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Symmetry protected topological orders and the group cohomology of their symmetry group

Chen, Xie and Gu, Zheng-Cheng and Liu, Zheng-Xin and Wen, Xiao-Gang (2013) Symmetry protected topological orders and the group cohomology of their symmetry group. Physical Review B, 87 (15). Art. No. 155114. ISSN 1098-0121. doi:10.1103/PhysRevB.87.155114.

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Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H^(1+d)[G,UT(1)], the Borel (1+d)-group-cohomology classes of G over the G module UT(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological θ term in continuous nonlinear σ models to lattice nonlinear σ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear σ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models. As a result, the symmetry G must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use H^(1+d)[U(1)⋊ Z^(T)_(2),U_T(1)] to obtain interacting bosonic topological insulators (protected by time reversal Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H^(1+d)[Z^(T)_(2),U_T(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use H1+d[U(1)×Z2T,UT(1)] to construct the SPT phases of integer spin systems with time-reversal and U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H,G_Ψ,H^(1+d)[G_Ψ,U_T(1)]), where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.

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Additional Information:© 2013 American Physical Society. Received 5 January 2013; published 4 April 2013. X.G.W. would like to thank Michael Levin for helpful discussions and for sharing his result of bosonic SPT phases in (2 + 1)D.71 This motivated us to calculate H1+d [U(1),U(1)], which reproduced his results for d = 2.Wewould like to thank Geoffrey Lee, Jian-Zhong Pan, and Zhenghan Wang for many very helpful discussions on group cohomology for discrete and continuous groups. Z.C.G. would like to thank Dung-Hai Lee for discussion on the possibility of discretized Berry phase term in (1 + 1)D. This research is supported by NSF Grants No. DMR-1005541, No. NSFC 11074140, and No. NSFC 11274192. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research. Z.C.G. is supported by NSF Grant No. PHY05-51164.
Funding AgencyGrant Number
NSFNSFC 11074140
NSFNSFC 11274192
Government of Canada Industry CanadaUNSPECIFIED
Province of Ontario Ministry of ResearchUNSPECIFIED
Issue or Number:15
Classification Code:PACS: 71.27.+a, 02.40.Re
Record Number:CaltechAUTHORS:20130509-104730026
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38383
Deposited On:09 May 2013 21:30
Last Modified:09 Nov 2021 23:37

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