# Abstract:

Unlike purely bosonic superfluids, fermionic superfluids can undergo phase transitions into different kinds of normal phases, loosely characterized by whether Cooper pairs survive or not. The paired normal state has a distinct topological order from the unpaired state, and may be described as a liquid of vortices. We explore the normal phases in the context of fermion systems near unitarity, where the superfluid is destroyed by fast rotation. The obtained quantum Hall state in three dimensions is always a paired state at zero temperature, while in two dimensions both kinds of normal phases are possible at T=0. The second order phase transition between paired and unpaired normal phases is driven by confinement/deconfinement of pi-flux quanta. The stability of vortex liquids in neutral cold atom gases is expected to be enhanced by the scale-free scattering of particles near unitarity. The analysis proceeds by making a 1/N expansion in a model with SP(N) symmetry. We show that in the semiclassical regime of large-N the paired quantum Hall liquids are unstable to density waves of incoherent Cooper pairs, bearing some similarity with electronic quantum Hall states in high Landau levels. We discuss the phase diagram and characterization of the order parameter. At the end, the transport properties of these vortex liquids will be discussed, with a look toward the Nernst effect in cuprate superconductors.