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Take 2D quantum hall insulator for example. The typical argument goes like this:

We have a Hamiltonian that has translation symmetry in both directions on a infinite lattice, and we assign a integer number(Chern number) to each of its band. The Chern number is a topological invariant(against homotopy, I suppose), hence remains constant if the Hamiltonian is varied continuously and as long as it is well defined.

Attach a the quantum hall insulator(which has non-zero Chern-numbered bands) to a trivial insulator(of which all bands have 0 Chern number), and the band of the former must be concatenated with the latter. However this means the well-definedness of Chern number has to break down somewhere, or else one band could have two different Chern numbers.

The place where the well-definedness breaks down must be a band gap closing, hence we have conducting (edge) states.

My question is, the moment we decide to attach two insulators together, we must have broken the translation symmetry along at least one direction, so Chern number is already not well defined, how can we proceed with the argument that follows after?

I am aware that there exists more serious arguments about bulk-edge relation like the ones by Yasuhiro Hatsugai, but I haven't got around to read it and I will in the near future. But can someone comment on the previous heuristic argument I phrased? Just how hard is it to make it a rigorous argument, if possible at all? Of course if I'm told Hatsugai's paper is exactly formalizing this heuristic argument I'll get to read it as soon as possible.

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Usually people assume that the transition is smooth so that locally the system looks translationally invariant locally. And I believe this type of argument sometimes breaks down if you have a very sharp boundary. (The chiral edge of the QHI is protected by other arguments as well though, so it doesnt really care.) –  BebopButUnsteady Mar 23 at 19:05
    
@BebopButUnsteady: But if the tranlation invariance is merely local, we have no Bloch theorem and hence no torus as the base manifold. Is there a way of fixing this? –  Jia Yiyang Mar 24 at 4:38

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