Falicov-Kimball model and the problem of valence and metal-insulator transitions

Metal-insulator and valence transitions in the spinless Falicov-Kimball model are studied exactly by numerical calculations and perturbation theory. We have found that the model can describe the following valence transitions: (i) Transitions from an integer-valence ground state (the density of the localized f electrons ${\mathrm{n}}_{\mathrm{f}}$=1) into an inhomogeneous intermediate-valence ground state (${\mathrm{n}}_{\mathrm{f}}$≠1). (ii) Transitions from one inhomogeneous intermediate-valence state with ${\mathrm{n}}_{\mathrm{f}}$ into another inhomogeneous intermediate-valence state with ${\mathrm{n}}_{\mathrm{f}}^{\prime }$≠${\mathrm{n}}_{\mathrm{f}}$. (iii) Transitions from an inhomogeneous intermediate-valence state into a homogeneous intermediate-valence state. In the general case these transitions have a staircase structure and for sufficiently strong interactions U they are very sharp.