The Ising model on curved lattices

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.