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Physics Bulletin

Exactly Solved Models in Statistical Mechanics

M A Moore

Published Apr 1, 1983• 4,409 citations
Publisher

Abstract

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

DOI
10.1088/0031-9112/34/4/045

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Introduction

M. Gaudin

From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.

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The Onsager Revolution

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We have mentioned in §1.5 the shattering blow which Onsager's exact calculations for a two-dimensional Ising model delivered to the classical theories, and the challenge it posed in relation to critical behaviour. But Onsager's work had much wider ramifications since detailed aspects of his calculations stimulated important new areas of research and progress. His topological interpretation of the duality transformation discovered by Kramers and Wannier greatly extended its scope, and soon led, in conjunction with his formulation of the star-triangle transformation, to exact solutions for other twodimensional lattices (see e.g. Domb 1960 pp. 211-20). The star-triangle transformation itself initiated a search which revealed other useful transformations of a similar kind (see e.g. Syozi 1972). The simplification of the calculations by the use of spinors led to the discovery of other simplified methods of calculation each of which had its own specific advantages (Temperley 1972, Schultz, Mattis and Lieb 1964). More remotely the existence of exact solutions to the Ising model in two dimensions suggested that other realistic models might give rise to exact solutions. Over 20 years elapsed before a breakthrough occurred (Lieb 1967). Subsequently this area of research has generated results of great interest and importance (Baxter 1982).

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Fifty Years of the Exact Solution of the Two-Dimensional Ising Model by Onsager

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The exact solution of the two-dimensional Ising model by Onsager in 1944 represents one of the landmarks in theoretical physics. On the occassion of the fifty years of the exact solution, we give a historical review of this model. After briefly discussing the exact solution by Onsager, we point out some of the recent developments in this field. The exact solution by Onsager has inspired several developments in various other fields. Some of these are also briefly mentioned.

arXiv:cond-mat/9511003•Nov 1, 1995•26 citations

Some comments on developments in exact solutions in statistical mechanics since 1944

R. J. Baxter

Onsager and Kaufman calculated the partition function of the Ising model exactly in 1944 and 1949. Since then there have been many developments in the exact solution of similar, but usually more complicated, models. Here I shall mention a few, and show how some of the latest work seems to be returning once again to the properties observed by Onsager and Kaufman.

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Two-Dimensional Ising Model as a Soluble Problem of Many Fermions

T. D. Schultz, Daniel Charles Mattis, Élliott H. Lieb

The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.

Reviews of Modern Physics•Jul 1, 1964•935 citations

Star-Triangle and Inversion Relations in Statistical Mechanics

J.-M. Maillard

The study of the two-dimensional models in lattice statistics and the related exactly integrable 1+1 dimensional field theory has developed intensively over the past ten years and this evolution, this developement does not seem to decrease. There are several reasons why one can be interested in such models: the need to leave the framework of perturbation theory with the hope to describe particle spectrum and scattering, the fact that some 1+1 dimensional exactly solved models have physical applications for some important solid state theories (Solitons in magnets [61,11], Peierls-Frohlich model, Anderson localization, Kondo effects [19,37,1]),the importance of exactly solved two-dimensional models of lattice statistics to understand the critical phenomena (universality...). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Birkhäuser Boston eBooks•Jan 1, 1985•0 citations
R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.
From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.
We have mentioned in §1.5 the shattering blow which Onsager's exact calculations for a two-dimensional Ising model delivered to the classical theories, and the challenge it posed in relation to critical behaviour. But Onsager's work had much wider ramifications since detailed aspects of his calculations stimulated important new areas of research and progress. His topological interpretation of the duality transformation discovered by Kramers and Wannier greatly extended its scope, and soon led, in conjunction with his formulation of the star-triangle transformation, to exact solutions for other twodimensional lattices (see e.g. Domb 1960 pp. 211-20). The star-triangle transformation itself initiated a search which revealed other useful transformations of a similar kind (see e.g. Syozi 1972). The simplification of the calculations by the use of spinors led to the discovery of other simplified methods of calculation each of which had its own specific advantages (Temperley 1972, Schultz, Mattis and Lieb 1964). More remotely the existence of exact solutions to the Ising model in two dimensions suggested that other realistic models might give rise to exact solutions. Over 20 years elapsed before a breakthrough occurred (Lieb 1967). Subsequently this area of research has generated results of great interest and importance (Baxter 1982).
The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.
The study of the two-dimensional models in lattice statistics and the related exactly integrable 1+1 dimensional field theory has developed intensively over the past ten years and this evolution, this developement does not seem to decrease. There are several reasons why one can be interested in such models: the need to leave the framework of perturbation theory with the hope to describe particle spectrum and scattering, the fact that some 1+1 dimensional exactly solved models have physical applications for some important solid state theories (Solitons in magnets [61,11], Peierls-Frohlich model, Anderson localization, Kondo effects [19,37,1]),the importance of exactly solved two-dimensional models of lattice statistics to understand the critical phenomena (universality...). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.