# 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 Zheng-xin Liu and Xiao-Gang Wen}, journal={Physical Review B}, year={2011}, volume={84}, pages={235141} }

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… Expand

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#### References

SHOWING 1-3 OF 3 REFERENCES

But this is not a problem because we can redefineT (CZX) = iT (CZX) and the extra minus sign would disappear

The (d + 1)-cohomology group of Z 2 H d+1 (G,U(1)) is trivial for odd d and is a Z 2 group for even d

This is a reasonable restriction because we want to study systems at fixed point where the correlation length in the bulk is zero