Ground-state analysis of the Falicov-Kimball model on complete graphs

The ground-state nature of the Falicov-Kimball model of mobile electrons an d fixed nuclei on complete graphs is investigated. We give a pedagogic deri vation of the eigenvalue problem and present a complete account of the grou nd-state energy both as a function of the number of electrons and nuclei an d as a function of the total number of particles for any value of interacti on U is an element of R. We also study the energy gap and show the existenc e of a phase transition characterized by the absence of gap at the half-fil led band for U < 0. The model in consideration was proposed and partially s olved by Farkasovsky for finite graphs and repulsive on-site interaction U < 0. Contrary to his proposal, we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this mode l on bipartite complete graphs and examine how sharp the Kennedy-Lieb varia tional estimate is as compared with the exact ground state.