${\mathbb Z}_N$ symmetry breaking in projected entangled pair state models

Abstract

We consider Projected Entangled Pair State (PEPS) models with a global ZN symmetry, which are constructed from ZN -symmetric tensors and are thus ZN -invariant wavefunctions, and study the occurence of long-range order and symmetry breaking in these systems. First, we show that longrange 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 Z3-symmetric model, and show that the entanglement Hamiltonian derived from the symmetry broken states is quasilocal (unlike the one derived from the symmetric state), reinforcing the locality of the entanglement Hamiltonian for gapped phases.

Matrix product states and projected entangled pair states: Concepts, symmetries, theorems

“…In numerical studies, however, it has been always found that if the state jψi has a finite correlation length, then the boundary Hamiltonian is quasilocal (Cirac et al, 2011). With a suitable treatment of sectors, this also holds for symmetrybreaking (Rispler, Duivenvoorden, and Schuch, 2015;Rispler, Duivenvoorden, and Schuch, 2017) and topological (Schuch et al, 2013) phases. More specifically, we can always expand it as…”