Excitonic insulator and the extended Falicov-Kimball model away from half-filling

We consider an extended spinless Falicov-Kimball model at an arbitrary doping level, focusing on the range of parameter values where a uniform excitonic insulator is stabilized at half-filling. In the two-dimensional case, we compare the properties of possible uniform phases and construct the Hartree--Fock phase diagrams, which include sizable phase separation regions. It is seen that the excitonic insulator can appear as a component phase in a mixed-phase state in a broad interval of doping levels. In addition, in a certain range of parameter values the excitonic metal (doped excitonic insulator) is identified as the lowest-energy uniform phase. We suggest that this phase, which is unstable with respect to phase separation, may be stabilized when the phase separation is suppressed by the long-range Coulomb interaction. Overall, we find that excitonic correlations can affect the behavior of the system relatively far away from half-filling.