Hard hexagon model

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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 z_c = (11 + 5^3/2)/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 < z_c by

$\displaystyle z={\frac {-xH(x)^{5}}{G(x)^{5}}}$

$\kappa ={\frac {H(x)^{3}Q(x^{5})^{2}}{G(x)^{2}}}\prod _{{n\geq 1}}{\frac {(1-x^{{6n-4}})(1-x^{{6n-3}})^{2}(1-x^{{6n-2}})}{(1-x^{{6n-5}})(1-x^{{6n-1}})(1-x^{{6n}})^{2}}}$

$\rho =\rho _{1}=\rho _{2}=\rho _{3}={\frac {-xG(x)H(x^{6})P(x^{3})}{P(x)}}$

where

$G(x)=\prod _{{n\geq 1}}{\frac {1}{(1-x^{{5n-4}})(1-x^{{5n-1}})}}$

$H(x)=\prod _{{n\geq 1}}{\frac {1}{(1-x^{{5n-3}})(1-x^{{5n-2}})}}$

$P(x)=\prod _{{n\geq 1}}(1-x^{{2n-1}})=Q(x)/Q(x^{2})$

$Q(x)=\prod _{{n\geq 1}}(1-x^{n}).$

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

$\displaystyle z={\frac {G(x)^{5}}{xH(x)^{5}}}$

$\kappa ={\frac {G(x)^{3}Q(x^{5})^{2}}{H(x)^{2}}}\prod _{{n\geq 1}}{\frac {(1-x^{{3n-2}})(1-x^{{3n-1}})}{(1-x^{{3n}})^{2}}}$

$\rho _{1}={\frac {H(x)Q(x)(G(x)Q(x)+x^{2}H(x^{9})Q(x^{9}))}{Q(x^{3})^{2}}}$

$\rho _{2}=\rho _{3}={\frac {x^{2}H(x)Q(x)H(x^{9})Q(x^{9})}{Q(x^{3})^{2}}}$

$R=\rho _{1}-\rho _{2}={\frac {Q(x)Q(x^{5})}{Q(x^{3})^{2}}}.$

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 = e^2πiτ, then q^−1/60G(x), x^11/60H(x), x^−1/24P(x), z, κ, ρ, ρ₁, ρ₂, and ρ₃ are modular functions of τ, while x^1/24Q(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).

References

Andrews, George E. (1981), "The hard-hexagon model and Rogers-Ramanujan type identities", Proceedings of the National Academy of Sciences of the United States of America 78 (9): 5290–5292, doi:10.1073/pnas.78.9.5290, ISSN 0027-8424, PMC 348728, PMID 16593082, MR 629656
Baxter, Rodney J. (1980), "Hard hexagons: exact solution", Journal of Physics. A. Mathematical and General 13 (3): L61–L70, doi:10.1088/0305-4470/13/3/007, ISSN 0305-4470, MR 560533
Baxter, Rodney J. (1982), Exactly solved models in statistical mechanics, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-083180-7, MR 690578
Joyce, G. S. (1988), "Exact results for the activity and isothermal compressibility of the hard-hexagon model", Journal of Physics. A. Mathematical and General 21 (20): L983–L988, doi:10.1088/0305-4470/21/20/005, ISSN 0305-4470, MR 966792

External links

Weisstein, Eric W., "Hard Hexagon Entropy Constant", MathWorld.

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