Dimers and the Critical Ising Model on Lattices of genus>1

TL;DR

This paper investigates the partition functions of dimers and the critical Ising model on genus two lattices, revealing their dependence on boundary conditions and their relation to conformal field theory via theta functions.

Contribution

It introduces a method to connect lattice combinatorics with conformal field theory on higher genus surfaces using discrete holomorphic structures and theta functions.

Findings

Determinants depend on boundary conditions and relate to genus two theta functions.

The period matrix of the continuum limit is computed using discrete holomorphic structures.

Results bridge lattice models with conformal field theory on higher genus Riemann surfaces.

Abstract

We study the partition function of both Close-Packed Dimers and the Critical Ising Model on a square lattice embedded on a genus two surface. Using numerical and analytical methods we show that the determinants of the Kasteleyn adjacency matrices have a dependence on the boundary conditions that, for large lattice size, can be expressed in terms of genus two theta functions. The period matrix characterizing the continuum limit of the lattice is computed using a discrete holomorphic structure. These results relate in a direct way the lattice combinatorics with conformal field theory, providing new insight to the lattice regularization of conformal field theories on higher genus Riemann Surfaces.