Z2 -projective translational symmetry protected topological phases

-projective translational symmetry protected topological phases},
  author={Y. X. Zhao and Yue-Xin Huang and Shengyuan A. Yang},
  journal={Physical Review B},
Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel physics. We show that the ${\mathbb{Z}}_{2}$ projectively represented translational symmetry operators adopt a distinct anticommutation relation. As a result, each energy band is twofold degenerate, and carries a varying spinor structure for translation operators in momentum space, which cannot be… 
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