Kohmoto from TKNN(Thouless-Kohmoto-Nightingale-deNijs) who described the topology of the integer quantum hall effect always stressed the importance of the 2D Brillouin zone being a donut due to periodic boundary conditions.
--> http://www.sciencedirect.com/science/article/pii/0003491685901484
Now I don't really see why this is relevant. Shouldn't the zeros of the wavefunction always lead to a Berry phase?. What would happen if we have a sphere instead of a donut? I think I am missing a major point here, because I can't see how the Gauss-Bonnet theorem that connects topology to geometry plays a role. For a sphere with no holes the Gaussian curvature gives us 4$\pi$, for a donut if gives us 0. In both cases Stokes' theorem should still give us a nonzero value?
Charles Kane then uses this argument to compare a donut with the quantum hall state and a sphere with an insulator. He then writes down the Gauss-Bonnet theorem and immediately talks about topological insulators, and again I don't see the connection or is it just an analogy and I shouldn't waste any time on this?
If there is a connection, I would like to know if there is a simple explanation for using the Gauss-Bonnet theorem in the context of topological insulators. I'm even more confused because Xiao-Gang-Wen said in a recent post here, that a topological insulator is NOT due to topology but due to symmetries...
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