Dual Transformations in Many-Component Ising Models

Abstract

Dual transformations in many-component Ising models in two dimensions on a square lattice are studied. The models considered include those of Ashkin and Teller and of Potts. In certain cases the dual transformation is a relation between the partition function of a lattice at high and low temperatures and can be used to determine a unique critical temperature if one exists. Dual transformations are considered both from a topological and an algebraic point of view. The topological arguments are a natural extension of those used by Onsager for the 2-component Ising model. The transfer matrices for these models are constructed, and it is shown how the dual transformation arises in this formulation of the problem. The algebras generated by these models are investigated and provide a generalization of the spinor algebra introduced by Kaufman in the 2-component Ising model.