Quantum spin systems, considered in previous chapters, describe only spin dynamics of correlated electrons. However, it is interesting and important to also understand charge dynamics of correlated electron systems.

Usually there are two main energetical scales in the behaviour of electrons: the width of the band of itinerant electrons (and related to it characteristic velocity of electrons or the Fermi energy of electrons) and the strength of the Coulomb repulsion between electrons. If the former is much larger than the Coulomb repulsion, then electrons can be considered as a free lattice gas of itinerant electrons with Bloch-like wave functions. The weak interaction between electrons can be treated in the framework of perturbation theories. This kind of theory is well developed. The other limiting case, which is studied even better than the previous situation, is the atomic (localized) behaviour of electrons, where the effect of the Coulomb interaction is considered exactly, and the hopping of electrons between lattice sites can be considered perturbatively. In such a case wave functions of electrons are of Wannier-type rather than Bloch-like. However, the most interesting situation pertains to the case in which the energy of the hopping of electrons from site to site of the crystal lattice (which characteristic energy is the bandwidth of electrons) is of the same order as the strength of the repulsion between electrons. Here correlation effects and itinerant effects interfere with each other, which results in a reach behaviour of such systems: they can reveal metal-insulator phase transitions, heavy fermion...