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Band structures from *A Short Course on Topological Insulators*

I'd like to know what the band structure (or gap) of a SSH topological insulator means. In particular, what does the K mean in the x-axis? I've seen different points from the Brillouin Zone from other topological insulators, but why does a SSH topological insulator only have K? Thanks.

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    $\begingroup$ I'm not an expert, but I'm guessing it is because it is a one-dimensional structure. Other points in the Brillouin Zone define symmetries in a 3D crystal. $\endgroup$ Nov 15, 2019 at 17:05
  • $\begingroup$ Do you mean the $k$, indicated in (a) as the wavenumber? $\endgroup$
    – Jon Custer
    Nov 15, 2019 at 17:45
  • $\begingroup$ For non-experts like me who want to learn more about this, the figure shown in the OP appears to be from "A short course on topological insulators" (arxiv.org/abs/1509.02295) $\endgroup$ Nov 16, 2019 at 1:46

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In this case, k is the crystal momentum which is a good quantum number that labels the eigenstates of the Hamiltonian in the independent-particle approximation. You may be confusing it with K, the symmetry point in the band structure of higher-dimensional lattices. For those lattices its more complicated to plot the energy as a function of the crystal momentum and so one chooses special paths in the first Brillouin zone that go along high-symmetry points to sample the band structure of the material. In this case, because its only a one dimensional system, the dispersion relation only depends upon a single number (instead of three for a 3D lattice) so one can plot the dispersion relation as a function of crystal momentum without the need of sampling special points in the first Brillouin zone.

For this topological insulator, the closing of the gap at the zone boundary is a hallmark of the topological phase transition. In changing the parameters of the Hamiltonian (v and w in the plot), you can eventually close the gap which will generate zero energy modes that are localized at the edges of the SSH chain.

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