Stability of Frustration-Free Hamiltonians

TL;DR

This paper proves the stability of spectral gaps in frustration-free Hamiltonians under perturbations, introduces a new stability condition called 'Local Topological Quantum Order,' and links it to entanglement entropy area laws.

Contribution

It establishes a necessary and sufficient condition for spectral gap stability in frustration-free Hamiltonians, extending prior work on topological quantum order.

Findings

Spectral gap stability under quasi-local perturbations

Introduction of 'Local Topological Quantum Order' condition

Connection between stability and entanglement entropy area law

Abstract

We prove stability of the spectral gap for gapped, frustration-free Hamiltonians under general, quasi-local perturbations. We present a necessary and sufficient condition for stability, which we call "Local Topological Quantum Order" and show that this condition implies an area law for the entanglement entropy of the groundstate subspace. This result extends previous work by Bravyi et al., on the stability of topological quantum order for Hamiltonians composed of commuting projections with a common zero-energy subspace. We conclude with a list of open problems relevant to spectral gaps and topological quantum order.