# Hard hexagon model

In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjacent.

The model was solved by Baxter (1980), who found that it was related to the Rogers–Ramanujan identities.

## The partition function of the hard hexagon model

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

$\displaystyle \mathcal Z(z) = \sum_n z^n g(n,N) = 1+Nz+ \tfrac{1}{2}N(N-7)z^2+\cdots$

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

$\kappa(z) = \lim_{N\rightarrow\infty} \mathcal Z(z)^{1/N} = 1+z-3z^2+\cdots$

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

$\rho= z\frac{d\log(\kappa)}{dz} =z-7z^2+58z^3-519z^4+4856z^5+\cdots.$

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

$\rho_1 = 1-z^{-1}-5z^{-2}-34z^{-3}-267z^{-4}-2037z^{-5}-\cdots$
$\rho_2=\rho_3 = z^{-2} + 9z^{-3} + 80z^{-4} + 965z^{-5}-\cdots.$

## Solution

The solution is given for small values of z < zc by

$\displaystyle z=\frac{-xH(x)^5}{G(x)^5}$
$\kappa = \frac{H(x)^3 Q(x^5)^2} {G(x)^2} \prod_{n\ge 1} \frac{(1-x^{6n-4})(1-x^{6n-3})^2(1-x^{6n-2})} {(1-x^{6n-5})(1-x^{6n-1})(1-x^{6n})^2}$
$\rho =\rho_1=\rho_2=\rho_3= \frac{-xG(x)H(x^6)P(x^3)}{P(x)}$

where

$G(x) = \prod_{n\ge 1}\frac{1}{(1-x^{5n-4})(1-x^{5n-1})}$
$H(x) = \prod_{n\ge 1}\frac{1}{(1-x^{5n-3})(1-x^{5n-2})}$
$P(x) = \prod_{n\ge 1}(1-x^{2n-1}) = Q(x)/Q(x^2)$
$Q(x) = \prod_{n\ge 1}(1-x^n).$

For large z > zc the solution (in the phase where most occupied sites have type 1) is given by

$\displaystyle z=\frac{G(x)^5}{xH(x)^5}$
$\kappa = \frac{G(x)^3 Q(x^5)^2} {H(x)^2} \prod_{n\ge 1} \frac{(1-x^{3n-2})(1-x^{3n-1})} {(1-x^{3n})^2}$
$\rho_1 = \frac{H(x)Q(x)(G(x)Q(x)+x^2H(x^9)Q(x^9))}{Q(x^3)^2}$
$\rho_2=\rho_3 = \frac{x^2H(x)Q(x)H(x^9)Q(x^9)}{Q(x^3)^2}$
$R=\rho_1-\rho_2= \frac{Q(x)Q(x^5)}{Q(x^3)^2}.$

The functions G and H turn up in the Rogers–Ramanujan identities, and the function Q is closely related to the Dedekind eta function. If x = e2πiτ, then q−1/60G(x), x11/60H(x), x−1/24P(x), z, κ, ρ, ρ1, ρ2, and ρ3 are modular functions of τ, while x1/24Q(x) 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 (of quite high degree) (Joyce 1988).