The functional integration measure D[r] -- Fix ddr(x)/Zo is normalized such that the partition function of the free Gaussian manifold Z(b = 0; L) = 1. By dimensional analysis, the partition function (14.2) only depends on the dimensionless coupling constant g = b LC
(14.5)
Z(b; L) = Z(g; L = 1) = Z(g)
(14.6)
via and is defined as a series
(x)
Z(g) = ~
z, g".
(14.7)
n=0
Of course, Z (g) also depends on the shape of the manifold .A4. Let us assume that Z(g) is analytical around the origin for -Tr < arg(g) < Jr, and has a discontinuity along the negative real axis. This assumption is natural, since for g < 0, the membrane is collapsed and perturbation expansion is performed around an unstable classical state. Then we can write Zn as a dispersion integral
Zn =
dg - n - l ~l~ g Z(g) -- f0 -~176 ~dg g -n 1 Im(Z(g + i0+)). Y/"
(14.8)
2
Polymerized membranes, a review
437
To obtain the behaviour for large n, it turns out that it is sufficient to evaluate the integral in (14.8) in a saddle point approximation. Indeed, we shall show that, at least for 0 < e < D, the integral at large n is dominated by the discontinuity of Z(g) at small negative g. Moreover, Z(g) is dominated by a saddle point when re-expressed as a functional integral over properly defined auxiliary fields. The Hamiltonian (14.3) is nonlocal and involves a distribution of the field r. It is convenient to rewrite the path integral as
Z(b;L)=fDtr]fZ~[V]e -~'tr'v:b'L]
(14.9)
with the new effective Hamiltonian
7-l'[r, V; b,L] =
d~
~(Vr(x)
+ V(r(x))
- ~
ddr V(r) 2.
(14.10) This representation is nothing but the generalization of the well-known formulation of the Edwards model as a model of free random walks in an (imaginary) annealed random potential. As above, Z is a function of the dimensionless coupling g and we replace b ~ g and L ~ 1 as in (14.5) and (14.6). As argued before, we aim at calculating the partition function for small negative g. For that purpose, it is convenient to rescale the coordinates and the potential V (r) x ~
l
(_g)Tra--; x,
r ~
2-D
(_g)2(D-,~ r,
-D
V -+ (_g)~7-~ V,
(14.11)
-I
so that we now consider a membrane with size/~ = (-g)zr:7 and volume ~) = /~D = (_g)7~_~
(14.12)
This yields the rescaled Hamiltonian 7-/'resc[r, V;/~] =
dDx
(Vr(x)) 2 + V(r(x))
- ~
ddr V(r) 2.
(14.13) The integral over r for fixed potential V defines the free energy density g[ V] of a 'phantom' (i.e. non-self-avoiding) membrane in the external potential V e - 9~'[vl =
a~ 89
2+
V(r(x)).
(14.14)
The partition function finally becomes
Z(g) = f D [ V l e - ~'[gIVl+ 89f ddr V(r) 2] .
(14.1 5)
438
K.J. Wiese
The crucial point of this formulation is that according to (14.12), as long as 0 < e < D,
(14.16)
the limit g --+ 0 - corresponds to the thermodynamic limit when the volume ]) --+ oo. In this limit the free energy density E[ V] has a finite limit so that the volume appears only as a global prefactor in the exponential of (14.15). Hence in the large V limit the integral (14.15) is dominated by a saddle point Vinst, which is an extremum of the effective energy S[ V] for an infinite and flat membrane. The latter is defined as
if
S[VI = g[Vl + ~
dar V(r) 2,
(14.17)
where E[VI is defined in (14.14) as the free energy density of an infinite flat membrane in the potential V. This saddle point Vinst(r) is the nontrivial instanton, since the action S of the trivial extremum V (r) = 0 is real and does not contribute to the discontinuity of Z(g). Moreover, as the instanton is obtained through the thermodynamic limit L --, oo, it is independent of the shape of the initial membrane. This implies that the large-order behaviour of perturbation theory is universal, and does not depend on the internal geometry of the membrane. Let us now derive the saddle-point equations: The variation of the free energy density is in general
~g[v] ~V(r)
= (6[r]) v,
(14.18)
where ~[r] is the normalized density of the membrane ~ [ r ] = p[r] _ _1 f dDx~a(r -- r(x)) V VJ
(14.19)
(which has a finite limit when the volume becomes infinite), and ( ) v denotes the expectation value for the phantom membrane in the potential V, as defined in (14.14). Hence extremizing S[V] leads to the variational equation for the instanton potential Vinst 0 = (~[f])Vins t "['- Vinst(r).
