  Motivated by the search for quantum liquid crystal phases in a gas of
ultracold atoms and molecules, we study the density wave and nematic
instabilities of dipolar fermions on the two-dimensional square lattice (in the
$x-y$ plane) with dipoles pointing to the $z$ direction. We determine the phase
diagram using two complimentary methods, the Hatree-Fock mean field theory and
the linear response analysis of compressibility. Both give consistent results.
In addition to the staggered ($\pi$, $\pi$) density wave, over a finite range
of densities and hopping parameters, the ground state of the system first
becomes nematic and then smectic, when the dipolar interaction strength is
increased. Both phases are characterized by the same broken four-fold (C$_4$)
rotational symmetry. The difference is that the nematic phase has a closed
Fermi surface but the smectic does not. The transition from the nematic to the
smectic phase is associated with a jump in the nematic order parameter. This
jump is closely related to the van Hove singularities. We derive the kinetic
equation for collective excitations in the normal isotropic phase and find that
the zero sound mode is strongly Landau damped and thus is not a well defined
excitation. Experimental implications of our results are discussed.
