  The two-dimensional Falicov-Kimball model in the presence of a perpendicular
magnetic field is investigated by the dynamical mean-field theory. Within the
model the interplay between electron correlations and the fine electron
structure due to the magnetic field is essentially emerged. Without electron
correlations the magnetic field induces the electron structure to the so-called
Hofstadter butterfly. It is found that when electron correlations drives the
metal-insulator transition, they simultaneously smear out the fine structure of
the Hofstadter butterfly. In a long-range ordered phase, the electron
correlation induced gap preserves the fine structure, but it separates the
Hofstadter butterfly into two wings.
