Hubbard model of strong correlations

A state-of-the-art review is given of the investigations of the Hubbard model, which is the main model used in the theory of strongly correlated electron systems. It is shown how the main correlation effects — the metal—insulator phase transition, the appearance of localised magnetic moments, and the breakdown of the Fermi-liquid behaviour — appear in the case of strong electron correlations when $U\sim W$, where $U$ is the Coulomb repulsion parameter and $W$ is the band width. Different approaches to the problem, not based on perturbation theory in terms of the parameter $UW$ or $WU$, are presented. A new method in the theory of strongly correlated systems, in which the limit of infinite number of dimensions of space, $d=\infty$, is the main feature of the review. The physical results obtained by this approach are compared with those deduced by traditional methods. An overview of the behaviour of strongly correlated systems is given for wide ranges of the model parameters $U$ and $W$, and of the electron density $n$.