The question is not so much about the theorem, but more about what it means in this context: see this link.

So yes, because of Bloch's theorem the Hamiltonian eigenstates in a crystalline system can be written as
\begin{align}
\psi_{n,\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{r}}u_{n,\vec{k}}(\vec{r}),
\end{align}
and so the Berry connection can be defined:
\begin{align}
A_{n}(\vec{k})=i\langle n(\vec{k})|\nabla_{\vec{k}}|n(\vec{k})\rangle,
\end{align}
but what in the world is $|n(\vec{k})\rangle$?

I've read a few articles on topological insulators and they always seem to start off with the Bloch wavefunction $e^{i\vec{k}\cdot\vec{r}} u_k(\vec{r})$, and then somehow they magically get the ket $|u(\vec{k})\rangle$ from which the Berry connection is defined... is $|u(\vec{k})\rangle$ the column vector comprised of the Fourier coefficients of $u_\vec{k}(\vec{r})$ w.r.t. $e^{i\vec{G}\cdot\vec{r}}$ or what?

This post imported from StackExchange Physics at 2014-03-24 04:14 (UCT), posted by SE-user nervxxx