Topological quantum order: stability under local perturbations

TL;DR

This paper proves that certain topologically ordered quantum phases remain stable under small local perturbations, maintaining spectral gaps and localized eigenstates, which is crucial for quantum information applications.

Contribution

It establishes a rigorous stability result for topological phases of matter under local perturbations using Hamiltonian flow and Lieb-Robinson bounds.

Findings

Spectral gaps are preserved under small perturbations.

Eigenstates associated with topological order remain exponentially localized.

The stability holds for a broad class of quantum spin Hamiltonians.

Abstract

We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H_0 we prove that there exists a constant threshold \epsilon>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions the perturbed Hamiltonian H=H_0+\epsilon V has well-defined spectral bands originating from O(1) smallest eigenvalues of H_0. These bands are separated from the rest of the spectrum and from each other by a constant gap. The band originating from the smallest eigenvalue of H_0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb-Robinson bound.