Instabilities of a birefringent semi-metal

Birefringent fermions arise as massless fermionic low energy excitations of a particular tight binding model for spinless fermions on a square lattice which have two "speeds of light" [M. P. Kennett, ${\it et \, al.}$, Phys. Rev. A ${\bf 83}$, 053636 (2011)]. We use mean field theory to study phases that can arise when there are nearest neighbour and next-nearest neighbour repulsive interactions in this model and demonstrate robustness of the birefringent semi-metal phase in the presence of weak interactions and identify transitions to staggered density and quantum anomalous Hall ordered phases. We consider the effect of coupling birefringent fermions to a magnetic field, and find analytic expressions for the corresponding Landau levels and demonstrate that their integer Quantum Hall effect displays additional plateaux beyond those observed for regular Dirac fermions, such as in graphene. We briefly discuss a tight-binding construction that leads to three dimensional birefringent fermions.