Algebraic description of a two-dimensional system of charged particles in an external magnetic field and periodic potential

Abstract

Properties of the magnetic translation operators for a charged particle moving in a crystalline potential and a uniform magnetic field show that it is necessary to consider all inequivalent irreducible projective representations of the the crystal lattice translation group. These considerations lead to the concept of magnetic cells and indicate the periodicity of physical properties with respect to the charge. It is also proven that a direct product of such representations describe a system of two (many, in general) particles. Therefore, they can be applied in description of interacting electrons in a magnetic field, for example in the fractional quantum Hall effect.