Recently, topological concepts are popular in solid state physics, and berry connection and berry curvature are introduced in band theory. The integration of berry curvature, i.e. chern number, is quantized because Brillouin zone is a torus.(adsbygoogle = window.adsbygoogle || []).push({});

However, I cannot justify the argument that Brillouin zone is a torus. Here is my analysis:

1. I admit that ##\psi_k## and ##\psi_{k+G}## can be identified. However, since ##\psi_k(x+R) = e^{ikR}\psi_k(x)##, ##\psi_k##s do not live in the same Hilbert space because they fulfill different boundary conditions. (##R## is lattice vector; ##G## is reciprocal lattice vector)

2. ##u_k##s, whose relation with ##\psi_k## is ##\psi_k(x)=e^{ikx} u_k(x)##, do live in the same Hilbert place because they are periodic with respect to lattice vector. However, since ##\psi_k = \psi_{k+G}##, ##u_k(x)=e^{ikG}u_{k+G}(x)##, which cannot be identified.

So, how is Brillouin zone a torus?

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# A Why is brillouin zone a torus?

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