The Hubbard Model

Introduction

What it is
History of it
Utility

The model referred to as the Hubbard model appeared in the literature for the first time in 1963, in two subsequent independent papers -- the first by Gutzwiller, and the second by Hubbard -- as an attempt to describe in a simplified way the effect of correlations for $d$ -electrons in a transition metals, In deed in the latter case the feature that electrons form a (narrow) delocalized $d$ -band is in competition with the quasi-atomic behavior originated from correlations, which would make plausible an atomic description of the problem. The model hamiltonian $H$ consists of two contributions,

H=\sum _{\mathbf {i} ,\mathbf {j} }\sum _{\sigma }t_{\mathbf {i} ,\mathbf {j} }\left(a_{\mathbf {i} ,\sigma }^{\dagger }a_{\mathbf {j} ,\sigma }+h.c.\right)+U\sum _{\mathbf {i} }n_{\mathbf {i} ,\uparrow }n_{\mathbf {i} ,\downarrow }

, (1.1.1)

a kinetic term describing the motion of electrons between neighboring sites (the hopping integral $t_{\mathbf {i} ,\mathbf {j} }$ is usually restricted to nearest-neighbors, and is assumed translationally invariant, namely $t_{\mathbf {i} ,\mathbf {j} }=-t,t>0$ ), and an on-site term, which approximates the interactions among electrons, whose strength is given by the parameter $U$ . $U>0$ corresponds to repulsive Coulomb interaction, whereas $U<0$ could eventually describe an effective attractive interaction mediated by the ions. $\mathbf {i} ,\mathbf {j}$ label the sites of a $D$ -dimensional lattice $\Lambda$ , $\sigma =\uparrow ,\downarrow$ denotes the spin, and $a_{\mathbf {i} ,\sigma }^{\dagger },a_{\mathbf {j} ,\sigma }$ are the electrons creation and annihilation operators, with $n_{\mathbf {i} ,\sigma }=a_{\mathbf {i} ,\sigma }^{\dagger }a_{\mathbf {i} ,\sigma }$ .