Hard Hexagon Model - The Partition Function of The Hard Hexagon Model

The Partition Function of The Hard Hexagon Model

For a triangular lattice with N sites, the partition function is

where g(n, N) is the number of ways of placing n particles on distinct lattice sites such that no 2 are adjacent. The variable z is called the activity and larger values correspond roughly to denser configurations. The function κ is defined by

so that log(κ) is the free energy per unit site. Solving the hard hexagon model means (roughly) 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 zc = (11 + 53/2)/2 = 11.0917.... 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

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