Critical properties of the Ising model in hyperbolic space

TL;DR

This paper investigates the critical properties of the Ising model in hyperbolic space, revealing mean-field behavior and the significant role of boundary conditions, using Monte Carlo and series expansion techniques.

Contribution

It provides the first comprehensive analysis of the Ising model's critical behavior in hyperbolic space, demonstrating mean-field universality and clarifying the relationship with Bethe lattice results.

Findings

Critical exponents are of mean-field type.

Boundary conditions significantly influence bulk properties.

Results contrast with some field theory predictions.

Abstract

The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two and three dimensional hyperbolic spaces using Monte Carlo and high and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the 'asymptotic' limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.