Locality at the Boundary Implies Gap in the Bulk for 2D PEPS

Proving that the parent Hamiltonian of a Projected Entangled Pair State (PEPS) is gapped remains an important open problem. We take a step forward in solving this problem by showing two results: first, we identify an approximate factorization condition on the boundary state of rectangular subregions...

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Bibliographic Details
Published in:Communications in mathematical physics 2019-03, Vol.366 (3), p.895-926
Main Authors: Kastoryano, Michael J., Lucia, Angelo, Perez-Garcia, David
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Martingales
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Projectors
Quantum Physics
Quantum theory
Relativity Theory
Theoretical
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Summary:Proving that the parent Hamiltonian of a Projected Entangled Pair State (PEPS) is gapped remains an important open problem. We take a step forward in solving this problem by showing two results: first, we identify an approximate factorization condition on the boundary state of rectangular subregions that is sufficient to prove that the parent Hamiltonian of the bulk 2D PEPS has a constant gap in the thermodynamic limit; second, we then show that Gibbs state of a local, finite-range Hamiltonian satisfy such condition. The proof applies to the case of injective and MPO-injective PEPS, employs the martingale method of nearly commuting projectors, and exploits a result of Araki (Commun Math Phys 14(2):120–157, 1969 ) on the robustness of one dimensional Gibbs states. Our result provides one of the first rigorous connections between boundary theories and dynamical properties in an interacting many body system.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-019-03404-9