A Short Proof of Stability of Topological Order under Local Perturbations

Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological quantum order for which the unperturbed Hamiltonian H(0) can be written as a sum of local pairwise commuting projectors on a D-dimensional lattice. We consider a perturbed Hamiltonian H = H(0) + V involving a generic perturbation V that can be written as a sum of short-range bounded-norm interactions. We prove that if the strength of V is below a constant threshold value then H has well-defined spectral bands originating from the low-lying eigenvalues of H(0). These bands are separated from the rest of the spectrum and from each other by a constant gap. The width of the band originating from the smallest eigenvalue of H(0) decays faster than any power of the lattice size.