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I am a beginner of studying topological insulator. I want to ask some general question in this area to clarify my understanding. May be I am asking wrong, hope you can point me out.

If certain material is topological non trivial, we are saying that it has non-zero topological index and it can be shown by direct computation. Also it correspond to the number of edge state. But when we watch the band diagram of certain material or system, how can we immediately say that it has topological edge state if there are some band connect the valence band to conduction band? To be more precise, How can we sure that the one crossing the Fermi level is contributed by the edge state?

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  • $\begingroup$ AFAIK the Fermi-level-crossing state is drawn as diagonal, breaking $k$-symmetry. As long as the material is isotropic, one would need an edge to explain corresponding symmetry breaking. $\endgroup$
    – dominecf
    Dec 30, 2020 at 8:54

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