  We consider the possibility of topological quantum phase transitions of
ultracold fermions in optical lattices, which can be studied as a function of
interaction strength or atomic filling factor (density). The phase transitions
are connected to the topology of the elementary excitation spectrum, and occur
only for non-zero angular momentum pairing (p-wave, d-wave and f-wave), while
they are absent for s-wave. We construct phase diagrams for the specific
example of highly anisotropic optical lattices, where the proposed topological
phase transitions are most pronounced. To characterize the existence of these
topological transitions, we calculate several measurable quantities including
momentum distribution, quasi-particle excitation spectrum, atomic
compressibility, superfluid density, and sound velocities.
