  We study systems of fully polarized ultracold atomic gases obeying Fermi
statistics. The atomic transition interacts dispersively with a mode of a
standing-wave cavity, which is coherently pumped by a laser. In this setup, the
intensity of the intracavity field is determined by the refractive index of the
atomic medium, and thus by the atomic density distribution. Vice versa, the
density distribution of the atom is determined by the cavity field potential,
whose depth is proportional to the intracavity field amplitude. In this work we
show that this nonlinearity leads to an instability in the intracavity
intensity that differs substantially from dispersive optical bistability, as
this effect is already present in the regime, where the atomic dipole is
proportional to the cavity field. Such instability is driven by the matter
waves fluctuations and exhibits a peculiar dependence on the fluctuations in
the atomic density distribution.
