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 is an element of>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H-0+is an element of V has well-defined spectral bands originating from low-lying eigenvalues of H-0. These bands are separated from the rest of the spectra 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. (C) 2010 American Institute of Physics. [doi:10.1063/1.3490195]