  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.
