Effect of the induced interaction on the superfluid-transition temperature of ultracold Fermi gases within the $T$-matrix approximation

In this paper, taking into account the effect of the induced interaction, we calculate the superfluid-transition temperature of ultracold Fermi gases within the framework of the non-self-consistent $T$-matrix approximation. The result shows that the transition temperature is reduced by about $10\%$ at unitarity due to the induced interaction. In the Bardeen-Cooper-Schrieffer (BCS) limit, the critical temperature is about $0.45T_c^{\text{BCS}}$ and it is the same as the result of the usual Gorkov and Melik-Barkhudarov correction, whereas in the Bose-Einstein condensate (BEC) limit, the influence of the induced interaction can be neglected and the critical temperature turns out to be the transition temperature of ideal Bose gases. Our results show that the induced interaction cannot be neglected in the study of the BCS-BEC crossover.

Figure 1

The induced interaction considered by GMB [12]. Solid lines and dashed lines represent free fermion propagators and the interactions between atoms, respectively.

Figure 2

The induced interaction in the generalized GMB approximation considered by Yu and co-workers [16].

Figure 3

Self-energy correction $\Sigma (\mathbf{k},i{\omega }_n)$ in the nTMA. The solid line represents the free fermion propagator.

Figure 4

Particle-particle scattering matrix $\Gamma (\mathbf{q},i{\nu }_n)$ in the nTMA. The solid and wavy lines represent the free fermion propagator and the effective interaction in Eq. (8), respectively.

Figure 5

Transition temperatures of ultracold Fermi gases as a function of ${\left(k_F{a}_s\right)}^{-1}$. The solid line is our result obtained considering the induced interaction in the framework of nTMA, the dotted-dashed line is the result obtained from extended GMB in the NSR approach by Yu and co-workers [16], and the dashed line is the result obtained from nTMA by Tsuchiya and co-workers [17].