  We investigate a spinless Fermi gas trapped in a honeycomb optical lattice
with attractive nearest-neighbor interactions. At zero temperature, mean-field
theory predicts three quantum phase transitions, two being topological. At low
interactions, the system is semi-metallic. Increasing the interaction further,
the semi-metal destabilizes into a fully gapped superfluid. At larger
interactions, a topological transition occurs and this superfluid phase becomes
gapless, with Dirac-like dispersion relations. Finally, increasing again the
interaction, a second topological transition occurs and the gapless superfluid
is replaced by a different fully gapped superfluid phase. We analyze these
different quantum phases as the temperature and the lattice filling are varied.
