Chapter 5 - The Hubbard model

Abstract

This chapter focuses on the simple Hubbard model. This model describes the s-like narrow energy band; s-like means the band that can accommodate up to two electrons and the general model for interacting electrons moving in crystals. The basic approximation introduced by Hubbard in the Hamiltonian ignores in the potential energy, the electron–electron interaction on different lattice sites, which is supposed to be much smaller than the on-site Coulomb repulsion. Experimental data support this assumption. Usually, one has U = 3–5 eV. Sometimes, when the band is quite narrow, which makes screening more difficult, the on-site Coulomb repulsion is as much as 10 eV, while Uij for i≠j is of the order of the fraction of 1 eV that is much less than U. After ignoring all inter-site Coulomb interactions and taking into account the Pauli exclusion principle that holds for electrons on the same lattice site, the first term is the kinetic energy of electrons, K, in the tight-binding approximation and the second term is the potential energy, V. The electron hopping is controlled by the tij –hopping integral between the ith and jth lattice site.