Band structure, Brillouin zone, and condensation of states for an itinerant electron in a magnetic quantum dot

The model of a hopping electron on a two-dimensional lattice placed in a quantised magnetic field is considered. The symmetry of the system is described by the magnetic translation group. Its maximal abelian subgroup defines magnetic Brillouin zone while irreducible representations determine an energy band structure. It turns out, due to an equivalence relation between irreducible representations, that for the characterization of the energy band it is sufficient to use only some quasimomenta from magnetic Brillouin zone. These quasimomenta constitute the magnetic band Brillouin zone.