Finite size corrections for the Ising model on higher genus triangular lattices

TL;DR

This paper investigates how the topology of higher genus surfaces affects finite size corrections in the Ising model, revealing universal, shape-dependent corrections at criticality linked to Riemann theta functions and conformal field theory.

Contribution

It introduces a novel analysis of finite size corrections on genus two surfaces, connecting lattice models with Riemann surface theory and conformal invariance.

Findings

Universal shape-dependent correction expressed via Riemann theta functions

Reproduction of conformal field theory partition function invariants

Numerical determination of the period matrix for the Riemann surface

Abstract

We study the topology dependence of finite size corrections to the Ising model partition function by considering the model on a triangular lattice embedded on a genus two surface. At criticality we observe a universal shape dependent correction, expressible in terms of Riemann theta functions, that reproduces the modular invariant partition function of the corresponding conformal field theory. The period matrix characterizing the moduli parameters of the limiting Riemann surface is obtained by a numerical study of the lattice continuum limit. The same results are reproduced using a discrete holomorphic structure.