Renormalization group fixed points, universal phase diagram, and 1/N expansion for quantum liquids with interactions near the unitarity limit

It has long been known that particles with short-range repulsive interactions in spatial dimension d=1 form universal quantum liquids in the low density limit: all properties can be related to those of the spinless free Fermi gas. Previous renormalization group (RG) analyses demonstrated that this universality is described by an RG fixed point, infrared stable for d<2, of the zero density gas. We show that for d>2 the same fixed point describes the universal properties of particles with short-range attractive interactions near a Feshbach resonance; the fixed point is now infrared unstable, and the relevant perturbation is the detuning of the resonance. Some exponents are determined exactly, and the same expansion in powers of (d-2) applies for scaling functions for d<2 and d>2. A separate exact RG analysis of a field theory of the particles coupled to `molecules' finds an alternative description of the same fixed point, with identical exponents; this approach yields a (4-d) expansion which agrees with the recent results of Nishida and Son (cond-mat/0604500). The existence of the RG fixed point implies a universal phase diagram as a function of density, temperature, population imbalance, and detuning; in particular, this applies to the BEC-BCS crossover of fermions with s-wave pairing. Our results open the way towards computation of these universal properties using the standard field-theoretic techniques of critical phenomena, along with a systematic analysis of corrections to universality. We also propose a 1/N expansion (based upon models with Sp(2N) symmetry) of the fixed point and its vicinity, and use it to obtain results for the phase diagram.