The Falicov-Kimball model in external magnetic field: orbital effects

We study thermodynamic properties of the two-dimensional (2D) Falicov-Kimball model in the presence of external magnetic field perpendicular to the lattice. The field is taken into account by the Peierls substitution in the hopping term. In the non-interacting case the field dependent energy spectrum forms the famous Hofstadter butterfly. Our results indicate that for arbitrary nonzero interaction strength and arbitrary magnetic field there is a gap in the energy spectrum at sufficiently low temperature. The gap vanishes with increase of temperature for weak coupling, however, it persists at high temperatures if the coupling is strong enough. Numerical results have been obtained with the help of Monte Carlo technique based on a modified Metropolis algorithm.