  The birefringent fermions possess a spectrum with two distinct Fermi
velocities. Here based on the lattice model, we use the mean-field method to
investigate the interaction-induced phase transitions of the birefringent
fermions. We consider both the short-range nearest-neighbor (NN) and
next-nearest-neighbor (NNN) repulsive interactions and calculate the phase
diagrams under different conditions. We find that the NN interactions can
induce the asymmetric charge density wave order, while the NNN interactions can
drive the asymmetric quantum anomalous Hall order (AQAH) in the large limit of
anisotropy in the hopping integrals, \beta\rightarrow 1. The AQAH order is
characterized by the unequal loop currents connecting the NNN sites and by the
appearance of a gap between the two conduction (valence) bands. Such asymmetric
fermionic orders can be attributed to the specific lattice geometry of the
birefringent fermions. The implications of our results for experiments are also
discussed.
