Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations
Author(s)
Chen, Xie; Liu, Zheng-Xin; Wen, Xiao-Gang
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Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z[subscript 2] symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z[subscript 2] group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z[subscript 2] symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite Z[subscript 2] symmetry-protected topological order.
Date issued
2011-12Department
Massachusetts Institute of Technology. Department of PhysicsJournal
Physical Review B
Publisher
American Physical Society
Citation
Chen, Xie, Zheng-Xin Liu, and Xiao-Gang Wen. “Two-dimensional Symmetry-protected Topological Orders and Their Protected Gapless Edge Excitations.” Physical Review B 84.23 (2011): 235141-1-235141-13. ©2011 American Physical Society
Version: Final published version
ISSN
1098-0121
1550-235X