Dispersion of first sound in a weakly interacting ultracold Fermi liquid

TL;DR

This paper investigates the dispersion of first sound in a weakly interacting ultracold Fermi liquid, providing an exact second-order hydrodynamic analysis that predicts a positive dispersion shift, suitable for experimental validation.

Contribution

It offers an exact analytical solution for second-order collision effects on sound dispersion in ultracold Fermi liquids, advancing the understanding of hydrodynamics in quantum gases.

Findings

Frequency shift of sound wave above linear prediction

Positive dispersion coefficient computed exactly

Ultracold Fermi gases as ideal testbeds for second-order hydrodynamics

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 $1/a^2T^2$. 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.