Scale breaking and fluid dynamics in a dilute two-dimensional Fermi gas

We study two observables related to the anomalous breaking of scale invariance in a dilute two dimensional Fermi gas, the frequency shift and damping rate of the monopole mode in a harmonic confinement potential. For this purpose we compute the speed of sound and the bulk viscosity of the two dimensional gas in the high temperature limit. We show that the anomaly in the speed of sound scales as $(2P-\rho c_s^2)/P\sim z/[\log(T/E_B)]^2$, and that the bulk viscosity $\zeta$ scales as $\zeta/\eta \sim z^2/[\log(T/E_B)]^6$. Here, $P$ is the pressure, $c_s^2$ is the speed of sound, $\eta$ is the shear viscosity, $z$ is the fugacity, and $E_B$ is the two-body binding energy. We show that our results are consistent with the experimental results of Vogt et al. [Phys. Rev. Lett. 108, 070404 (2012)]. Vogt et al. reported a frequency shift $\delta\omega/\omega$ of the order of a few percent, and a damping rate smaller than the background rate $\Gamma/\omega_0\sim 5$.