Topological quantum order: Stability under local perturbations

Journal Article · Wed Sep 15 00:00:00 EDT 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3490195· OSTI ID:21476500

Bravyi, Sergey ^[1]; Hastings, Matthew B ^[2]; Michalakis, Spyridon ^[3]

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1. IBM Watson Research Center, Yorktown Heights, New York 10594 (United States)
2. Microsoft Research Station Q, CNSI Building, University of California, Santa Barbara, California 93106 (United States)
3. T-4 and CNLS, LANL, Los Alamos, New Mexico 87544 (United States)

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 H{sub 0}, we prove that there exists a constant threshold {epsilon}>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H{sub 0}+{epsilon}V has well-defined spectral bands originating from low-lying eigenvalues of H{sub 0}. 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 H{sub 0} 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.

• OSTI ID: 21476500
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 9; Other Information: DOI: 10.1063/1.3490195; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DISTURBANCES
EIGENVALUES
HAMILTONIANS
INTERACTION RANGE
LATTICE FIELD THEORY
SPIN
STABILITY
TOPOLOGY
ANGULAR MOMENTUM
CONSTRUCTIVE FIELD THEORY
DISTANCE
FIELD THEORIES
MATHEMATICAL OPERATORS
MATHEMATICS
PARTICLE PROPERTIES
QUANTUM FIELD THEORY
QUANTUM OPERATORS