(14.20)
Let us postpone the solution of (14.20) and first ask what the consequences of the existence of an instanton for the large-order behaviour are. Denoting by Sinst the action for the instanton S[Vinst], we deduce from (14.12) and (14.15) that for small negative g, the discontinuity of Z(g) behaves as
[
o
Im(Z(g)) ~ exp _(_g)~-~-7 Sinst
]
(14.21)
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Polymerized membranes, a review
439
and the integral representation for Zn (14.8) can be evaluated by the saddle-point method at large n. This saddle point is at
,Sinst BC
~
] 1- t / D
n(1 - e / D )
(14.22)
and replacing the integral in (14.8) by its value at gc gives the large n behaviour at leading order 1 -
Zn "~" (-- C) n (?/!)l-e/D
,
C --
e/D]l-e/D
~inst
(14.23)
Let us briefly discuss this result. For 0 < e < D, perturbation theory is divergent with alternating signs 9 For e = 0, one recovers the typical factorial behaviour (--C)nn! of field theories, provided that Sinst remains UV finite. As we shall see in the next subsection, our result (14.23) coincides for D = 1 with the large-order behaviour deduced from the q~4-model with n = 0 components. The reasoning seems to break down at e - D, but we shall see that in fact the factor of C, when considered as a function of D and e, is regular at e = D and can be continued to the region e >_ D. Thus, the asymptotics (14.23), although derived for 0 < e < D, are valid in the whole physical domain 0 < e < 2D. A more rigorous argument is as follows: (14.21) and (14.22) are still valid for e > D; the instanton then governs the behaviour of the discontinuity of Z ( g ) at large g. This means that the saddle point of (14.23) for large n now is at large negative g. To go beyond these estimates, one must (i) compute the instanton and its action, and (ii) integrate the fluctuations around the instanton in (14.15). If one assumes that this calculation goes along the same lines as in standard field theory, one must first isolate the zero modes, i.e. the collective coordinates of the instanton. As we shall see later, the instanton Vinst is rotationally invariant and is characterized by its position in d-dimensional space only. Thus it has d zero modes, each of them gives a factor of V 1/2 (by a standard collective coordinates argument), and the remaining fluctuations 6• V (orthogonal to the translational variations ~u V ~ OVinst/Or #) give a finite determinant A. Therefore we expect the semiclassical estimate for the discontinuity to be I m ( Z ( g ) ) ~ .4 -1/2 ' ~ e -'~'Sin~t
(14.24)
and that the large n behaviour is more precisely Zn -- .A, nd/2 (-- C) n (n !) l - e / D [1 + . . . ] .
(14.25)
Finally we shall see that the action of the instanton remains finite in the limit e --+ 0. As in standard 4~4-theory, one expects UV divergences to appear only
440
K.J. Wiese
for fluctuations around the instanton, and that these divergences are cancelled by the same renormalizations as in perturbation theory. This would imply that our large-order estimate (14.23) is also valid for the renormalized theory at e = 0, in particular for the renormalization group functions which enter into the eexpansion of the scaling exponents. Renormalization, however, has to be taken into account when evaluating the constant .,4' in (14.25).
14.2
The p o l y m e r case and physical i n t e r p r e t a t i o n of the i n s t a n t o n
Before discussing membranes, let us study in detail the special case D = 1, where the model reduces to the Edwards model for polymers. Using the wellknown mapping between the problem of a Brownian walk in a potential V (r) and quantum mechanics of a single particle in the same potential, the free energy density E[ V] of a linear chain fluctuating in a potential V (r) is in the thermodynamic limit given by the lowest eigenvalue E0 of the operator H =
A 2
+ V(r),
(14.26)
where A is the Laplacian in d dimensions. The derivation of the above equation is similar to what has been done in Section 4.4. Thus we have E[ V ] - E0.
(14.27)
Denoting by qJ0(r) the ground state wave function, and using (14.20) and the standard result from first-order perturbation theory (3[r]) v =
6E0[V] 6H - (q'ol I q ' o ) - IqJo(r)l 2 3 V (r) 3 V (r)
(14.28)
we obtain the instanton potential
Vinst(r) = -
(qt0(r)) 2.
(14.29)
The eigenvalue equation H qJ0 = E0 ~0 becomes nonlinear: 1
- A qJo + E0 g'0 + ~ 0 3 = 0. 2
(14.30)
Since qJ0 obeys the normalization condition IIq,0112
-- f ddr qJ0(r) 2
= 1,
(14.31)
the wave function q'0 and the ground state energy E0 are fully determined by (14.30) and (14.31). Equation (14.30) has nontrivial normalizable solutions for
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Polymerized membranes, a review
441
2 < d < 4 and E0 < 0. In addition, the ground state qJ0 is rotationally symmetric, i.e. does not vanish at finite r. The action for the instanton (14.17) finally reads
if
Sinst = E0 + ~
d dr q/0 4.
(14.32)
To make contact with the instanton analysis in the Landau-Ginzburg-Wilson (LGW) 4~4-theory with n - 0 components, note that (14.30) and (14.31) hold if and only if qJo and E0 are extrema of the action
1 tlj4 ] 2
S'[qJ, E] -- E + f dar [2 (Vqj)2 - E qj2
This is the standard LGW action with negative coupling associated with mass m 2 -- - 2 E . Moreover, at the extrema, the two actions are equal:
Sinst [q/0, E0] = S' [tP0, E0].
(14.33)
q14 and (14.34)
The relation becomes clearer by the change of variables I
9 (r)-
(4-2E) ~ r189
(14.35)
The action S' then reads d
S ' [ ~ , E] = E + with SLGW[4~] =
/
dar
SLCW[4~]
(14.36)
4 02 1 ]
(14.37)
4- d
[1
2 (V~)2 +
2
- ~4~ 4 .
We can extremize (14.36) with respect to E and 4) independently, and denoting by 4)0 and Eo these extremizing solutions, we get
, Eo --
- 2 SLGW[q~0]~~-~.
(14.38)
The change of variables in (14.35) was constructed such that the instanton action takes the simple form
Sinst = St[q/0, E0] =
( d _ 1) SLGW[q~0]~rcr-~.
(14.39)
Since for polymers (D -- 1) d / 2 - 1 -- 1 - e / D , we can use (14.23) to write the large-order constant C of the Edwards model as 1 -
C
=
SLGW[~0].
(14.40)
442
K.J. Wiese 20
1
15 o
10
'~.'* "
exact
5 0
1
2
3
4
d
Fig. 37 1/C as obtained from a numerical solution of (14.40), compared to the variational bound derived later in (14.57).
This result could have been derived directly from the standard field-theoretical formulation of the Edwards model as an n = 0 component (q~2)2 model. The equation for the instanton derived from the action (14.37) admits a regular solution r for any 0 < d < 4, so that nothing special occurs at the point d = 2 (i.e. e = D = 1) as one might have expected from (14.23). Let us note that since the 'mass' in (14.37) is equal to 4 - d, it is positive for d < 4 but vanishes at the critical dimension d - 4, so that the instanton solution ~0 still exists for d = 4. In Fig. 37 we plot c - l ( d ) for 0 < d < 4, as obtained from numerical integration. Note that for d > 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
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Polymerized membranes, a review
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
Polymerized membranes, a review
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 << 1 the true solution ~0 is well approximated by (14.59) (at least as long as Irl2(4 - d) << 1) with an instanton size r0 which vanishes as d ~ 4 as r0 ~
1
9
(14.60)
~/1 In(4 - d)l The corresponding instanton potential Vinst = -I~012 is also singular in the limit of d ~ 4 (it may be considered as a Dirac-like 8-function), and is very poorly approximated by the Gaussian variational solution at D = 1, d = 4 for the
2
Polymerized membranes, a review
447
potential V T M = -- 167r2e-16zr2r2
(14.61)
with positive width. As usual with variational methods, the approximation for the ground state energy is much better than that for the wave function.
14.4.2 Consequencesfor the e-expansion Of course, one is interested in the consequences of these large-order estimates for the e-expansion of the scaling exponents for self-avoiding membranes and polymers. Let us recall that in renormalized perturbation theory one computes the renormalization group ~-function ~ (g) as a power series in g of the form* /~(g) = --e g + BI g2 + O(g3).
