Induced $P$-wave superfluidity within the full energy- and momentum-dependent Eliashberg approximation in asymmetric dilute Fermi gases

We consider a very asymmetric system of fermions with an interaction characterized by a positive scattering length only. The minority atoms pair and form a Bose-Einstein condensate of dimers, while the surplus fermions interact only indirectly through the exchange of Bogoliubov sound modes. This interaction has a finite range, the retardation effects are significant, and the surplus fermions will form a $P$-wave superfluid. We compute the $P$-wave pairing gap in the BCS and Eliashberg approximations with only energy-dependence approximations, and demonstrate their inadequacy in comparison with a full treatment of the momentum and energy dependence of the induced interaction. The pairing gap computed with a full momentum and energy dependence is significantly larger in magnitude, and that makes it more likely that this new exotic paired phase could be put in evidence in atomic trap experiments.

Figure 1

(Color online) In these panels the ratio of dimer versus the surplus fermions varies as $y=n_b/n_f=2$, 1, and 0.5 from the upper panel down. The solid black line is the pairing gap computed using the BCS approximation as in Ref. [6]. The solid (red) and dotted (red) lines with circles are the pairing gaps computed with full momentum and energy dependence, while the solid (blue) and dotted (blue) lines with squares are pairing gaps obtained in the simple Eliashberg approximation. The solid line is for the pairing gap ${\Delta }_{10}$ with $\left(l,m\right)=\left(1,0\right)$, while the dashed line is for ${\Delta }_{11}$ with $\left(l,m\right)=\left(1,1\right)$. Note that in the Eliashberg approximation ${\Delta }_{11}>{\Delta }_{10}$, while in the case of the BCS approximation there is no dependence of the pairing gap on the magnetic quantum number. The pairing gap is evaluated at zero energy $\omega =0$ and on the Fermi surface $\left|\mathit{p}\right|=\hslash k_f$. In all cases we plotted the magnitude of the pairing gap, after the angular dependence $Y_{1m}\left(\hat{\mathit{p}}\right)$ has been factored out. The region of smaller values of $k_{f}a$ is shown in insets.