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

Abstract

We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the x−y plane, with parameters vx=vy, 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 kz=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→∞, we find energies En∝n3/2, which means the nth step as a function of the chemical potential roughly occurs at a value μ∝n3/2. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case vy=λvx, with λ≠1. First, we consider a perturbative treatment around λ≈1 and find that energies En∝n3/2 still hold as n→∞. 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. Published by the American Physical Society 2024