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There are similar topological invariants for band structures in one dimension, but an important difference is that these invariants always require some symmetry in the band Hamiltonians, for example particle-hole symmetry. In such cases, typically the invariant is given by $$ C=\int\frac{dk}{2\pi} a_k \text{ mod }1 $$


I am new to this and just happened to have similar questions. Here are some of my thoughts -- not sure they are correct and I would certainly welcome some discussion. I think it is important to remember what is drawn in your attached figure are Wannier orbitals rather than band states. To answer your first question -- it is not shown. The figure you ...

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