Thermodynamics of One-Dimensional Solvable Models Exactly solvable models are very important in physics. They are important not just from a theoretical point of view but also from the experimentalist's perspective, because in such cases theoretical results and experimental results can be compared without ambiguity. This is a book about an important class of exactly solvable models in physics. The subject area is the Bethe-ansatz approach for a number of one- dimensional models, and the setting up of equations within this approach to determine the thermodynamics of these systems. It is a topic that crosses the boundaries between condensed matter physics, mathematics and field theory. The derivation and application of thermodynamic Bethe-ansatz equations for one-dimensional models are explained in detail. This technique is indispensable for physicists studying the low-temperature properties of one-dimensional substances. This book, written by one of the top physicists in this field, and the originator of much of the work in the field, will be of great interest to theoretical condensed matter physicists. THERMODYNAMICS OF ONE-DIMENSIONAL SOLVABLE MODELS Minoru Takahashi Institute for Solid State Physics, University of Tokyo, Tokyo, Japan CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge C132 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521551434 © Cambridge University Press 1999 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 This digitally printed first paperback version 2005 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Takahashi, Minoru, 1944- Thermodynamics of one-dimensional solvable models / Minoru Takahashi. p. cm. Includes bibliographical references and index. ISBN 0 521 55143 9 (hb) 1. Bethe-ansatz technique. 2. Statistical thermodynamics. 3. Mathematical physics. I. Title. QC20.7.b47T35 1999 530.15-dc2l 98-34997 CIP ISBN-13 978-0-521-55143-4 hardback ISBN-10 0-521-55143-9 hardback ISBN-13 978-0-521-01979-8 paperback ISBN-10 0-521-01979-6 paperback Contents Preface page xi Part one: Thermodynamics of non-interacting systems and ground states of interacting systems 1 1 Free energy and correlation functions of the XY model 1 1.1 The isotropic XY model 1 1.1.1 Introduction and historical overview 1 1.1.2 Energy eigenvalues of the Hamiltonian and the partition function 3 1.1.3 Correlation functions 4 1.2 The anisotropic XY model 7 1.2.1 The subspace a = 1 8 1.2.2 The subspace a = -1 9 1.2.3 The free energy 9 2 Systems with a delta function potential 10 2.1 The boson problem 10 2.1.1 The c = 0 case 10 2.1.2 The c = oo case 11 2.1.3 Scattering states of Bose particles with finite interaction 12 2.1.4 Periodic boundary conditions 14 2.1.5 Linear integral equation for the distribution of quasi-momenta 15 2.1.6 Bound states in the case c < 0 16 2.1.7 Elementary excitations for repulsive bosons 18 2.2 The fermion problem 20 2.2.1 The c = 0 case 20 2.2.2 The two-body problem 21 2.2.3 The three-body problem 22 2.2.4 The M = 1 and arbitrary N case 25 2.2.5 The arbitrary M and N case 25 2.2.6 Periodic boundary conditions 26 v vi Contents 2.2.7 The ground state for c > 0 28 2.2.8 The ground state for c < 0 29 2.2.9 Expansion from small n j 31 2.2.10 A unified form of the integral equations 32 2.3 Bosons and fermions with arbitrary spin 32 3 The isotropic Heisenberg model 34 3.1 The ferromagnetic case 34 3.2 The string solution of an infinite system 37 3.3 The Hulthen solution for an antiferromagnet 37 3.4 The des Cloizeaux-Pearson mode of an antiferromagnet 39 3.5 The magnetic susceptibility and magnetization curve for J < 0 41 3.6 Wiener-Hopf type integral equation 44 4 The XXZ model 46 4.1 Symmetry of the Hamiltonian 46 4.2 The Bethe-ansatz wave function 47 4.3 The string solution for A > 1 48 4.4 The lowest energy state for A < -1 49 4.5 The magnetization curve for a field in the z-direction at A < -1 51 4.6 The lowest energy state for fixed M and -1 < A < 1 52 4.7 The magnetization curve for a field in the z-direction at -1 0 77 5.6 Long range order 79 5.7 Elementary excitations 80 5.7.1 The nearly degenerate ground state 80 5.7.2 Spinon excitations 80 Contents vii 5.7.3 Spin-wave excitations 81 5.7.4 Spin-wave bound states 83 6 The Hubbard model 85 6.1 Symmetry of the Hamiltonian 85 6.1.1 Particle-hole symmetry 85 6.1.2 S U(2) symmetry 87 6.2 The Bethe-ansatz equation for the one-dimensional Hubbard model 88 6.2.1 The wave function for a