A semi-classical analysis of Dirac fermions in 2+1 dimensions

TL;DR

This paper develops a semiclassical framework for analyzing massless Dirac fermions in 2+1 dimensions, comparing classical trajectories and spectra with quantum results, and demonstrating good qualitative agreement.

Contribution

It introduces a novel semiclassical approach for Dirac fermions in 2+1D, generalizing spin matrices and analyzing the classical limit with applications to magnetic fields.

Findings

Semiclassical spectrum matches exact quantum spectrum for free particles.

Qualitative features of quantum spectra are reproduced semiclassically in magnetic fields.

Quantitative agreement is reasonable for spin-1/2 particles.

Abstract

We investigate the semiclassical dynamics of massless Dirac fermions in 2+1 dimensions in the presence of external electromagnetic fields. By generalizing the $\alpha$ matrices to the spin-$S$ matrices and doing a certain scaling, we formulate a $S\rightarrow\infty$ limit where the spin and the orbital degrees become classical. We solve for the classical trajectories for a free particle on a cylinder and a particle in a constant magnetic field. We compare the semiclassical spectrum, obtained by Bohr-Sommerfeld quantization with the exact quantum spectrum for low values of. For the free particle, the semiclassical spectrum is exact. For the particle in a constant magnetic field, the semiclassical spectrum reproduces all the qualitative features of the exact quantum spectrum at all $S$. The quantitative fit for $S=1/2$ is reasonably good.