# $\mathbb{Z}_N$ symmetry breaking in Projected Entangled Pair State models

## TL;DR

This paper investigates how $ ext{Z}_N$ symmetry breaking manifests in PEPS models, linking long-range order to transfer operator degeneracy, and characterizes symmetry-broken states via transfer operator fixed points, supported by numerical analysis.

## Contribution

It provides a theoretical framework connecting symmetry breaking in PEPS to transfer operator degeneracy and characterizes stable states through boundary conditions, with numerical validation.

## Key findings

- Long-range order correlates with transfer operator degeneracy.
- Symmetry-broken states are characterized by fixed points of the transfer operator.
- Entanglement Hamiltonian from broken states is quasi-local.

## Abstract

We consider Projected Entangled Pair State (PEPS) models with a global $\mathbb Z_N$ symmetry, which are constructed from $\mathbb Z_N$-symmetric tensors and are thus $\mathbb Z_N$-invariant wavefunctions, and study the occurence of long-range order and symmetry breaking in these systems. First, we show that long-range order in those models is accompanied by a degeneracy in the so-called transfer operator of the system. We subsequently use this degeneracy to determine the nature of the symmetry broken states, i.e., those stable under arbitrary perturbations, and provide a succinct characterization in terms of the fixed points of the transfer operator (i.e.\ the different boundary conditions) in the individual symmetry sectors. We verify our findings numerically through the study of a $\mathbb Z_3$-symmetric model, and show that the entanglement Hamiltonian derived from the symmetry broken states is quasi-local (unlike the one derived from the symmetric state), reinforcing the locality of the entanglement Hamiltonian for gapped phases.