The Ising Model on $\mathbb S^2$

TL;DR

This paper introduces a 2D Ising model on a triangulated sphere designed to approximate the conformal field theory in the continuum limit, revealing geometric constraints and validated by Monte Carlo simulations.

Contribution

It presents a novel lattice construction of the Ising model on curved manifolds that aligns with the exact CFT, including geometric constraints and simulation validation.

Findings

Monte Carlo results agree with the Ising CFT on $\\mathbb S^2$

Derivation reveals geometric constraints for lattice theories on curved surfaces

Method can be extended to other quantum theories on curved manifolds

Abstract

We define a 2-dimensional Ising model on a triangulated sphere, $\mathbb S^2$, designed to approach the exact conformal field theory (CFT) in the continuum limit. Surprisingly, the derivation leads to a set of geometric constraints that the lattice field theory must satisfy. Monte Carlo simulations are in agreement with the exact Ising CFT on $\mathbb S^2$. We discuss the inherent benefits of using non-uniform simplicial lattices and how these methods may be generalized for use with other quantum theories on curved manifolds.