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arXiv:1010.0710 · Journal of Statistical Mechanics Theory and Experiment

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

R. J. Baxter

Published Nov 23, 2010• 6 citations
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Abstract

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.

DOI
10.1088/1742-5468/2010/11/p11037

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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.
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 low-temperature series expansion for the partition function of the two-dimensional Ising model on a square lattice can be determined exactly for finite lattices using Kaufman's generalization of Onsager's solution. The exact distribution function for the energy can then be determined from the coefficients of the partition function. This provides an exact solution with which one can compare energy histograms determined in Monte Carlo simulations. This solution should prove useful for detailed studies of statistical and systematic errors in histogram reweighting.
The computation of classical Ising partition functions, coming from statistical physics, is a natural generalization of binary optimization. This is a notoriously hard problem in general, which makes it an especially interesting task to consider in the search for practical quantum advantage in near term quantum computers. In this work we view classical Ising models (on certain graphs) as quantum imaginary time evolution, which is enabled by the use of the transfer matrix mapping. We study this mapping from two points of view: (1) following Onsager and Kaufman’s original solution of the 2D Ising model, which serves as a starting point, we consider more general models and the possibility of a similar Lie-theoretic solution; (2) we consider quantum algorithms for the computation of partition functions and thermal averages via transfer matrices, which can be implemented either with block encodings inside larger unitaries or by approximating the state trajectories with unitary operators.