Phy5670/HubbardModel 2

Introduction

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 $\mathbf {\Lambda }$, $\sigma =\uparrow ,\downarrow$ denotes the spin, and $a_{\mathbf {i} ,\sigma }^{\dagger },a_{\mathbf {j} ,\sigma }$ are the electrons creation and annihilation operators, with $n_{\mathbf {i} ,\sigma }=a_{\mathbf {i} ,\sigma }^{\dagger }a_{\mathbf {i} ,\sigma }$. Hamiltonian (1.1.1) is expected to be capable of describing the main collective features of above materials, namely itinerant magnetism and metal-insulator (Mott) transition. Indeed, for $U=0$, $H$ reduces to a system of non-interacting moving electrons, while for $t=0$ (atomic limit) the electrons are fully localized, and at half-filling the ground state contains exactly one electron per site, i.e. the system is insulating. The later feature still holds for finite $t$ and $U=\infty$, and the corresponding system has been shown to be an anti-ferromagnetic insulator. A first question is then for which correlation strength one has the Mott transition, and under which conditions the system exhibits (anti-) ferromagnetic long-range order. Besides, one would like to know how these features depend on temperature, as well as on the filling band. The latter can be controlled, as usual, through the chemical potential $\mu$, by adding to $H$ a term $-\mu \sum _{\mathbf {i} ,\sigma }n_{\mathbf {i} ,\sigma }$, and fixing then $\mu$ so that the expectation value of the electron number operator per site is equal to $2\delta$. At half-filling and $T=0$, $\mu =U/2$. Notice that the case $\delta >{\frac {1}{2}}$ can be discussed similarly to the case $\delta <{\frac {1}{2}}$ by considering holes instead of electrons, and changing the sign of $t$. A considerable amount of work has been devoted to the solution of the Hubbard model since its introduction in physics. Nevertheless, exact results are still very rare, and there validity is mainly confined to the one-dimensional case, in which the metal-insulator transition is absent at any $T\neq 0$, according to the Mermin-Wagner theorem on the absence of long-range order in one- (and two-) dimensional systems. Even at $T=0$ the exact solution exhibits no metal-insulator transition for any $U>0$ and $\delta$, and the magnetic properties of the model turn out to be peculiar of the dimension, in particular that of having always a total average spin equal to zero. Many diverse approaches have been proposed in order to gain information about the behavior of such an oversimplified system. See section 1.2 for details.