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Complete classification of 1D gapped quantum phases in interacting spin systems

Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen

TL;DR

This work provides a complete framework for classifying gapped 1D spin systems under arbitrary symmetry constraints, unifying symmetry breaking and symmetry-protected topological (SPT) order via matrix product states. By exploiting projective representations and boundary degrees of freedom, it extends SPT classifications to combinations of parity, time reversal, and on-site symmetries, and proves that phases are determined by the unbroken subgroup and its SPT order. The authors also show how symmetry breaking coexists with, and is constrained by, symmetry fractionalization, and apply the framework to 1D fermionic systems through Jordan-Wigner mapping. The results yield a comprehensive catalog of possible 1D gapped phases and provide concrete tools for identifying and distinguishing them in interacting systems and fermionic analogs.

Abstract

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or on-site unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B \textbf{83}, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.

Complete classification of 1D gapped quantum phases in interacting spin systems

TL;DR

This work provides a complete framework for classifying gapped 1D spin systems under arbitrary symmetry constraints, unifying symmetry breaking and symmetry-protected topological (SPT) order via matrix product states. By exploiting projective representations and boundary degrees of freedom, it extends SPT classifications to combinations of parity, time reversal, and on-site symmetries, and proves that phases are determined by the unbroken subgroup and its SPT order. The authors also show how symmetry breaking coexists with, and is constrained by, symmetry fractionalization, and apply the framework to 1D fermionic systems through Jordan-Wigner mapping. The results yield a comprehensive catalog of possible 1D gapped phases and provide concrete tools for identifying and distinguishing them in interacting systems and fermionic analogs.

Abstract

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or on-site unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B \textbf{83}, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.

Paper Structure

This paper contains 20 sections, 53 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review: Matrix product states and SPT classification
  3. Classification with combined symmetry
  4. Parity + On-site $G$
  5. Time reversal + On-site $G$
  6. Parity + Time reversal
  7. Parity + Time reversal + On-site $G$
  8. General classification for SPT phases
  9. Classification with symmetry breaking
  10. Matrix product representation of symmetry breaking
  11. Classification with combination of symmetry breaking and symmetry fractionalization
  12. Application: 1D fermion SPT phases
  13. Fermion Parity Symmetry Only
  14. Fermion Parity and $T^2=1$ Time Reversal
  15. Fermion Parity and $T^2 \neq I$ Time Reversal
  16. ...and 5 more sections

Figures (3)

  • Figure 1: (Color online) (a) The possible phases for class of Hamiltonians without any symmetry. (b) The possible phases for class of Hamiltonians with some symmetries. Each phase is labeled by the phase separating mechanisms involved. The shaded regions in (a) and (b) represent the phases with long range entanglement. SRE stands for short range entanglement, LRE for long range entanglement, SB for symmetry breaking, SF for symmetry fractionalization.
  • Figure 2: (Color online) Representative states for different SPT phases. Each box represents one site, containing four spins. Every two connected spins form an entangled pair.
  • Figure 3: (Color online) Representative states for different SPT phases. Each box represents one site, containing six spins. Every two connected spins form an entangled pair.