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

Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen
Phys. Rev. B 87, 155114 – Published 4 April 2013

Abstract

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 H1+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 H1+d[U(1)Z2T,UT(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 H1+d[Z2T,UT(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: (GH,GΨ,H1+d[GΨ,UT(1)]), where GH is the symmetry group of the Hamiltonian and GΨ the symmetry group of the ground states.

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  • Received 5 January 2013

DOI:https://doi.org/10.1103/PhysRevB.87.155114

©2013 American Physical Society

Authors & Affiliations

Xie Chen1,2, Zheng-Cheng Gu3,4, Zheng-Xin Liu5,2, and Xiao-Gang Wen6,2,5

  • 1Department of Physics, University of California, Berkeley, California 94720, USA
  • 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
  • 4Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
  • 5Institute for Advanced Study, Tsinghua University, Beijing, 100084, People's Republic of China
  • 6Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

See Also

Classification of symmetry enriched topological phases with exactly solvable models

Andrej Mesaros and Ying Ran
Phys. Rev. B 87, 155115 (2013)

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Vol. 87, Iss. 15 — 15 April 2013

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