Renormalization-group fixed points, universal phase diagram, and1∕Nexpansion for quantum liquids with interactions near the unitarity limit

Abstract

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 $(d\ensuremath{-}2)$ 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 $(4\ensuremath{-}d)$ 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 $\mathrm{Sp}(2N)$ symmetry] of the fixed point and its vicinity, and use it to obtain results for the phase diagram.