Kovtun-Son-Starinets Conjecture and Effects of Mass Imbalance in the Normal State of an Ultracold Fermi Gas in the BCS-BEC Crossover Region

Abstract

We theoretically assess the conjecture proposed by Kovtun, Son, and Starinets, stating that the ratio $\eta/s$ of the shear viscosity $\eta$ to the entropy density $s$ has the lower bound as $\eta/s\ge\hbar/(4\pi k_{\mathrm{B}})$. In the normal state of a mass-imbalanced ultracold Fermi gas, consistently including strong-coupling corrections to both $\eta$ and $s$ within the self-consistent $T$-matrix approximation, we evaluate $\eta/s$ over the entire BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover region, in the presence of mass imbalance. We find that $\eta/s$ achieves the minimum value $4.5\times \hbar/(4\pi k_{\mathrm{B}})$, not at the unitarity, but slightly in the BEC regime, $(k_{\mathrm{F}}a_s)^{-1}\simeq 0.4>0$ (where $a_s$ is the $s$-wave scattering length, and $k_{\mathrm{F}}$ is the Fermi momentum). In contract to the previous expectation, we find that this lower bound is almost independent of mass imbalance: Our results predict that all the mass-balanced $^6$Li-$^6$Li and $^{40}$K-$^{40}$K mixtures and the mass-imbalanced $^{40}$K-$^{161}$Dy mixture give almost the same lower bound of $\eta/s$. We also point out that the two quantum phenomena, Pauli blocking and bound-state formation, are crucial keys for the lower bound of $\eta/s$.