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

Figure 1

The exact RG flow of Eq. (2), as discussed in Ref. 5. Here $u$ is a measure of the short-range two-body interaction between the particles, and the RG applies in the limit of low density of either Bose, Fermi, or Bose-Fermi quantum liquids. (a) For $d<2$, the infrared stable fixed point at $u=u^{*}>0$ describes quantum liquids of either bosons or fermions with repulsive interactions which are generically universal in the low density limit. In $d=1$ this fixed point is described by the spinless free Fermi gas (“Tonks” gas), for all statistics and spin of the constituent particles. (b) For $d>2$, the infrared unstable fixed point at $u=u^{*}<0$ describes the Feshbach resonance which is obtained for the case of attractive interactions. The relevant perturbation $(u-u^{*})$ corresponds to the detuning from the resonant interaction.

Figure 2

Universal phase diagram at zero temperature $(T=0)$ and balanced densities $(h=0)$ for the two-component Fermi gas in $d=3$. The vacuum state (shown hatched) has no particles, and is present for $\mu <0$ and $\nu >0$, or for $\nu <0$ and $\mu <-{\nu }^2∕\left(2m\right)$, where $m$ is the mass of a fermion. The position of the $\nu <0$ phase boundary is determined by the energy of the two-fermion bound state. The density of particles vanishes continuously at the second order quantum phase transition boundary of the superfluid phase, which is indicated by the thin continuous line. The quantum multicritical point at $\mu =\nu =h=T=0$ (denoted by the filled circle) is the RG fixed point which is the basis for the analysis in this paper.

Figure 3

(Color online) Universal phase diagram at $T=0$, $h\ne 0$, and $N=\infty$. The axes have been scaled by $\mid h\mid$ to the dimensionless parameters $\mu ∕\mid h\mid$ and $\nu ∕\sqrt{2m\mid h\mid }$. The density in the superfluid is balanced, except in the shaded (red) region representing a “magnetized superfluid,” which also has a single $(1\times N)$ Fermi surface of (say) up spin fermions (for $h>0$). The 1FS and 2FS phases are nonsuperfluid states with $N$ and $2N$ Fermi surfaces, respectively. The thick line is a first order quantum phase transition, while the thin lines are second order transitions.

Figure 4

The same universal, zero temperature phase diagram as in Fig. 3, but for $\mu >0$. Now we have scaled the axes by $\mu$, and they measure detuning and the field, respectively. The first order phase transition between the superfluid and normal phases occurs at $h=h_c\left(\nu \right)$, plotted as the thick line; $h_c\left(0\right)=0.807124\mu$. The dashed faint line denotes the “upper critical field” $h_{c2}\left(\nu \right)$ (equal to the fermion BCS pairing gap), below which the superfluid may be found at least as a metastable state. Similarly, the dotted faint line denotes the “lower critical field” $h_{c1}\left(\nu \right)$, above which the normal state may be found at least as a metastable state.

Figure 5

(Color online) Same as in Fig. 4, but for $\mu <0$. The shaded (red) region is a “magnetized superfluid.”

Figure 6

(Color online) $1∕N$ correction to pressure in the normal state $[h>h_{c1}\left(\nu \right)]$ at zero temperature for $\mu >0$. In the limit $N\to \infty$ the faded pink portions of the curves should be discarded since they lie within the superfluid phase.

Figure 7

(Color online) A typical plot of $\mathrm{Im}\left\{K_2^{\prime }(k,\omega )\right\}$ vs frequency (here $\mu =1$, $m=0.5$, $h=0.2$, and $k=1$). There are up to five kink frequencies, listed in Eq. (B8), including the end points. $\mathrm{Im}\left\{K_2^{\prime }(k,\omega )\right\}$ is nonzero only in a finite interval.

Figure 8

(Color online) The complex path of $\omega$ along which $\ln K_2(k,\omega )$ is integrated out in Eq. (B10) (the radius of the semicircle is infinite). The dashed red line shows the branch-cut of the logarithm.