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Journal of Mathematical Physics

Relation between the Onsager and Pfaffian Methods for Solving the Ising Problem. II. The General Lattice

C. A. Hurst

Published Jan 1, 1966• 2 citations
Publisher

Abstract

The considerations of a previous paper are extended to the case of a general lattice. It is shown algebraically why the Pfaffian and Onsager methods of solution of the Ising problem coincide only when the lattice is planar, and that the problem is then a linear one. When the lattice is nonplanar the Pfaffian method breaks down due to the appearance of unwanted negative signs, and it is shown how the Onsager method compensates for this at the expense of making the problem nonlinear.

DOI
10.1063/1.1704819

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The considerations of a previous paper are extended to the case of a general lattice. It is shown algebraically why the Pfaffian and Onsager methods of solution of the Ising problem coincide only when the lattice is planar, and that the problem is then a linear one. When the lattice is nonplanar the Pfaffian method breaks down due to the appearance of unwanted negative signs, and it is shown how the Onsager method compensates for this at the expense of making the problem nonlinear.
The partition function for the Ising problem for a general class of plane lattices is evaluated exactly using the Pfaffian method. A classification of plane lattices in terms of basic lattices with vertices decorated by sublattices is given, and the most general class which can be solved by the Pfaffian method described. The solutions given in this paper are for the special case when neighbouring lattice points of a basic rectangular lattice are connected by single bonds only. All of the exact solutions known so far are obtained as special cases. The critical points are defined as singularities of an analytic function and equations to determine their location are found. The derivatives of the partition function are evaluated in terms of elliptic integrals.
It is shown that the transfer matrices for various Ising lattices in two dimensions commute with certain linear operators. The problem of finding an explicit form for the largest eigenvector is considerably simplified. The expansion coefficients appearing in the eigenvectors found as the solution of a set of nonlinear difference equations are Pfaffians. The connection between this type of solution and other solutions is clarified. This form for the eigenvector also simplifies the calculation of correlation functions. Some geometrical aspects of the Ising model are discussed.
Introduction The planar Ising model has become one of the most important statistical mechanical systems for the study of phase transitions and critical phenomena. Although there are many rigorous results, such as correlation inequalities, Peierls argument and the Yang-Lee circle theorem to name but three [reviewed by Griffiths, 1971], which are dimensionindependent in their validity and which lead to results of considerable interest, only the planar model to date benefits from the added insights which stem ultimately from Onsager's tour de force [Onsager, 1944]. It is not the purpose of this article to enter into a general review – for this the reader is referred elsewhere [Gallavotti, 1972] but rather to discuss two more mathematical aspects of the development of Onsager's solution. The first item is the Yang-Baxter system of equations for the planar Ising model in zero field with transfer in the (1,1) direction. This work shows that the Clifford-algebraic structure of the exact solution is a natural consequence of the star-triangle equations. The second item is a Fredholm system which turns out to be of crucial importance in understanding surface and interface problems, as well as the pair correlation function.