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.

Non-Local Finite-Size Effects in the Dimer Model

“…The effects caused by the insertion of monomers have also been reconsidered recently [10,11,12]. Finite-size effects also have a long history, starting in [4,13], with many subsequent works [14,15,16,17,18,19,20,21,11].…”

Exact finite-size corrections for the spanning-tree model under different boundary conditions

“…[7] and Kronecker's double series K 0,0 2p+2 (izξ ) in terms of the elliptic θ functions are given in Refs. [3,5,7] for p = 1, 2, 3, and 4. It is easy to see from Eq. (61) that for the spanning tree on finite square lattices under periodic-boundary conditions, f 0 (zξ ) does not contain the corner free energy f corner given by Eq.…”

“…In the quest to improve our understanding of realistic systems of finite extent, two-dimensional models play crucial roles in statistical mechanics as they have long served as a testing ground to explore the general ideas of finite-size scaling under controlled conditions. Of particular importance in such studies are exact results where the analysis can be carried out without numerical errors, the Ising model [1][2][3], the dimer, and spanning tree model [4][5][6][7] being the most prominent examples.…”

“…Our objective in this paper is to study the finite-size properties of a spanning tree on the plane square lattice under five different sets of boundary conditions using the same techniques developed in Refs. [3,5] and [7]. The paper is organized as follows.…”

“…The Kronecker's double series K 0,0 2p (τ ) and K 1 2 , 1 2 2p (τ ) can all be expressed in terms of the elliptic θ functions only [3,5,7].…”