Hard Hexagon Model - Solution

Solution

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

\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}

where

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

\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}

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).

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