Dispersion of first sound in a weakly interacting ultracold Fermi liquid

Abstract

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 departure as ${\omega }_q=c_{1}q(1+\alpha q^2{\tau }^2)$ where $c_1$ and $q$ are the speed and wavenumber of the sound wave and the typical collision time $\tau$ scales as $1/a^2{T}^2$. 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.

arXiv comment10 pages, 3 figures

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