Topological Quantum Order: Stability Under Local Perturbations

Topological Quantum Order: Stability Under Local Perturbations
Spiros Michalakis, Los Alamos National Laboratory
We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. We focus on 2D models of TQO for which the unperturbed Hamiltonian H_0 can be written as a sum of local pairwise commuting projectors. A perturbed Hamiltonian H=H_0+ epsilon V involves a generic perturbation V that can be written as a sum of short-range bounded-norm interactions. Let lambda_1, ldots, lambda_g be the set of g lowest eigenvalues of H, where g is the ground state egeneracy of H_0. We prove that if epsilon is below a constant threshold value then all eigenvalues lambda_1, ldots, lambda_g coincide up to exponentially small corrections and separated from the rest of the spectrum of H by a constant gap. This result implies that H and H_0 can be connected by an adiabatic path without closing the gap. Our proof goes by setting up Hamiltonian flow equations transforming a generic perturbation to a special "block-diagonal" perturbation that preserves the ground subspace of H_0. These flow equations are analyzed using Lieb-Robinson bounds and the theory of relatively bounded operators. Our technical tools might be useful for analyzing other stability problems for quantum spin systems.

http://www.cs.caltech.edu/seminars/abstracts/09-10/michalakis-s.html

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