Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic Cubic Dirac Semi-Metal and the peculiar case of star-shaped classical orbits

TL;DR

This paper investigates the Landau levels and Hall conductivity in anisotropic cubic Dirac semi-metals, revealing unique degeneracies, energy scaling, and star-shaped classical orbits that could serve as experimental signatures.

Contribution

It provides an exact solution for Landau levels in isotropic cases, analyzes anisotropic effects perturbatively and semi-classically, and uncovers star-shaped orbits indicating weakly localized states.

Findings

Zero energy level degeneracy is three times higher in isotropic case.

Landau level energies scale as n^{3/2} for large n.

Star-shaped classical orbits emerge in the anisotropic case, indicating weak localization.

Abstract

We study an anisotropic cubic Dirac semi-metal subjected to a constant magnetic field. In the case of an isotropic dispersion in the $x$-$y$ plane, with parameters $v_{x}=v_{y}$, it is possible to find exact Landau levels, indexed by the quantum number $n$, using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero energy state in the case $k_z=0$) has a degeneracy that is three times that of other states. This degeneracy manifests in the Hall conductivity as a step at zero chemical potential that is 3/2 the size of other steps. Moreover, as $n\to\infty$ we find energies $E_n\propto n^{3/2}$, which means the $n$-th step as a function of chemical potential roughly occurs at a value $\mu\propto n^{3/2}$. We propose that these exciting features could be used to identify cubic Dirac semi-metals experimentally. Subsequently, we analyze the anisotropic case $v_{y}=\lambda v_{x}$ with $\lambda\neq 1$. First, we consider a perturbative treatment around $\lambda\approx 1$ and find that energies $E_n\propto n^{3/2}$ still holds as $n\to\infty$. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semi-classical treatment that reveals interesting star-shaped orbits in phase space that close at infinity. This property is a manifestation of weakly localized states. Despite being infinite in length, these orbits enclose a finite phase space volume and permit finding a simple semi-classical formula for the energy, which again has the form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semi-metals should leave similar experimental imprints.