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 a 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 [Phys. Rev. Lett. 97, 050403 (2006)]. 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 crossover between the Bose-Einstein condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) superfluid of s-wave paired fermions. 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.