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

By Xie Chen, Z. Gu, X. Wen. 2011

304 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/38e0e39ec34f3ecbd6b38d183da5114efacb3cbe

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Gapless Symmetry-Protected Topological Order (2017, background): “[22] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, “Complete classification of one-dimensional gapped quantum phases in interacting spin systems,” Phys.”
Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states (2012, background): “[59] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, Complete Classification of One-Dimensional Gapped”
Quantum spin liquid states (2016, background): “…Sσ describes two plausible phases, the weak coupling phase, in which the ground state is antiferromagnetically ordered, 1 For S = 1 chains, the S = 1/2 end states are protected by a weaker Z2 × Z2 symmetry (Chen et al., 2011a,b). and the strong coupling phase, in which the ground state is gapped.”
Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems (2014, background): “In addition, the theoretical underpinning of tensor-network states in terms of the area law of entanglement entropy [38,39] makes them suitable for theoretical results as profound as the complete classification of gapped quantum phases [40,41].”
Classification of Interacting Topological Floquet Phases in One Dimension (2016, methodology): “Consequently, in the absence of periodic driving, such projective representations fully characterize the set of ground-state SPT phases [32,41,42].”
Measurement as a Shortcut to Long-Range Entangled Quantum Matter (2022, background): “More recently, it was found that performing measurements on SRE states in certain symmetry-protected-topological (SPT) phases [5, 6] can lead to a large class of LRE states in topologically ordered phases [7, 8].”