What is invalidated when turning on many body interactions in a crystal? I have just started to think about strongly interacting particles and Fermi liquid theory, and I have two questions.
For non interacting particles moving in an potential field, we know that the eigenstate evolves in time with a simple phase. Now, if we suppose that we can exactly solve the $N$ interacting body Hamiltonian, would its eigenstate evolve with a simple phase too?
For interacting particles, is Bloch's theorem still valid? In other words, is the band structure description qualitatively correct?
 A: The time-evolution will be more complex. In some cases the collective correlated motion of many particles can be describe by a coherent oscillation similar to the case of a single particle propagation. In this case the quasi-particles appear. For instance, an exciton in a semiconductor material is a correlated motion of electrons and holes, but it can be described similar to the electron in the hydrogen atom. Stationary many-particle interactions in periodic structures can be described in terms of the band structure as well, when the mean-field theory is applied and when the many-body interactions can be treated as a small perturbation. In this case the band structure parameters related to the single-electron problem has to be renormalized. Namely, the exchange interactions make the band gap smaller. Also, exchange-correlation interactions can affect the effective mass.
To be short, the band structure can contain information about many-body interactions at the mean-field theory level.
The band structure itself does not provide full information about the many-electron system, it may be considered as an auxiliary characteristic. For instance, in order to compute something that can be measured, one has also know the distribution function together with the band structure.
