The Eight-Vertex Model
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.
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- 1.As may be expected, since the Ising model with just a two-site interaction and an external field is unsolved.Google Scholar
- 2.Any one of the vertex energies e 1, e 3, e 5 and e 7 can be set to zero without loss of generality. However, this is not advantageous, except in special cases, because of the symmetry of the eight-vertex free energy transformations.Google Scholar
- 3.These are the high-temperature zero-field graphs of Sect. 7.4.Google Scholar
- 4.The correspondence between spin configurations and polygon graphs on dual lattices was used in Volume 1, Sects. 8.2 and 8.3, in deriving the Ising model dual transformation.Google Scholar
- 5.This terminology is used because of the mathematical equivalence to a system of non-interacting fermions (Lieb and Wu 1972).Google Scholar