  We investigate the phase diagram of dipolar fermions with aligned dipole
moments in a two-dimensional (2D) bilayer. Using a version of the
Singwi-Tosi-Land-Sjolander scheme recently adapted to dipolar fermions in a
single layer [M. M. Parish and F. M. Marchetti, Phys. Rev. Lett. 108, 145304
(2012)], we determine the density-wave instabilities of the bilayer system
within linear response theory. We find that the bilayer geometry can stabilize
the collapse of the 2D dipolar Fermi gas with intralayer attraction to form a
new density wave phase that has an orientation perpendicular to the density
wave expected for strong intralayer repulsion. We thus obtain a quantum phase
transition between stripe phases that is driven by the interplay between strong
correlations and the architecture of the low dimensional system.
