The Eight-Vertex Model

Abstract

The name vertex model is used to denote a lattice model in which the microstates are represented by putting an arrow on each edge (line connecting a pair of nearest-neighbour sites) of the lattice. Such models can be constructed on any lattice, but those for the square lattice have received the most attention. It is clear that the most general model of this type on the square lattice is the sixteen-vertex model, where the different vertex types correspond to all 2⁴ possible directions of the arrows on the four edges meeting at a vertex. This model, which can be shown to be equivalent to an Ising model with two, three and four site interactions and with an external field (Suzuki and Fisher 1971), is unsolved.¹ By applying the ice rule which restricts the vertex types to those with the same number of arrows in as out, the model becomes the six-vertex model which was solved by Lieb (1967a, 1967b, 1967c). It is discussed in Volume 1, Chap. 10. In the present chapter we consider the model where the ice rule is replaced by the rule restricting the vertex type to those with an even number of arrows pointing in and out. This allows the eight vertex types shown in the first line of Fig. 5.1. The first six of these vertex types correspond to those of the six-vertex model. In the eight-vertex model, vertices with four inward or four outward arrows are also permitted, The new vertices are labelled 7 and 8.