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

TL;DR

This paper establishes a connection between boundary properties of 2D PEPS and the spectral gap of their parent Hamiltonians, demonstrating that certain boundary conditions imply a constant gap in the bulk.

Contribution

It introduces an approximate boundary factorization condition that guarantees a spectral gap for 2D PEPS parent Hamiltonians and shows that Gibbs states of local Hamiltonians satisfy this condition.

Findings

Boundary factorization condition implies bulk gap

Gibbs states of local Hamiltonians satisfy the boundary condition

Provides rigorous link between boundary theories and bulk spectral properties

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 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.