The hard hexagon model occurs within the framework of the grand canonical ensemble, where the total number of particles is allowed to vary naturally, and is fixed by a chemical potential. In the hard hexagon model, all valid states have zero energy, and so the only important thermodynamic control variable is the ratio of chemical potential to temperature μ/. The exponential of this ratio, z = exp is called the activity and larger values correspond roughly to denser configurations. For a triangular lattice with N sites, the grand partition function is where g is the number of ways of placing n particles on distinct lattice sites such that no 2 are adjacent. The function κ is defined by so that log is the free energy per unit site. Solving the hard hexagon model means finding an exact expression for κ as a function of z. The mean density ρ is given for small z by The vertices of the lattice fall into 3 classes numbered 1, 2, and 3, given by the 3 different ways to fill space with hard hexagons. There are 3 local densities ρ1, ρ2, ρ3, corresponding to the 3 classes of sites. When the activity is large the system approximates one of these 3 packings, so the local densities differ, but when the activity is below a critical point the three local densities are the same. The critical point separating the low-activity homogeneous phase from the high-activity ordered phase is with golden ratioφ. Above the critical point the local densities differ and in the phase where most hexagons are on sites of type 1 can be expanded as
Solution
The solution is given for small values of z < zc by where For large z > zc the solution is given by The functionsG and H turn up in the Rogers–Ramanujan identities, and the function Q is the Euler function, which is closely related to the Dedekind eta function. If x = e2πiτ, then q−1/60G, x11/60H, x−1/24P, z, κ, ρ, ρ1, ρ2, and ρ3 are modular functions of τ, while x1/24Q is a modular form of weight 1/2. Since any two modular functions are related by an algebraic relation, this implies that the functions κ, z, R, ρ are all algebraic functions of each other .