Locality at the boundary implies gap in the bulk for 2D PEPS

Michael J. Kastoryano; Angelo Lucia; David Perez-Garcia

doi:10.1007/s00220-019-03404-9

Locality at the boundary implies gap in the bulk for 2D PEPS

Michael J. Kastoryano, Angelo Lucia, David Perez-Garcia

4 Citationer (Scopus)

Abstract

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.

Originalsprog	Engelsk
Tidsskrift	Communications in Mathematical Physics
Vol/bind	366
Udgave nummer	3
Sider (fra-til)	895–926
ISSN	0010-3616
DOI	https://doi.org/10.1007/s00220-019-03404-9
Status	Udgivet - 1 mar. 2019

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10.1007/s00220-019-03404-9

https://arxiv.org/abs/1709.07691

Citationsformater

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AU - Perez-Garcia, David

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AB - 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.

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Locality at the boundary implies gap in the bulk for 2D PEPS

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