Classifying symmetric and symmetry-broken spin chain phases with anomalous group actions

TL;DR

This paper develops a classification framework for quantum spin chains with local group symmetries, capturing both symmetric and symmetry-broken phases, and connects operator algebraic invariants with matrix product state techniques.

Contribution

It introduces invariants for classifying spin chain phases with local decomposable group actions, linking algebraic and MPS approaches, and covers symmetry protected topological phases.

Findings

Derived invariants for 1D SPT phases

Established correspondence between algebraic and MPS invariants

Explicit GNS representation of MPUs and MPSs

Abstract

We consider the classification problem of quantum spin chains invariant under local decomposable group actions, covering matrix product unitaries (MPUs), using an operator algebraic approach. We focus on finite group symmetries hosting both symmetric and symmetry broken phases. The local-decomposable group actions we consider have a 3-cocycle class of the symmetry group associated to them. We derive invariants for our classification that naturally cover one-dimensional symmetry protected topological (SPT) phases. We prove that these invariants coincide with the ones of [J. Garre Rubio et al, Quantum 7, 927 (2023)] using matrix product states (MPSs) techniques, by explicitly working out the GNS representation of MPSs and MPUs, resulting in a useful dictionary between both approaches that could be of independent interest.