Large-$N$ expansion for unitary superfluid Fermi gases

We analyze strongly interacting Fermi gases in the unitary regime by considering the generalization to an arbitrary number $N$ of spin-$1∕2$ fermion flavors with $\mathrm{Sp}\left(2N\right)$ symmetry. For $N\to \infty$ this problem is exactly solved by the Bardeen-Cooper-Schrieffer–Bose-Einstein condensate mean-field theory, with corrections small in the parameter $1∕N$. The large-$N$ expansion provides a systematic way to determine corrections to mean-field predictions, allowing the calculation of a variety of thermodynamic quantities at (and in proximity to) unitarity, including the energy, the pairing gap, and the upper-critical polarization (in the case of a polarized gas) for the normal to superfluid instability. For the physical case of $N=1$, among other quantities, we predict in the unitarity regime, the energy of the gas to be $\xi =0.28$ times that for the noninteracting gas and the pairing gap to be 0.52 times the Fermi energy.

Figure 1

(Color online) Chemical potential $\mu$ and order parameter $\Delta$ as a function of ${\left(k_F{a}_s\right)}^{-1}$. The dashed lines are the $N\to \infty$ results for $\mu$ and $\Delta$. The diamond symbols include the $O(1∕N)$ corrections, evaluated at $N=1$. The solid lines are a guide to the eye. The star symbols at unitarity are the results of quantum Monte Carlo calculation from Ref. 19.