SOME RIGOROUS RESULTS ON THE GROUND-STATES OF THE FALICOV-KIMBALL MODEL

The Falicov-Kimball model consists of nuclei which are not allowed to hop and spinless electrons which can hop between nearest neighbor sites. There is an on-site interaction between electrons and nuclei. We consider the model in two dimensions with a large attractive potential. For the neutral model with densities between 1/4 and 1/2 we prove that the configuration of the nuclei in the ground state must consist of parallel lines of lattice sites which are either completely occupied by nuclei or completely free of nuclei. (The angle of the lines with respect to the lattice depends on the density. Some mild assumptions on the ground state are needed for this result.) For densities 1/3, 1/4 and 1/5 we prove that the ground state configuration of the nuclei is indeed that which had been conjectured [8]. For the nonneutral model we show that if the model is close to neutrality in the sense that both the electron and nuclear densities are close to 1/2, 1/3, 1/4 or 1/5 then the configuration of the nuclei in the ground state is close to the nuclear ground state for the neutral model with density 1/2, 1/3, 1/4 or 1/5.