A canonical form for Projected Entangled Pair States and applications

TL;DR

This paper establishes a canonical form for injective PEPS, characterizes their symmetries, and applies these results to prove constraints on symmetric PEPS, including a Lieb-Shultz-Mattis type theorem.

Contribution

It introduces a canonical form for injective PEPS and uses it to analyze symmetry constraints and non-injectivity conditions in PEPS.

Findings

Injective PEPS tensors are unique up to trivial gauge transformations.

SU(2) invariant PEPS with half-integer spins cannot be injective.

PEPS with Wilson loops cannot be injective.

Abstract

We show that two different tensors defining the same translational invariant injective Projected Entangled Pair State (PEPS) in a square lattice must be the same up to a trivial gauge freedom. This allows us to characterize the existence of any local or spatial symmetry in the state. As an application of these results we prove that a SU(2) invariant PEPS with half-integer spin cannot be injective, which can be seen as a Lieb-Shultz-Mattis theorem in this context. We also give the natural generalization for U(1) symmetry in the spirit of Oshikawa-Yamanaka-Affleck, and show that a PEPS with Wilson loops cannot be injective.