Transport in $p$-wave-interacting Fermi gases

The scattering properties of spin-polarized Fermi gases are dominated by $p$-wave interactions. Besides their inherent angular dependence, these interactions differ from their $s$-wave counterparts as they also require the presence of a finite effective range in order to understand the low-energy properties of the system. In this article we examine how the shear viscosity and thermal conductivity of a three-dimensional spin-polarized Fermi gas in the normal phase depend on the effective range and the scattering volume in both the weakly and the strongly interacting limits. We show that, although the shear viscosity and the thermal conductivity both explicitly depend on the effective range near resonance, the Prandtl number which parametrizes the ratio of momentum to thermal diffusivity does not have an explicit interaction dependence both at resonance and for weak interactions in the low-energy limit. In contrast to $s$-wave systems, $p$-wave scattering exhibits an additional resonance at weak attraction from a quasi-bound state at positive energies, which leads to a pronounced dip in the shear viscosity at specific temperatures.

Figure 1

Shear and thermal scattering rates as functions of the fugacity $z$. In all calculations we set the effective range ${\beta }^{1/2}R=100$. The lines from bottom to top indicate the direction towards resonance, $|\beta E_b|\to 0$. The solid lines are the full numerical solution to the Boltzmann equation, while the dashed lines represent the corresponding high-temperature approximation which is linear in $z$.

Figure 2

BEC-BCS crossover for the ratio of scattering times ${\tau }_{\eta }/{\tau }_{\kappa }$ in the high-temperature limit with ${\beta }^{1/2}R=100$. In this limit the ratio of scattering times coincides with the Prandtl number. The left-hand side corresponds to the BCS side, while the right-hand side is the BEC side. The solid, dashed, and dashed-dotted lines correspond to the RTA prediction, the BRTA prediction at resonance, and the BRTA prediction for weak interactions, respectively. The vertical dashed lines correspond to the points where the Prandtl number saturates the RTA value, $\beta E_b\approx 0$ and $7/2$.

Figure 3

Ratio of shear to thermal scattering times as a function of the fugacity $z$ (a) and the binding energy $E_b$ (b). The black solid line, the black dashed line, and the black dashed-dotted line correspond to the high-temperature results in the RTA, the resonant limit in the BRTA, and the weakly interacting limit in the BRTA, respectively. The lines from bottom to top indicate the direction towards (a) resonance, $|\beta E_b|\to 0$, and (b) the low-temperature limit.

Figure 4

Shear viscosity as a function of temperature for fixed density and various values of the binding energy $E_b$ on the BCS side ($E_b>0$). In this figure, $T_F$ and $E_F$ are the Fermi temperature and the Fermi energy, while the solid and the dashed lines correspond to the RTA calculation and the high-temperature limit, respectively. The dip in the shear viscosity is associated with an increased scattering at the quasi-bound-state energy. This dip in the shear viscosity does not appear at resonance, $E_b=0$, as zero-energy scattering is strongly suppressed for $p$-wave interactions. Similar physics will occur for the thermal conductivity.

Figure 5

Prandtl number on the BEC side as a function of the fugacity $z$ and the density through the Fermi temperature $T_F$, for various values of the binding energy $\beta E_b$. For the panel (a) [ panel (b)], the lines from bottom to top denote towards (away from) resonance. In general, decreasing the interaction decreases the Prandtl number, while lowering the temperature (increasing the fugacity or increasing the density) tends to an ultimate decrease of the Prandtl number by about $25\%$.

Figure 6

Comparison of the full angular average of Eq. (B4) (black line) and the solution from Eq. (B7) (red dashed line) for various values of $a=\beta (Q^2/8+p^2/2-\mu )$ and $b=\beta Qp/2$. The approximation is quite successful at capturing the full structure of the angular average for the desired parameter regime, $a,b\lesssim 1$.

Figure 7

Comparing the predictions of the scattering rates ${\tau }_{\eta ,\kappa }$ from the relaxation time approximation (RTA), beyond the relaxation time approximation (BRTA), and the high-temperature approximation for the BRTA. For this simulation, $\beta E_b=0$ and ${\beta }^{1/2}R=100$. The scattering times in the BRTA are larger than the RTA values, which are smaller than the high-temperature results. This is consistent with the basic structure of the BRTA.