The Hubbard model plays a special role in condensed matter theory as it is considered to be the simplest Hamiltonian model one can write in order to describe anomalous physical properties of some class of real materials. Unfortunately, this model is not exactly solved except in some limits and therefore one should resort to analytical methods, like the Equations of Motion Approach, or to numerical techniques in order to attain a description of its relevant features in the whole range of physical parameters (interaction, filling and temperature). In this paper, the Composite Operator Method, which exploits the above mentioned analytical technique, is presented and systematically applied in order to get information about the behaviour of all relevant properties of the model (local, thermodynamic, single-and two-particle properties) in comparison with many other analytical techniques, the above cited known limits and numerical simulations. Within this approach, the Hubbard model is also shown to be capable of describing some anomalous behaviour of cuprate superconductors.

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Theories and models of many-electron systems. PACS. 71.10.Fd -Lattice fermion models (Hubbard model, etc.). PACS. 71.27.+a -Strongly correlated electron systems; heavy fermions.Abstract. -In this paper, we show that a system of localized particles, satisfying the Fermi statistics and subject to finite-range interactions, can be exactly solved in any dimension. In fact, in this case it is always possible to find a finite closed set of eigenoperators of the Hamiltonian. Then, the hierarchy of the equations of motion for the Green's functions eventually closes and exact expressions for them are obtained in terms of a finite number of parameters. For example, the method is applied to the two-state model (equivalent to the spin-1/2 Ising model) and to the three-state model (equivalent to the extended Hubbard model in the ionic limit or to the spin-1 Ising model). The models are exactly solved for any dimension d of the lattice. The parameters are self-consistently determined in the case of d = 1.c EDP Sciences

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