The Ising model on curved lattices

TL;DR

This paper reviews recent findings on finite size corrections to the Ising model free energy on curved lattices, highlighting universal terms linked to topology and curvature, confirmed through numerical methods.

Contribution

It demonstrates the numerical observation of universal finite size correction terms on curved lattices and relates them to conformal field theory predictions.

Findings

Logarithmic correction matches theoretical predictions by Cardy and Peschel.

Constant term expressed via Riemann theta functions.

Universal finite size corrections observed across various topologies.

Abstract

We review recent results concerning finite size corrections to the Ising model free energy on lattices with non-trivial topology and curvature. From conformal field theory considerations two distinct universal terms are expected, a logarithmic term determined by the system curvature and a scale invariant term determined by the system shape and topology. Both terms have been observed numerically, using the Kasteleyn Pfaffian method, for lattices with topologies ranging from the sphere to that of a genus two surface. The constant term is shown to be expressible in terms of Riemann theta functions while the logarithmic correction reproduces the theoretical prediction by Cardy and Peschel for singular metrics.