Vertex models for the hard-square and hard-hexagon gases, and critical parameters from the scaling transformation

Abstract

In a general formulation of hard-core lattice gas models the hard-square and hard-hexagon models are expressed in terms of vertex models. It is shown that the hard-square gas is a 16-vertex model on the square lattice, and in one of its representations is equivalent to a lattice ramrod model first considered by Nagle (1968). A lattice transformation of the hard-hexagon model generates a special case of the 64-vertex model on the triangular lattice. In these two reductions clear differences between these two hard-core models emerge, and it is probably not surprising that they exhibit quite different critical behaviour. It is also shown that the block-site scaling transformation can be used to provide an alternative computational scheme for obtaining accurate numerical estimates of critical point parameters for these models. The method is illustrated by calculations performed on the hard-square gas.