Confined Dirac Fermions in a Constant Magnetic Field

TL;DR

This paper provides an exact solution to the Dirac equation in (2+1) dimensions with a magnetic field and oscillator potential, revealing energy spectra and degeneracies relevant to relativistic quantum Hall systems.

Contribution

It introduces an exact analytical solution for Dirac fermions in combined magnetic and oscillator potentials, highlighting spectral properties and degeneracies.

Findings

Energy spectrum split into positive and negative energies

Infinite degeneracy for negative azimuthal quantum number k

Relevance to relativistic quantum Hall effect

Abstract

We obtain an exact solution of the Dirac equation in (2+1)-dimensions in the presence of a constant magnetic field normal to the plane together with a two-dimensional Dirac-oscillator potential coupling. The solution space consists of a positive and negative energy solution, each of which splits into two disconnected subspaces depending on the sign of an azimuthal quantum number, k = 0, \pm 1, \pm 2,... and whether the cyclotron frequency is larger or smaller than the oscillator frequency. The spinor wavefunction is written in terms of the associated Laguerre polynomials. For negative k, the relativistic energy spectrum is infinitely degenerate due to the fact that it is independent of k. We compare our results with already published work and point out the relevance of these findings to a systematic formulation of the relativistic quantum Hall effect in a confining potential.