Exactly Solved Models In Statistical Mechanics

Exactly Solved Models in Statistical Mechanics Rodney J. Baxter F.R.S. Department of Theoretical Physics, Research School of Physical Sciences, The Australian National University, Canberra, A.C.T., Australia ACADEMIC PRESS Harcourt Brace Jovanovich, Publishers London • San Diego - New York • Berkeley Boston • Sydney • Tokyo - Toronto ACADEMIC PRESS LIMITED 24-28 Oval Road London NW1 7DX United States Edition published by ACADEMIC PRESS INC. San Diego, CA 92101 Copyright C) 1982 by ACADEMIC PRESS LIMITED Second printing 1984 Third Printing 1989 First published as a paperback 1989 All Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers British Library Cataloguing in Publication Data Baxter, R. J. Exactly solved models in statistical mechanics. 1. Statistical mechanics I. Title 530.1'3 QC174.8 ISBN 0-12-083180-5 ISBN 0-12-083182-1 (P) LCCN 81-68965 Printed in Great Britain by St Edmundsbury Press Limited Bury St Edmunds, Suffolk PREFACE This book was conceived as a slim monograph, but grew to its present size as I attempted to set down an account of two-dimensional lattice models in statistical mechanics, and how they have been solved. While doing so I have been pulled in opposite directions. On the one hand I remembered the voice of the graduate student at the conference who said 'But you've left out all the working—how do you get from equation (81) to (82)?' On the other hand I knew from experience how many sheets of paper go into the waste-paper basket after even a modest calculation: there was no way they could all appear in print. I hope I have reached a reasonable compromise by signposting the route to be followed, without necessarily giving each step. I have tried to be selective in doing so: for instance in Section 8.13 I discuss the functions k(a) and g(Œ) in some detail, since they provide a particularly clear example of how elliptic functions come into the working. Conversely, in (8.10.9) I merely quote the result for the spontaneous staggered polarization P0 of the F-model, and refer the interested reader to the original paper: its calculation is long and technical, and will probably one day be superseded when the eight-vertex model conjecture (10.10.24) is verified by methods similar to those used for the magnetization result (13.7.21). There are 'down-to-earth' physicists and chemists who reject lattice models as being unrealistic. In its most extreme form, their argument is that if a model can be solved exactly, then it must be pathological. I think this is defeatist nonsense: the three-dimensional Ising model is a very realistic model, at least of a two component alloy such as brass. If the predictions of universality are corrected, then they should have exactly the same critical exponents. Admittedly the Ising model has been solved only in one and two dimensions, but two-dimensional systems do exist (see Section 1.6), and can be quite like three-dimensional ones. It is true that the two-dimensional Ising model has been solved only for zero magnetic Vi PREFACE field, and that this case is quite unlike that of non-zero field; but physically this means Onsager solved the most interesting and tricky case. His solution vastly helps us understand the full picture of the Ising model in a field. In a similar way, the eight-vertex model helps us understand more complicated systems and the variety of behaviour that can occur. The hard hexagon model is rather special, but needs no justification: It is a perfectly good lattice gas and can be compared with a helium monolayer adsorbed onto a graphite surface (Riedel, 1981). There is probably also a feeling that the models are `too hard' mat