Periodic ground states in simple models of itinerant fermions interacting with classical fields

A model of itinerant lattice electrons interacting with classical nuclei is studied. The electrons can interact between themselves via a Hubbard on-site term. It is shown that, in the ground state of the half-filled band, the nuclei form a periodic crystal. More precisely for a cubic lattice they form a chessboard state under some appropriate condition on the hopping matrix. This generalizes known results previously obtained for the Falicov-Kimball model. The case where fermions are replaced by hard-core bosons in this later model is also discussed. More generally models of fermions interacting with classical scalar or vector fields are briefly considered. The mathematical technique used is “reflection positivity” adapted to the case of Fermi statistics.