Complete classification of one-dimensional gapped quantum phases in interacting spin systems

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 by  X. Chen, Z.-C. Gu, and X.-G. Wen [Phys. Rev. B 83, 035107 (2011)]. 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.

Figure 1

(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

Representative states for different SPT phases. Each box represents one site, containing four spins. Every two connected spins form an entangled pair.

Figure 3

Representative states for different SPT phases. Each box represents one site, containing six spins. Every two connected spins form an entangled pair.