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To obtain a topological band insulator, we usually start with two bands with either spin up or down. If these bands now get 'inverted', they will cross. When there is coupling of these two bands such that we evidence a gap opening (anti-crossing), this is classified as a topological insulator, and it will present an edge state lying within the band gap.

  1. If we have only one spin (effectively), we would have the QAH (Auantum Anomalous Hall system)—shown in the figure below.
  2. If we have spin degeneracy in our system, we would essentially have the SQHE, which is described, for instance, by the BHZ model. enter image description here

My questions now regards having two crossing bands with opposite spins (see below). If after the addition of SOC, they present an anti-crossing (shown below), would this characterize a topological material?

I was taking a look at some references, and it seems this situation is described in the following paper: https://www.science.org/doi/epdf/10.1126/science.1187485

So my first guess would be that this also presents a non-trivial topology. Thoughts? Thanks!

enter image description here

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    $\begingroup$ I only stumbled upon this while looking for something else. So please pardon that I won't give a more elaborate answer. But I think what you describe is how graphene with p- and d-orbitals becomes a spin-Hall insulator (not the Haldane or Kane-Mele model, but an LCAO model with SOC between the d-orbitals). $\endgroup$
    – Fred
    Commented Feb 4, 2023 at 14:16
  • $\begingroup$ Could you provide any reference for that? $\endgroup$
    – sined
    Commented Feb 5, 2023 at 15:35
  • $\begingroup$ On the measurement: journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.046403 and here's a thesis that covers the band gap: ediss.sub.uni-hamburg.de/handle/ediss/9221 There's definitely more sources. In particular, there's probably shorter sources. But I hope these will help you. $\endgroup$
    – Fred
    Commented Feb 6, 2023 at 14:10

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