Exact Results for Two-Dimensional Ising Models

Abstract

We already have exact results for the one-dimensional Ising model (Chap. 2 and Sect. 4.1) and exact transcriptions between models, for instance from the zero-field Ising ferromagnet to the zero-field antiferromagnet (Sect. 4.2) or from the Ising model to the simple lattice fluid or two-component mixture (Chaps. 5 and 6). In Sects. 8.2-8.6, dual and star-triangle transformations are derived, connecting the partition function for the zero-field Ising model on a given lattice to that of the zero-field Ising model on a related lattice at a different temperature.¹ Some associated relations connecting nearest-neighbour correlations are obtained, a particularly important one being derived in Sect. 8.6. We then derive exact results for critical conditions and thermodynamic functions for the two-dimensional zero-field Ising model. There are a large number of approaches to this, all mathematically intricate.² The original method of Onsager (1944), Kaufman (1949) and Kaufman and Onsager (1949) depended on transfer matrices, of which the 2 × 2 matrix appearing in the theory of the one-dimensional Ising model in Sect. 2.4 is a simple example. Other workers have employed the Pfaffian (see Volume 2, Chap. 8, Green and Hurst 1964, McCoy and Wu 1973) or combinatorial methods (Kac and Ward 1952, Vdovichenko 1965a, 1965b). We shall use a method due to Baxter and Enting (1978), which is closely related to the preceding transformation theory and has the advantage of giving results simultaneously for the square, triangular and honeycomb lattices. The exact expressions for configurational energy and heat capacity depend on elliptic integrals and these protean quantities appear in the literature in a number of different forms. We have tried to use expressions valid for the whole temperature range and closely related to the formulae given for the partition function or free energy.