Quasi-Locality Bounds for Quantum Lattice Systems. Part II.
Perturbations of Frustration-Free Spin Models with Gapped Ground States
- URL: http://arxiv.org/abs/2010.15337v3
- Date: Fri, 11 Jun 2021 22:53:37 GMT
- Title: Quasi-Locality Bounds for Quantum Lattice Systems. Part II.
Perturbations of Frustration-Free Spin Models with Gapped Ground States
- Authors: Bruno Nachtergaele, Robert Sims, Amanda Young
- Abstract summary: We study the stability with respect to a broad class of perturbations of gapped ground state phases of quantum spin systems.
Under a condition of Local Topological Quantum Order, the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential.
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- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We study the stability with respect to a broad class of perturbations of
gapped ground state phases of quantum spin systems defined by frustration-free
Hamiltonians. The core result of this work is a proof using the
Bravyi-Hastings-Michalakis (BHM) strategy that under a condition of Local
Topological Quantum Order, the bulk gap is stable under perturbations that
decay at long distances faster than a stretched exponential. Compared to
previous work we expand the class of frustration-free quantum spin models that
can be handled to include models with more general boundary conditions, and
models with discrete symmetry breaking. Detailed estimates allow us to
formulate sufficient conditions for the validity of positive lower bounds for
the gap that are uniform in the system size and that are explicit to some
degree. We provide a survey of the BHM strategy following the approach of
Michalakis and Zwolak, with alterations introduced to accommodate more general
than just periodic boundary conditions and more general lattices. We express
the fundamental condition known as LTQO by means of the notion of
indistinguishability radius, which we introduce. Using the uniform
finite-volume results we then proceed to study the thermodynamic limit. We
first study the case of a unique limiting ground state and then also consider
models with spontaneous breaking of a discrete symmetry. In the latter case,
LTQO cannot hold for all local observables. However, for perturbations that
preserve the symmetry, we show stability of the gap and the structure of the
broken symmetry phases. We prove that the GNS Hamiltonian associated with each
pure state has a non-zero spectral gap above the ground state.
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