On the hard-hexagon model and the theory of modular functions

Abstract

The mathematical properties of the exact solution of the hard-hexagon lattice gas model are investigated by using the Klein-Fricke theory of modular functions. In particular, it is shown that the order-parameter R and the reciprocal activity z' for the model can be expressed in terms of hauptmoduls that are associated with certain congruence subgroups of the full modular group Known modular equations are then used to prove that R (z') is an algebraic function of A connection is established between the singular points of this function and the geometrical properties of the icosahedron .