condensed matter - Why is there a band structure for strongly correlated systems?

The existence of band structure of a crystalline solid comes from the Bloch theorem, which relies on the independent-electron approximation. Why do people still talk about the band structure for a strongly-correlated system, e.g. superconductors and topological insulators? In such strongly-correlated systems, shouldn't the independent-electron approximation becomes entirely invalid, rendering the band structure of a solid meaningless?

Thanks for the help!

Answer

Usually, when talking of the "band structure" of such a system one either refers to the non-interacting band structure (which relates to the free Green functions occuring in many methods to handle the interactions, like perturbation expansions or DMFT), or to the sharp features usually visible in the spectral function (which is more or less experimentally accesible via ARPES), these features reduce to $\delta$-like peaks at the bands in the non-interacting case. When considering, for example, simple thermodynamic properties the latter concept of "band structure" can be a reasonable approximation.

A side note: Nearly all the early research on topological insulators (especially when you refer by this term also to systems like Chern insulators and symmetry protected topological phases, the famous BHZ model, for example, is a non-interacting model) was on non-interacting systems. Symmetry protected topological phases do not require interacting electrons.