Energy bands for finite two-dimensional systems in a quantised magnetic field: the symmetry of the model

Abstract

Analysis of properties of an electron (electrons) in a periodic two- dimensional potential and quantised magnetic field is presented. The key role in this description is played by the symmetry of the model determined by the magnetic trans- lation group, together with the symmetric and unitary group according to the duality of Weyl. The magnetic translation group is described in detail along with its irre- ducible representations which form the Brillouin zone in magnetic field. Together with the other two mentioned groups, it allows to characterize the system provid- ing some good quantum numbers. These groups facilitate determination of the band structure through the diagonalisation of the eigenvalue problem in the base adapted to the considered symmetry. The difference between the Brillouin zone with and without magneticfieldispointedout.Discussionisconcentrated onthefinitetwo-dimensional systems, closed by use of Born-von Karman boundary conditions.