A Short Proof of Stability of Topological Order under Local Perturbations

By Sergey Bravyi, M. B. Hastings. 2010

192 citations recorded by the source corpus.

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 H0 can be written as a sum of local pairwise commuting projectors on a D-dimensional lattice. We consider a perturbed Hamiltonian H = H0 + 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 H0. 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 H0 decays faster than any power of the lattice size.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/699760a67163062a17939e9689b0febb1b8abd93

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Locality in Quantum Systems (2010, background): “The 4 ground states will no longer be exactly degenerate, but the difference in energy between them will be exponentially small as a function of system size (see [11, 12] for general upper bounds on the energy splitting).”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “The LTQO condition For frustration free Hamiltonians, stability of the spectral gap is implied by a conditions known as LTQO (local topological quantum order) condition (Bravyi et al., 2010; Bravyi and Hastings, 2011; Michalakis and Pytel, 2011).”