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

Jose Garre Rubio, Andras Molnar, Yoshiko Ogata

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.

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

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.
Paper Structure (23 sections, 18 theorems, 295 equations)

This paper contains 23 sections, 18 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. General theory
  3. Setting and main result
  4. The $3$rd group cohomology index from local decomposable finite group actions
  5. Index of the symmetry acting on the split states
  6. Proof of Theorem \ref{['mainthm']}
  7. Translation invariant case
  8. Representations of oxg
  9. Reduction to MPS
  10. Matrix product state approach and its GNS representation
  11. The finite size MPS and MPU setup
  12. Graphical notation
  13. The main tools for the finite size MPS setup
  14. Connection between the $3$-cocycle index of finite MPU and the one in Section \ref{['betas']}
  15. Third cohomology index of the finite size MPU representations
  16. ...and 8 more sections

Key Result

Theorem 1

There exists a ${\mathcal{T}}$-valued invariant $I(\boldsymbol{\omega})$ on $\boldsymbol{\omega}\in {\mathcal{S}}$ of the classification $\sim_G$.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • Lemma 1
  • ...and 22 more