Topological quantum order: Stability under local perturbations

By Sergey Bravyi, M. B. Hastings, Spyridon Michalakis. 2010

388 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/6ab3d559fe417f19c575d0c590df8e4d141344f3

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Citations in the corpus

Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems (2011, unknown): “More recently, stability under small perturbations of the interaction was proved for a class of models with topologically ordered ground states [7]; these include e.”
Locality of temperature (2013, background): “[44] S.”
A Short Proof of Stability of Topological Order under Local Perturbations (2010, methodology): “Recently the authors of [3] presented a rigorous proof of the gap stability for a large class of Hamiltonians describing TQO.”
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).”
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).”
Fracton Topological Order, Generalized Lattice Gauge Theory and Duality (2016, background): “The topologically ordered fracton phase and the Ising paramagnet survive up to a finite t/h and J/h, as both phases are gapped and stable to perturbations [39].”
Topological order in an exactly solvable 3D spin model (2010, background): “As was shown in [17], conditions TQO-1,2 together are sufficient for zero-temperature stability.”
Quantum error correction for quantum memories (2013, unknown): “This feature has been rigorously proven for stabilizer codes by Bravyi, Hastings, and Michalakis (2010) under a slightly sharpened form of the quantum error-correction conditions referred to as local topological order.”
Quantum memories at finite temperature (2014, background): “Probably the most well known, but also most misinterpreted result regarding perturbations in topological ordered systems is that of (Bravyi et al., 2010a).”
Stability of Frustration-Free Hamiltonians (2011, methodology): “Motivated by this issue, soon after the results of Klich and Bravyi et al., a series of papers appeared that focused on the beneficial effects of Anderson-type localization of the undesired excitations in the presence of impurities, or external magnetic fields [24, 43, 45].”