Is Bloch theorem suitable for correlated electrons? I have been studying solid state physics recently. For Bloch theorem which states that for crystal with periodic symmetry, we have
$$\psi_{nk}(r)=e^{ikr}u_{nk}(r).$$
According to the textbook, this theorem is deduced under the independent electrons approximation, as the deduction starts from the one electron Schrödinger Equation.
But I wonder if this theorem stills holds for correlated electrons. Because for correlated electrons, we should still have the potential $U(r)$ to obey
$$U(r)=U(r+R),$$ although in the case of correlated electrons $U(r)$ contains the interaction between different electrons. Therefore, in my opinion, Bloch theorem should stills hold for correlated electrons. Can anyone give some comments on my opinion?
 A: Consider a model of two 1D particles in a background periodic potential. When the particles are independent, the potential energy of the system will look like
$$U(x_1,x_2)=f(x_1)+f(x_2).$$
Since we know that the background potential $f$ is periodic, $U$ is then periodic in both $x_1$ and $x_2$. Moreover, it's periodic in a countable infinity of directions, not only in $x_1$ and $x_2$: e.g., it is periodic in $x_1-x_2$ and $x_1+x_2$ directions. But if you consider interaction between the particles, it'll likely depend on $x_1-x_2$, so the potential energy will now look like
$$U(x_1,x_2)=f(x_1)+f(x_2)+Q(x_1-x_2).$$
This interaction term $Q$ now spoils most of the periodicity. The only direction in which this potential energy remains periodic is $x_1+x_2$. This is related to the fact that despite the electrons interact, their center of mass (COM) is still "free". So you can use Bloch theorem, but only for the motion of the COM.
This readily generalizes to the case of $N$ particles and $D$ dimensions. After all you'll get that for interacting identical particles you have only $D$ wavenumbers — those related to motion of the COM, and all the other dimensions of configuration space are not periodic, thus not describable by Bloch's theorem.
A: I want to answer this by myself. Yesterday, I found some materials about the question. http://www.rug.nl/research/portal/files/2861460/c3.pdf
In general, the answer is yes. Bloch theorem still holds if we consider the interaction between electrons.
