Topological quantum order: stability under local perturbations

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.