Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations
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Chen-2012-Two-dimensional symmetry-protected.pdf
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Author(s)
Chen, Xie
Liu, Zheng-Xin
Wen, Xiao-Gang
Date Issued
December 2011
Journal
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
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Physics
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DOI of Published Version
https://doi.org/10.1103/PhysRevB.84.235141
