On the stability of topological phases on a lattice

TL;DR

This paper investigates the robustness of topological phases, specifically anyonic models and the toric code, against local perturbations on lattice systems, establishing conditions for spectral gap stability.

Contribution

It introduces a cluster expansion method to analyze the stability of topological models under perturbations and identifies conditions for their spectral gap stability.

Findings

Spectral gap remains stable on a sphere with no ground state degeneracy.

Toric code is stable on a large torus with simultaneous infinite directions.

Toric code becomes unstable in the thin torus limit.

Abstract

We study the stability of anyonic models on lattices to perturbations. We establish a cluster expansion for the energy of the perturbed models and use it to study the stability of the models to local perturbations. We show that the spectral gap is stable when the model is defined on a sphere, so that there is no ground state degeneracy. We then consider the toric code Hamiltonian on a torus with a class of abelian perturbations and show that it is stable when the torus directions are taken to infinity simultaneously, and is unstable when the thin torus limit is taken.