Dispersion of first sound in a weakly interacting ultracold Fermi liquid: an exact calculation

At low temperature, a normal gas of unpaired spin-1/2 fermions is one of the cleanest realizations of a Fermi liquid. It is described by Landau's theory, where no phenomenological parameters are needed as the quasiparticle interaction function can be computed perturbatively in powers of the scattering length $a$, the sole parameter of the short-range interparticle interactions. Obtaining an accurate solution of the transport equation nevertheless requires a careful treatment of the collision kernel, as the uncontrolled error made by the relaxation time approximations increases when the temperature $T$ drops below the Fermi temperature. Here, we study sound waves in the hydrodynamic regime up to second order in the Chapman-Enskog's expansion. We find that the frequency $\omega_q$ of the sound wave is shifted above its linear depart as $\omega_q=c_1 q(1+\alpha q^2\tau^2)$ where $c_1$ and $q$ are the speed and wavenumber of the wave and the typical collision time $\tau$ scales as $1/a^2T^2$. Besides the shear viscosity, the coefficient $\alpha$ is described by a single second-order collision time which we compute exactly from an analytical solution of the transport equation, resulting in a positive dispersion $\alpha>0$. Our results suggest that ultracold atomic Fermi gases are an ideal experimental system for quantitative tests of second order hydrodynamics.