PHYSICA ELSEVIER Physica A 232 (1996) 616-624 Flux phase problem in the 2-D Falicov-Kimball model Ch. Gruber*, D. Ueltschi Institut de Physique Thkorique, EPFL, CH-IO15 Lausanne, Switzerland Abstract A lattice system of electrons in the presence of an external potential and under the influence of a magnetic field is considered. The energy of the electrons is given by the Falicov-Kimball hamiltonian, where the ions give the external potential. The problem is to find the configurations of ions and the magnetic fluxes which minimize the energy of the electrons at zero temperature. Our results, valid for large coupling constant, show that periodic configurations with periodic fluxes (possibly inhomogeneous) appear for some values of the chemical potentials. Keywords: Falicov-Kimball model; Flux phase problem 1. Introduction In 1969 Falicov and Kimball [1] considered experiments in rare-earth materials and proposed a model involving interactions between d- and f-band electrons. In their approximations, the d-electrons are localized and therefore considered as classical par- ticles, while the f-electrons are itinerant quantum particles. The modelization is done as follows: - the system is defined on a lattice, in this paper a two-dimensional square lattice; - the quantum particles are spinless fermions with kinetic energy represented by nearest-neighbour hopping; - the classical particles have hard-core repulsion; - the interaction is an on-site repulsion between the two kinds of particles. In the following, we consider a square lattice A; a configuration of ions is defined by s = {Sx}x~a, Sx = ±1 Vx E A, where Sx = +1 means that the site x is occupied by one ion, and Sx = -1 means that the site x is empty. For a given configuration s the * Corresponding author. E-mail: Gruber@eldpb.epfl,ch. 0378-4371/96/$15.00 Copyright (~ 1996 Elsevier Science B.V. All rights reserved PH S0378-4371 (96)00171-9