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When talking about topological insulator and talking about bulk-edge correspondence, it seems to be widely accepted conclusion that the band Chern number (winding number) is equal to, when the boundary becomes open, the amount of edge states. But why? For example, SSH model or Graphene (treat $k_x$ as parameter of a 1D chain and $k_y$ as the wave vector along this 1D chain). When the 1D chain is closed, in the topologically non-trivial phase, Chern number can be calculated to be, say, one. And after cutting the 1D chain open, there is one pair of gapless state living on the two edges. Why is Chern number equal to the amount of gapless edge state pairs? Is there any proof of that?

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Yes, there is a proof of that. The first one appeared by a beautiful paper of Hatsugai in 1993 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.71.3697 for a particular special classes of models of the IQHE. More general proofs ensued, but it does turn out that the proofs rely on some non-trivial math, the most simple way to present it seems to be rooted in the context of some basic facts of complex analysis, as for example is presented in this paper: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.125109

There you see for example that essentially the Cauchy integral formula lies at the heart of the bulk-boundary correspondence. More general proofs rely on more sophisticated math, e.g., K-theory or Fredholm theory.

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