In contrast, the Hubbard model applies to fermionic particles such as electrons, rather than bosons.

The Hubbard model is an approximate theory that can include these interactions.

Since then the method has been extended to other models in one dimension: Bose gas, Hubbard model, etc.

The Hubbard model is in general intractable under usual perturbation expansion techniques.

In particular, they are used to engineer one, two and three dimensional lattices for a Hubbard model with pre-specified parameters.

However there is no exact solution of the Hubbard model in more than one dimension.

If interactions between particles on different sites of the lattice are included, the model is often referred to as the 'extended Hubbard model'.

For electrons in a solid, the Hubbard model can be considered as an improvement on the tight-binding model, which includes only the hopping term.

The Hubbard model is based on the tight-binding approximation from solid state physics.

Expressed in terms of the Hubbard model, on the other hand, the Hamiltonian is now made up of two components.