Entanglement spectrum and boundary theories with projected entangled-pair states

TL;DR

This paper establishes an exact duality between the bulk properties of quantum spin systems and their boundary theories using PEPS, revealing how bulk phases relate to boundary Hamiltonian characteristics and providing a holographic framework.

Contribution

It introduces a novel duality mapping linking bulk entanglement spectra to boundary Hamiltonian spectra in quantum spin systems using PEPS.

Findings

Gapped bulk phases correspond to local boundary Hamiltonians.

Critical bulk behavior leads to diverging boundary interaction lengths.

Topologically ordered states have non-local boundary Hamiltonians.

Abstract

In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated to their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using Projected Entangled Pair States (PEPS). This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models, like a deformed AKLT [1], an Ising-type [2], and Kitaev's toric code [3], both in finite ladders and infinite square lattices. In the latter case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield non-local Hamiltonians. As our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary.