  We study density wave instabilities in a doubly-degenerate Fermi-Fermi
mixture with $SU(2)\times SU(2)$ symmetry on a square lattice. For sufficiently
large on-site inter-species repulsion, when the two species of fermions are
both at half-filling, two conventional ($s$-wave) number density waves are
formed with a $\pi$-phase difference between them to minimize the inter-species
repulsion. Upon moving one species away from half-filling, an unconventional
density wave with $d_{xy}$-wave symmetry emerges. When both species are away
from the vicinity of half-filling, superconducting instabilities dominate. We
present results of a functional renormalization-group calculation that maps out
the phase diagram at weak couplings. Also, we provide a simple explanation for
the emergence of the $d_{xy}$-density wave phase based on a four-patch model.
We find a robust and general mechanism for $d_{xy}$-density-wave formation that
is related to the shape and size of the Fermi surfaces. The density imbalance
between the two species of fermions in the vicinity of half-filling leads to
phase-space discrepancy for different inter-species Umklapp couplings. Using a
phase space argument for leading corrections in the one-loop renormalization
group approach to fermions, we show that the phase-space discrepancy in our
system causes opposite flows for the two leading intra-species Umklapp
couplings and that this triggers the $d_{xy}$-density-wave instability.
