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Phys Rev Lett. 2017 Dec 15;119(24):246402. doi: 10.1103/PhysRevLett.119.246402. Epub 2017 Dec 11.

(d-2)-Dimensional Edge States of Rotation Symmetry Protected Topological States.

Song Z1,2, Fang Z1,3, Fang C1.

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Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.
University of Chinese Academy of Sciences, Beijing 100049, China.
Collaborative Innovation Center of Quantum Matter, Beijing 100084, China.


We study fourfold rotation-invariant gapped topological systems with time-reversal symmetry in two and three dimensions (d=2, 3). We show that in both cases nontrivial topology is manifested by the presence of the (d-2)-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through the explicit construction of microscopic models having robust (d-2)-dimensional edge states.

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