PEPS as unique ground states of local Hamiltonians

TL;DR

This paper studies projected entangled pair states (PEPS) on various lattices, establishing conditions for their uniqueness as ground states of local Hamiltonians and exploring the presence or absence of energy gaps depending on lattice geometry.

Contribution

It introduces a boundary-bulk injectivity condition for PEPS that guarantees their uniqueness as ground states and analyzes gap properties related to lattice geometry.

Findings

Injectivity condition ensures PEPS are unique ground states

On certain lattices, parent Hamiltonians can be gapless

Boundary-bulk relation depends on lattice geometry

Abstract

In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.