(14.62)
Its zero at g* = e/Bl + O(e 2) is the IR fixed point which governs the scaling limit for large membranes. Other anomalous dimensions, like the dimension v(g) of the field r (which gives the fractal dimension of the membrane) can also be computed as a series in g. Their values at the fixed point g* give the scaling exponents of the membrane, and may be expanded as power series in e. By analogy with the ordinary Wilson-Fisher e-expansion for LGW field theories, let us assume that the large orders of the function/~(g) and of the other anomalous dimensions are given by the instanton estimate, and that they can be resummed by Borel techniques. We are not able at the moment to give any more precise argument for this last claim (which is still a conjecture even for the LGW theories). Then a simple calculation consists in estimating the 'optimal' order nopt beyond which the e-expansion starts to diverge. If we only know the first n terms of the expansion, we expect that for n < nopt 'ordinary' resummation procedures (like Pad6) will be sufficient. If n > nopt, or if one seeks higher precision, knowledge of the large orders and more sophisticated resummation methods are required. Assuming that for e = 0 the nth coefficient of/~(g) is of order (-C,g)nn!, and that we can approximate the fixed point g* by its first-order estimate e/Bl, the term of order n in the e expansion should behave as
(
--
n!.
(14.63)
BiJ
The optimal order nopt is obtained when the absolute value of (14.63) is the smallest, that is for Bl nopt e ~ J (14.64) C" *Strictly speaking g is now the renormalizedcoupling constant.
448
K.J. Wiese
With our choice of normalizations for the coupling constant b in the Hamiltonian (14.3), the one-loop coefficient of the fl-function is
BI---~[
4rt
]
$2~ 14 2 - D
F(22__~DD)
'
(14.65)
with So defined in 15. Let us replace C in (14.64) by the variational approximant CT M given by (14.57). Setting finally e = 0 in BI/C (since we are interested in the expansion around e = 0), we obtain the following variational estimate for the r.h.s, of (14.64) ~ nopte
16 (2_D)2
4
~ 1 I F (4-~D) 1 ~-:v I F(-~) 1-4 2 - D
(F (2--~OD))2 F(22__~DD)
1
.(14.66)
Let us recall that in practice the E-expansion is used as follows" in order to compute for instance the scaling exponent v* for a membrane with internal dimension D -- 2 in d-dimensional space, one starts from some point D' ~- D, e -- 0 (i.e. d' = 4D'/(2- D')), and uses an expansion in e and D - D' (or some more general expansion parameters) to evaluate v*(D'), which thus depends on the expansion point D'. v* is then taken as the best estimate v* (Dopt), as determined for instance by a minimal sensitivity criterion. Membranes (D -- 2) always correspond to e - 4, so setting e = 4 and replacing D by D' in (14.66) should give an estimate of the 'optimal order' nopt(D') for the e-expansion at D'. The result for nopt(D I) is plotted in Fig. 38. Some interesting comments can be made on this curve. For D' > 1.6, nopt(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
Polymerized membranes, a review
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
Polymerized membranes, a review
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)
Polymerized membranes, a review
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)
Polymerized membranes, a review
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'
.
.
.
.
.
yI
....
and this is defined as the integral over the MOPE coefficient, with all distances appearing in the MOPE coefficient, i.e. all distances included within the dashed lines, bounded by L. In general, this diagram will be diverging. Suppose the result scales like L e, then we can set
and we may write this as a Laurent series in e, which in the concrete example given here, starts with a term of order l/e:
The first term
+ _~ ,ist
ete nofor er -' , t.esecon t.ete nofor er
e ~ etc. In many cases, only the leading term in 1/e is needed, and we will denote the residue by
ThisE - should not be confounded with the term of order E = e l, which is denoted l i by ,.~,./"(-)..,i~1+}," (The latter does not appear in this review.)
Appendix C: Longitudinal and transversal projectors The longitudinal and transversal projectors are defined in Fourier representation as
eLj(k)
kik j "= k2
(C.1)
456
K.J. Wiese
kik j k2 .
P;J (k) .__ ~ij
(C.2)
They are projectors, i.e. they obey the rules (summing over repeated indices is understood)
PLj (k) PJLl (k) -- pill (k)
PTj (k) p~t (k) = PJrt (k)
(C.3)
We will mostly use the projectors to separate a matrix A ij (k) into its longitudinal and transversal part. Such a separation is possible if the matrix is symmetric and rotationally invariant. Setting
AiJ(k) = l(k)PLJ (k) + t(k)PTJ (k),
(C.4)
we can apply the longitudinal and transversal projectors onto A ij (k) and then take the trace, to obtain (d = ~ii i s the dimension)
9. l(k) = PLJ(k)AiJ(k),
t(k) = d-~-1 I e;J(k)AiJ(k)
(C.5)
For a two-dimensional matrix A ij (k), such a separation is always possible, as we show now. Choose as a basis k and o orthogonal to k, i.e. ok = 0. In this basis Aij(k) can be decomposed as follows:
Aij(k) -- (kAk)kikj
(k2)2
+
(kAo)kioj 'F (oAk)oikj (oAo)oioj k2o 2 + (02)2
= ki~oj + kj~oi +
~ij
k2
dO
with
qgj =
k-~o A ~2 ~ + -21k Ak (kz)2kj,
~ =
~o-~ A ~ = ( ~ij
k i k2J
aij.
