Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

The eight-vertex model is equivalent to a "solid-on-solid" (SOS) model, in which an integer height l_i is associated with each site i of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2 K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1≤l_i≤4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1≤l_i≤r-1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponents β, α, {Mathematical expression} are obtained.