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Birkhäuser Boston eBooks

Star-Triangle and Inversion Relations in Statistical Mechanics

J.-M. Maillard

Published Jan 1, 1985• 0 citations
Publisher

Abstract

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.

DOI
10.1007/978-1-4899-6650-6_15

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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.
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The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.
Conventional Monte Carlo simulation algorithms for models in statistical mechanics and quantum field theory are afflicted by problems caused by their locality. They become highly inefficient if investigations of critical or nearly-critical systems, i.e., systems with important large scale phenomena, are undertaken. We present two types of multiscale approaches that alleveate problems of this kind: stochastic cluster algorithms and multigrid Monte Carlo simulation algorithms. Another formidable computational problem in simulations of phenomenologically relevant field theories with fermions is the need for frequently inverting the Dirac operator. This inversion can be accelerated considerably by means of deterministic multigrid methods, very similar to the ones used for the numerical solution of differential equations. 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.
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.