Simplicial Lattice Study of the 2d Ising CFT

TL;DR

This paper develops a geometric framework for simulating the 2D critical Ising model on non-uniform simplicial lattices, demonstrating agreement with conformal field theory predictions and proposing broader applications for quantum field theory studies.

Contribution

It introduces a novel geometric formulation of the 2D Ising model on non-uniform lattices and establishes conditions for continuum limits, validated through Monte Carlo simulations.

Findings

Simulations agree with 2D Ising CFT predictions.

Derived geometric constraints for lattice continuum limits.

Applicable to complex manifolds like twisted tori and spheres.

Abstract

I derive a formulation of the 2-dimensional critical Ising model on non-uniform simplicial lattices. Surprisingly, the derivation leads to a set of geometric constraints that a lattice must satisfy in order for the model to have a well-defined continuum limit. I perform Monte Carlo simulations of the critical Ising model on discretizations of several non-trivial manifolds including a twisted torus and a 2-sphere and I show that the simulations are in agreement with the 2d Ising CFT in the continuum limit. I discuss the inherent benefits of using non-uniform simplicial lattices to study quantum field theory and how the methods developed here can potentially be generalized for use with other theories.