I am told that Brillouin Zone is a torus, which plays an important role in defining chern number and other topological invariants. The argument is that Bloch wave function ψk is periodical in reciprocal lattice vector: ψk+G=ψk
.
However, chern number (and also other invariants) is defined using the periodic part of Bloch wave function uk (ψk=eikruk), which is not periodic in k: uk+G=e−iGruk
. In many literature, to prove chern number is an integer, uk+G=eiθuk (notice that θ is r independent!) is used, which is not consistent with the equation above.
So, in what sense Brillouin zone is a torus and why chern number is an integer with this understanding?
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