(The latter equations follow the conventions used in Section 2.4.)
Appendix D: Derivation of the RG equations In this section, we give a derivation of the renormalization group functions. Starting from b = bo Zb I Z -d/2/z -e, (D. 1)
457
Polymerized membranes, a review
where b is the dimensionless renormalized coupling and b0 the dimensional bare one, the fl-function is given through the variation of the renormalized coupling, at fixed bare quantities, as
fl ( b ) "-- lz - ~0 ]0b .
(D.21
From the derivative of (D. 1) with respect to lz, we obtain
fl(b)
0
1 + b - ~ ln(Zb
zd/2) )
= -eb,
(D.3)
.
(D.4)
leading to /3(b) =
-eb 1 + b ~ ln(ZbZ a/2)
For infinite membranes, the scaling exponent v* is obtained from the largedistance behaviour of the correlation function
,(
C(Ix - Yl, b,/z) := ~-~ (r(x) - r(y))
2)b,u
(D.5)
as C (e, b,/z) ~ e 2v* .
(D.6)
To calculate v*, we first observe that the bare correlator C0(e, b0) = ZC(g, b, lz) is independent of the renormalization scale It. Therefore d d o = tz-duCo(e, bo) - ~dtz [Z(b)C(e, b, lz)]
= Z(b)
fl(b)=-:-. In Z(b) + #
C(e, b, #).
(D.7)
Since C(e, b, #) has canonical dimension D - 2 in units of #, it can be written as C(g, b,/z) -- # o - 2 f ( b ,
lzg).
(D.8)
We thus obtain 8 d ] D-2 0 -- fl(b)~-~ lnZ(b) + #~-~ # f ( b , I,Zg.)
=
fl(b)~--~ In Z(b) + D - 2 + e-~ + fl(b)-~
# o-2f(b,/ze).
(D.9)
Denoting
v(b) =
2-D 2
1 0 fl(b)-~ ln Z(b), 2
(D.10)
K. J. Wiese
458
and reinserting (D.8), (D.9) is written as
g. O--~OC(e, b, U) =
2v(b) - fl(b)-~
C(e, b, lz).
(D.11)
At the IR-fixed point b* with el(b*) = 0, the dimension of the field is thus
v* -- v(b*).
(D. 12)
Let us now consider the case of a grand-canonical ensemble of tethered membranes as introduced in Section 13. There, the scaling exponent v* relates the chemical potential t and radius of gyration R through
R ~ t
/O.
(D.13)
To obtain v*, we first observe that the dimensionless combination R2t krb-~ is a function of b and t//z D only, i.e.
2-D
(D. 14)
R2t-ff- = f (b, t/lzD).
Since in addition, t can be expressed as a function of to and b only, (D. 14) implies that
u-~.d ~
R2t~
--
fl(b) o__ ob _
ot-
R2t ~-
Ot
.
(D.15~
2-D
Next observe that R~t~--- is independent of the renormalization scale #. Replacing bare quantities by their renormalized counterparts, we obtain a relation for the total derivative with respect to ~, as
2-D
2-D]
(D.16)
RZt--Y- Z(b)Zt(b) --if- --0. Combining the latter relation with (D. 15) gives (
0 0 0 ( L~))[ D t o t - ~ ( b ) - ~ - fl(b)-~ In Z Z t
t R2 ~
]
= 0.
(D.17)
The scaling of the membrane is thus given by
v(b) =
2
2fl(b)~--~ In
ZZ
t
,
and the scaling exponent v* is as in (D.12) given by v* -- v(b*).
(D.18)
Polymerized membranes, a review
459
Appendix E: Reparametrization invariance In this appendix, we shall explore the consequences of a reparametrization
Za l/D,
!
X
> X -- X
(E. 1)
on the (renormalized) Hamiltonian of the grand-canonical ensemble (see Section 13)
7-I = 2 -
z D f ~l ( V r ( x ) )2 + blzeZb fx !
~d(r(x)- r(y))+ tZt~,
(E.2)
for a self-avoiding membrane (D = 1 for polymers). The Hamiltonian in (E.2) is in fact not invariant under this rescaling because of the cut-off implicit in the interaction. In order to achieve scale invariance, the cut-off, or equivalently the renormalization scale #, must also be rescaled to > /z t - - # Z ~ I / D .
