Electronic ferroelectricity in the falicov-kimball model

I show that a spontaneous electric polarization exists in the solution of the Falicov-Kimball model by mapping the strong coupling limit of this Hamiltonian into an xxz spin 1/2 model with a magnetic field. In this way, I determine the phase diagram of the strongly interacting model and show the existence of a transition to a mixed-valence regime containing two phases: an orbitally ordered state and a Bose-Einstein condensation of excitons with a built-in electric polarization.