Exact perturbative expansion of the transport coefficients of a normal low-temperature Fermi gas with contact interactions

Abstract

We compute the shear viscosity, thermal conductivity and spin diffusivity of a Fermi gas with short-range interactions in the Fermi liquid regime of the normal phase, that is at temperatures $T$ much lower than the Fermi temperature $T_{\rm F}$ and much larger than the superfluid critical temperature $T_c$. Given recent advances in the precision of cold atom experiments, we provide exact results up to second-order in the interaction strength. We extend the Landau-Salpeter equation to compute the collision amplitude beyond the forward-scattering limit, covering all collisions on the Fermi surface. We treat the collision kernel exactly, leading to significant corrections beyond relaxation-time or variational approximations. The transport coefficients, as functions of the $s$-wave scattering length $a$ and Fermi wavenumber $k_{\rm F}$, follow $(1+\gamma k_{\rm F}a)/a^2$ up to corrections of order $O(a^0)$, with a positive coefficient $\gamma$ for the viscosity and negative one for the thermal conductivity and spin diffusivity.