/z
(E.3)
The Hamiltonian then changes to 2-D
2- D
2 (Vr(xl)2 q- bldeZbZea/~
gd(r(x) -- r(yl) + tZtzffl~. (E.4)
Comparing to the original Hamiltonian then identifies the new renormalization group factors 2-D
Z' -- Z Z~-Tyl
Z t = Z t Za l Z;-
ZbZt
(E.5)
2+e/D.
As discussed in Section 13.3 (see (13.77) ft.), the renormalization group functions are left unchanged by the transformations in (E.5); the most useful case being
Z~ = Zt. Note that a similar transformation can also be performed in the case of a single (infinite) membrane; however, the space coordinate x then acquires an anomalous dimension. Also note that the MOPE is incompatible with such a transformation.
460
K. J. Wiese
Appendix F: Useful formulas
F.1
Momentum space integrals
The momentum space integrals over n vectors k l . . . . .
fk "" fk e-kikjAij = I
kn are
det(A~j)-d/2,
(El)
n
where AiSj - l(Aij + Aji) is the symmetrized version of respect to Aij then leads to
fI "'" fk kikje-kikjAq
=
_ O__~
OAiJ
n
Aij.
Deriving with
det(A~j)-d/2
(F.2)
and the same for higher moments. In the simple case where Aij is a number a this gives
fk k2e k2a = da-d/2-1 2)2e-k
fk(k For a 2 x 2 matrix
Aij
-"
2
2
a d(d + 2)a_Cl/2_2
(F.4)
4
All A21
f fk k2e-kik'Ai' ,
(E3)
2
A,12 ~, this simplifies to A22
/
(ES)
= d-A22det(A'Sj)-a/2-1 2
fk fk z klk2e-kikjAij = d2 A~2det(A~J)-d/2-1
(E6)
fk fk k2e-kikjAij dAlldet(A~J)-d/2-1
(E7)
~
i
2
o
2
F.2 Calculable manifold integrals The three basic solvable manifold integrals are
fx
fx,y; max(x,y)
L2D-vd
,Ixl
(E8)
IxlO-ud 2D - vd
(Ix12"-t-lyl2U) -d/2
1 F(2D--~)2L 2D-vd 2 - D F ( 22.~Do) 2D - vd x [1 + O ( ( 2 D -
vd)~
(F.9)
Polymerized membranes, a review
,y,z; max(x,y,z)
461
(Ixl2~lYl 2~ 4-lYl2~lzl 2~ + Izl2Vlxl2") -d/2
1
F (1 2--~DD)3 L 3D-2vd = ( 2 - D)2 F (32__~DD)3 D - 2vd [1 + O ((3D - 2 v d ) ~
F.3
(F.10)
Distributions
In our conventions listed in Appendix A,
A x l x - yl 2-o = ( 2 - D ) 8 o ( x - y)
(E11)
from which we deduce fx [gx ( I x - yl 2 - D - I x - zl2-O)] 2 - 2 ( 2 - D ) I y - zl 2-D.
(F.12)
A useful formula is (a = E-- b = fixed)
2-DL(1)b-D-~c'-D)22
-- a 2-D.
(F.13)
Another distribution which appears frequently is (a =fixed)
( 2 - D ) fc(a2-D(a~')2c-2)(..'-D= f ~tgx(~tgxc2-D) = a 2 mDlL Axc2-D = 2 - D a2 D
(El4)
Appendix G: Derivation of the Green function
It shall be shown that
g(x) : = - I x l z-o
(G.I)
is in the sense of distributions a solution of
--Ax g(x) -- ( 2 - D)SD(x). Proof. Let
go(x) : = -
( 2 - D) 2-D+rl Ixl ( 2 - O + r/)
(G.2) (G.3)
462
K. J. Wiese
Ixl and f
with r "=
f I "-- ] ( - A
--
(2-
go(x))f(x) D + 11)
-- 0 ( 2 - D) As f(x)
E
E C~(RD). Then
,/
dr rD-I--sD
/o l/ dr ~
,
dr2 ~--rr D-I Or
--Or
f (x)
d~r~-l f (x).
C~(]l~D) there are constants a, b and I such that If(r)-f(0)l If(r)-f(0)[
< av/'0
f(r)
Vr < Vr > ~/~ Yr>l.
(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)
Polymerized membranes, a review
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)
Polymerized membranes, a review
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
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