  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.
