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I want to calculate the Chern number of a surface of 3D time reversal insulator. I know the 3D topological insulator is characterized by the Z2 index annd that the Chern number can be used to characterize the 2D cases and see whether the system has non trivial edge state if it broke the time reversal symmetry. But is it possible if I want the calculate the Chern number of a specific 2D plane in Brillouin zone of a 3D structure?

The reason why I ask this is because I wrote a code to calculate Chern number of specific plane in brillouin zone of a 3D tight-binding structure from Wannier90 (I used the wannier90_hr.dat file to generate the tight binding matrix and solve the eigenvector directly to calculate the chern number).

Since the system is time reversal invariant, I expected that the Chern number should be equal to 0 in every plane of the Brillouin zone. I found some non integer nonsense result, but when I work my code in 2D model, I found it is completely correct. So I’m thinking whether my concept is wrong, whether it is nonsense to use this calculation on 2D plane of a 3D structure.

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  • $\begingroup$ You can calculate the Chern number of a manifold of bands on any 2D surface as long as there is a gap across the entire surface. This is true even if the 2D surface is embedded in 3D. For example, the "charge" of Weyl points in 3D is characterized by the Chern number on a surface enclosing them (typically in the form of a sphere around the Weyl point). $\endgroup$
    – ProfM
    Jan 13 at 8:13

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