A short proof of stability of topological order under local perturbations

TL;DR

This paper provides a concise alternative proof that topological quantum order remains stable under small, local perturbations, ensuring spectral gaps and well-defined low-energy bands in such quantum systems.

Contribution

It offers a shorter proof of the stability of topological order under local perturbations for models with commuting projectors, extending previous results.

Findings

Spectral bands remain well-defined under small perturbations.

The gap between low-lying eigenvalues and the rest of the spectrum is maintained.

The width of the lowest spectral band decreases faster than any power of the lattice size.

Abstract

Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological quantum order for which the unperturbed Hamiltonian $H_0$ can be written as a sum of local pairwise commuting projectors on a $D$-dimensional lattice. We consider a perturbed Hamiltonian $H=H_0+V$ involving a generic perturbation $V$ that can be written as a sum of short-range bounded-norm interactions. We prove that if the strength of $V$ is below a constant threshold value then $H$ has well-defined spectral bands originating from the low-lying eigenvalues of $H_0$. These bands are separated from the rest of the spectrum and from each other by a constant gap. The width of the band originating from the smallest eigenvalue of $H_0$ decays faster than any power of the lattice size.