PHYSICA ELSEVIER Physica A 232 (1996) 648-656 Periodic ground states in simple models of itinerant fermions interacting with classical fields N. Macris * Institut de Physique Th~orique, Ecole Polytechnique F~d~rale de Lausanne, CH-IOI5 Lausanne, Switzerland Abstract 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. 1. Introduction I report here on a simple quantum mechanical lattice model of itinerant electrons interacting with static nuclei, which displays periodic ground states and long range order at low temperatures. The hamiltonian is in second quantised form H = ~ t xvC t xoCyo+UZ ( nx - l ) sx+ f t ~ ( nxT , -- ~)(nx&l _ _~),1 (1.1) x,y,a=T,~ x x where x, y are sites of a finite D-dimensional cubic lattice A, c's are the usual creation and annihilation operators of electrons (fermions with spin one-half, a --T, J.), nx = nx~ + nxT the electron number at site x. The classical variable Sx takes the values +1 if a nucleus is present at x and -1 if not. The kinetic energy matrix txy is hermitian and we allow complex values txy = Itxyl exp( it~xy). (1.2) * E-mail: macris@eldp.epfl.ch. 0378-4371/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S0378-437 1(96)001 74-4