- #1

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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?