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

TL;DR

This paper provides a comprehensive classification of all gapped quantum phases in 1D spin systems, including symmetry protected topological phases, symmetry breaking, and long-range entanglement, extending previous work to combined symmetries.

Contribution

It extends the classification of 1D gapped phases to include combined symmetries and symmetry breaking, offering a complete framework for understanding these phases.

Findings

Classified all 1D gapped phases considering combined symmetries.

Clarified how symmetry fractionalization coexists with symmetry breaking.

Identified three main mechanisms generating diverse gapped phases.

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.