• DOI:10.1103/PhysRevB.84.235141
• Corpus ID: 55330505

Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations

@article{Chen2011TwodimensionalST,
  title={Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations},
  author={Xie Chen and Zhengxin Liu and Xiao-Gang Wen},
  journal={Physical Review B},
  year={2011},
  volume={84},
  pages={235141}
}

• Xie Chen, Zhengxin Liu, X. Wen
• Published 23 June 2011
• Physics, Mathematics
• Physical Review B

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{sub 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{sub 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{sub 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 usedmore » to construct a 2D fermionic model with onsite Z{sub 2} symmetry-protected topological order.« less