Origin of Hubbard sub-bands

Question

I would like to know the origin of the sub-bands in the non degenerate Hubbard model. Most of the texts consider the Hubbard model at half-filling. Is the emergence of the sub-band specific to the half-filling condition or is it also seen for arbitrary filling?

I am aware that the inclusion of the Coulombic potential is responsible for the splitting, but most texts I've come across just state this fact and the observation that the state at half-filling is insulating, and thus must correspond to a fully filled (sub-)band without giving much details on the physics behind the split.

I'm interested in knowing why this split occurs qualitatively. Is there any degeneracy that leads to this splitting, as in the case of conventional band theory?

Answer 1

At half filling we can on average populate each lattice site with one electron. If we have very large on interactions $U\gg J$ (e.g. stemming from coulomb interactions) it becomes preferable for the system to put one electron at each of the lattice site. This is known as a Mott insulator. This is because for every doubly occupied site you will have to pay interaction energy $U$.

The second band therefore represents states where some of the sites are doubly occupied and the band gap will be determined by both $U$ and $J$.

If you don't have half filling you will still see the both bands however, either one of them will be not completely filled and therefore you will find that you have a metal and not an insulating state.