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

TL;DR

This paper develops an algebraic framework for describing two-dimensional charged particles in a magnetic field and periodic potential, emphasizing magnetic translation operators and their representations, with applications to many-particle systems like the fractional quantum Hall effect.

Contribution

It introduces the concept of magnetic cells and uses irreducible projective representations to analyze multi-particle systems in magnetic fields, extending previous single-particle approaches.

Findings

Magnetic translation operators require all inequivalent irreducible projective representations.

The concept of magnetic cells explains the periodicity of physical properties.

Direct product representations describe multi-particle systems, relevant to phenomena like the fractional quantum Hall effect.

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.