finite system 88 6.2.2 Periodic boundary condition 90 6.2.3 Fredholm type integral equations for the ground state 90 6.2.4 Analytic solution for the ground state in the half-filled case 92 6.2.5 Spinon excitation in the half-filled case 92 6.2.6 Energy gap of the charge excitation 95 6.2.7 Susceptibility and magnetization curve of the half-filled case 97 6.3 11 U' expansion 101 6.4 Perturbation expansion in the half-filled case 102 6.5 Asymptotic expansion from U = 0 104 Part two: Finite temperature integral equations for un-nested systems 109 7 Repulsive delta function bosons 109 7.1 Uniqueness of the solution 109 7.2 Holes of quasi-momenta and their distribution function 110 7.3 Thermodynamic equilibrium 111 7.4 Elementary excitations 113 7.5 Some special limits 115 7.5.1 The c = oo limit 115 7.5.2 The c = 0+ limit 116 7.5.3 The T = 0+ limit 116 8 Thermodynamics of the XXX chain 117 8.1 String solution of an infinite XXX chain 117 8.2 String hypothesis for a long XXX chain 118 8.3 Thermodynamic Bethe-ansatz equations for the XXX chain 120 8.4 Some special cases and expansions 123 8.4.1 The J/T 0 case 123 8.4.2 High-temperature expansion or small J expansion 124 8.4.3 The low-temperature limit 126 8.4.4 The fugacity expansion 127 9 Thermodynamics of the XXZ model 130 9.1 Thermodynamic equations for the XXZ model for A > 1 130 viii Contents 9.2 Theory for the JAI < 1 XXZ model 133 9.2.1 String solution of the infinite XXZ model with JAI < 1 133 9.2.2 Scattering phase shift among strings 138 9.2.3 Bethe-ansatz equation for the XXZ model with JAI < 1 139 9.3 Some special limits 143 9.3.1 The T oc or J -* 0 limit 143 9.3.2 ThecaseJ >0, -10, 0 0, Jx < 0 and T -+ 0 limit 150 10.2.3 The JZ > 0, Jx > 0 and T -* 0 limit 150 10.2.4 The Jx = 0 case 151 11 Low-temperature thermodynamics 152 11.1 The XXZ model 152 11.1.1 The XXZ model at 2h > J(1 - A), A < 1 152 11.1.2 The case J(1 - A)/2 > h > 0 152 11.2 Roger's dilogarithm and specific heat at h = 0 153 11.2.1 Specific heat of the XXX antiferromagnet 153 11.3 The ferromagnetic chain and modified spin-wave theory 155 11.3.1 Numerical analysis of the thermodynamic Bethe-ansatz equation 155 11.3.2 Spin-wave calculation of the 1D ferromagnetic chain 155 11.4 The antiferromagnetic XXX model 156 11.5 The XYZ model at IJxI < Jy < JZ 157 Part three: Finite temperature integral equations for nested systems 159 12 S = 1/2 fermions with repulsive potential in the continuum 159 12.1 Derivation of the thermodynamic equations 159 12.2 Some special limits 164 12.2.1 c - * 0+ 164 12.2.2 coo 166 13 S = 1/2 fermions with an attractive potential 167 13.1 Derivation of the thermodynamic equations 167 13.2 Some special limits 171 13.2.1 c -* 0- 171 13.2.2 T --> 0+ 172 Contents ix 14 Thermodynamics of the Hubbard model 174 14.1 Strings of the Hubbard model 174 14.2 Thermodynamic Bethe-ansatz equation for the Hubbard model 177 14.3 Some special limits 179 14.3.1 The limit U oo 179 14.3.2 The limit U 0 180 14.3.3 The limit T -* 0 182 14.3.4 The limit t -* 0 183 Part four: The quantum transfer matrix and recent developments 185 15 The transfer matrix and correlation length 185 15.1 The transfer matrix for the Ising chain 185 15.2 The transfer matrix for the classical Heisenberg model 186 16 The spin 112 XXZ model in a magnetic field 189 16.1 The diagonal-to-diagonal transfer matrix 189 16.2 The limit of an infinite Trotter number 198 16.3 Analytical solutions for special cases 201 16.3.1 The Ising limit 201 16.3.2 The XY limit 202 16.3.3 The T = h = 0 case 203 16.4 Numerical calculations of the XXZ model 205 17 The XYZ model with no magnetic field 210 17.1 The transfer matrix for the XYZ model 210 17.1.1 Baxter's theory for the inhomogeneous eight-vertex model 211 17.1.2 Transcendental equations 212 17.1.3 The limit of M -* oo 215 17.2 Special cases and numerical methods 218 17.2.1 The T -* 0 limit 218 17.2.2 The Jx = 0 case (anisotropic XY chain) 220 17.3 Numerical calculations 221 18 Recent developments and related topics 223 18.1 Numerical analysis of the S = 1 chain 223 Appendix A The Young tableau and the theorem of Lieb and Mattis 226 Appendix B The number of string solutions 228 Appendix C The commuting transfer matrix and spectral parameter 231 Appendix D The matrix Q(v) 235 Appendix E Special functions 239 Bibliography 246 Index 251

Thermodynamics of One-Dimensional Solvable Models

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