Stability of Frustration-Free Hamiltonians

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.

21 pages, 1 figure, updated references, discussion includes note on stability of symmetry-protected and higher-energy sectors, and a path towards extending the result to frustrated, gapped Hamiltonians. Added new section on overview of the proof and updated the Introduction and Appendices