2010/01/03 by Sergey Bravyi, Matthew Hastings, Matthew B. Hastings +1 · 368 citations
Mathematics · Physics and Astronomy · #Bounded function #Eigenvalues and eigenvectors #Essential spectrum #Hamiltonian (control theory) #Lattice (music) #Perturbation (astronomy) #Quantum #Quantum chaos and dynamical systems #Quantum many-body systems #Topological Materials and Phenomena #Topological order #Topological quantum number #cond-mat.stat-mech #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1063/1.3490195
published in Journal of Mathematical Physics 51(9) (American Institute of Physics) · 41 pages, 1 figure
arxiv created 2010/01/03 · openalex publication_date 2010/09/01 · arxiv updated 2015/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 H0, we prove that there exists a constant threshold ϵ>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. 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 H0 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.