The Hubbard model

A famous model Hamiltonian which represents electron correlation in the tight-binding model for an array of one-electron atoms is

where N_i^↑,↓ are the numbers of spin-up and spin-down electrons, respectively, on the ith atom. The first term is the transfer term that creates the band of width W = 2Zt; the second is the Coulomb energy penalty involved in placing two electrons on the same atom. Electrons are localized when U/W > 1 because there are then no states available to accommodate the double-occupancy charge fluctuations that are indispensable for electronic conduction. Compounds with an integral number of electrons per atom which satisfy this condition are known as Mott–Hubbard insulators.

The second term can be rewritten using

The Stoner interaction −(/4)(N^↑ − N^↓)² is thereby identified as the spindependent part of the on-site Coulomb interaction; hence ≈ U. In the Hubbard model, the on-site correlations create a magnetic moment, and hopping between adjacent nondegenerate singly occupied orbitals provides an antiferromagnetic interaction.
A variant of the Hubbard model is the t– model, where the second term is replaced by