Exact finite-size corrections of the free energy for the square lattice dimer model under different boundary conditions

Abstract

We express the partition functions of the dimer model on finite square lattices under five different boundary conditions (free, cylindrical, toroidal, Möbius strip, and Klein bottle) obtained by others (Kasteleyn, Temperley and Fisher, McCoy and Wu, Brankov and Priezzhev, and Lu and Wu) in terms of the partition functions with twisted boundary conditions Z(alpha, beta) with (alpha, beta)=(1/2,0), (0,1/2) and (1/2,1/2). Based on such expressions, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above. We find that the aspect-ratio dependence of finite-size corrections is sensitive to boundary conditions and the parity of the number of lattice sites along the lattice axis. We have also established several groups of identities relating dimer partition functions for the different boundary conditions.