Exact finite-size corrections in the dimer model on a planar square lattice

Abstract

We consider the dimer model on the rectangular $2M\times 2N$ lattice with free boundary conditions. We derive exact expressions for the coefficients in the asymptotic expansion of the free energy in terms of the elliptic theta functions (${\theta }_2,{\theta }_3,{\theta }_4$) and the elliptic integral of second kind ($E$), up to 22nd order. Surprisingly we find that ratio of the coefficients $f_p$ in the free energy expansion for strip ($f_p^{\mathrm{strip}}$) and square ($f_p^{\mathrm{sq}}$) geometries $r_p=f_p^{\mathrm{strip}}/f_p^{\mathrm{sq}}$ in the limit of large $p$ tends to $1/2$. Furthermore, we predict that the ratio of the coefficients $f_p$ in the free energy expansion for rectangular ($f_p(\rho )$) for aspect ratio $\rho >1$ to the coefficients of the free energy for square geometries, multiplied by ${\rho }^{-p-1}$, that is $r_p={\rho }^{-p-1}{f}_p(\rho )/f_p^{\mathrm{sq}}$, is also equal to $1/2$ in the limit of $p\to \infty$. We find that the corner contribution to the free energy for the dimer model on rectangular $2M\times 2N$ lattices with free boundary conditions is equal to zero and explain that result in the framework of conformal field theory, in which the central charge of the considering model is $c=-2$. We also derive a simple exact expression for the free energy of open strips of arbitrary width.

Supplementary material appended to the paper

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