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

We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the $x\text{−}y$ 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 of $k_z=0$) has a degeneracy that is 3 times that of other states. This degeneracy manifests in the Hall conductivity as a step at a zero chemical potential 3/2 the size of other steps. Moreover, as $n\to \infty$, we find energies $E_n\propto n^{3/2}$, which means the $n\mathrm{th}$ step as a function of the chemical potential roughly occurs at a value $\mu \propto n^{3/2}$. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case $v_y=\lambda v_x$, with $\lambda \ne 1$. First, we consider a perturbative treatment around $\lambda \approx 1$ and find that energies $E_n\propto n^{3/2}$ still hold as $n\to \infty$. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semiclassical 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 semiclassical formula for the energy, which has the same form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semimetals should leave similar experimental imprints.