We study a rapidly rotating gas of unpolarized spin-1/2 ultracold fermions in
the two-dimensional regime when all atoms reside in the lowest Landau level.
Due to the presence of the spin degree of freedom both s-wave and p-wave
interactions are allowed at ultralow temperatures. We investigate the phase
diagram of this system as a function of the filling factor in the lowest Landau
level and in terms of the ratio between s- and p-wave interaction strengths. We
show that the presence of attractive interactions induces a wide regime of
phase separation with formation of maximally compact droplets that are either
fully polarized or composed of spin-singlets. In the regime with no phase
separation, we give evidence for fractional quantum Hall states. Most notably,
we find two distinct singlet states at the filling nu =2/3 for different
interactions. One of these states is accounted for by the composite fermion
theory while the other one is a paired state for which we identify two
competing descriptions with different topological structure. This paired state
may be an Abelian liquid of composite spin-singlet Bose molecules with Laughlin
correlations. Alternatively, it may be a known non-Abelian paired state,
indicated by good overlaps with the corresponding trial wavefunction. By fine
tuning of the scattering lengths it is possible to create the non-Abelian
critical Haldane-Rezayi state for nu =1/2 and the permanent state of Moore and
Read for nu =1. For purely repulsive interactions, we also find evidence for a
gapped Halperin state at nu=2/5.Comment: 12 pages, 9 figs (best viewed in color), published version with
additional evidence for a non-Abelian spin singlet state at filling nu=2/
