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A Short Proof of Stability of Topological Order under Local Perturbations

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Abstract

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.

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Authors and Affiliations

  1. IBM Watson Research Center, Yorktown Heights, NY, 10594, USA

    Sergey Bravyi

  2. Microsoft Research Station Q, CNSI Building, University of California, Santa Barbara, CA, 93106, USA

    Matthew B. Hastings

Authors
  1. Sergey Bravyi
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  2. Matthew B. Hastings
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Correspondence to Matthew B. Hastings.

Additional information

Communicated by I.M. Sigal

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Bravyi, S., Hastings, M.B. A Short Proof of Stability of Topological Order under Local Perturbations. Commun. Math. Phys. 307, 609–627 (2011). https://doi.org/10.1007/s00220-011-1346-2

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  • DOI: https://doi.org/10.1007/s00220-011-1346-2

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