The Hubbard model within the equations of motion approach

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.