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I'm reading this article about twisted TMD homobilayers (https://arxiv.org/abs/1807.03311) and there are certain topological properties that I don't understand:

On page 3, in the paragraph next to Fig 3 they said "Spin-valley locking implies that when the chemical potential is in the gap between the two topmost bands, the twisted homobilayer is not only a valley Hall insulator but also a quantum spin Hall insulator, i.e, a topological insulator". I don't get why they have this conclusion and the relation with the Chern numbers of the first, second band and the Fermi level. Up until now I only know about the TKNN Invariants which connect the Hall conductivity and the sum of bands' Chern number below the Fermi level, however a band with non-zero Chern number above the Fermi level has no physical sense to me.

Thank you in advance.

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  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Jun 16, 2022 at 23:33
  • $\begingroup$ Welcome to SE. Please keep it to one question per post, since this is a Q&A site. You could, however, include in each question a link to the others. $\endgroup$
    – Miyase
    Jun 16, 2022 at 23:34
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    $\begingroup$ I think it'd be useful for you to read up a bit on topological insulators. As a start, consider having a look at How is the topological Z2 invariant related to the Chern number? (e.g. for a topological insulator). In addition, $\endgroup$
    – Anyon
    Jun 18, 2022 at 17:40

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