The Ising Model in Curved Geometries*

The study of statistical mechanics in curved geometries has recently gained an increasing amount of attention. Particularly the Ising model in negatively curved spaces has been studied as a mean to understand exotic crystals, soft-matter and field theories in Anti-de Sitter spaces.
We analyze the Ising model on in the hyperbolic plane as well as 2+1-Anti-de Sitter (AdS) space using high temperature series-expansion and Monte-Carlo simulations. While series expansions have been performed on the hyperbolic plane before, we study a wider class of hyperbolic lattices and go to much higher order, allowing us to analyze the dependency of critical phenomena on the magnitude of curvature.
In the past Monte Carlo methods have been difficult to use as the curvature lead to severe boundary effects which persist even for large system sizes. We overcome this problem by constructing families of closed 2D surfaces of increasing area and compactified AdS spaces of increasing volume, akin to periodic boundaries in euclidean space.
It has been discussed into which universality class the Ising model in hyperbolic space falls. Our results strongly support that it falls into the mean-field universality class.