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arXiv:1308.2004 · Physical Review A

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

Clifford Chafin, Thomas Schäfer

Published Oct 28, 2013• 49 citations
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Abstract

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 $(2P\ensuremath{-}\ensuremath{\rho}{c}_{s}^{2})/P\ensuremath{\sim}z/{[\mathrm{ln}(T/{E}_{B})]}^{2}$ and that the bulk viscosity $\ensuremath{\zeta}$ scales as $\ensuremath{\zeta}/\ensuremath{\eta}\ensuremath{\sim}{z}^{2}/{[\mathrm{ln}(T/{E}_{B})]}^{6}$. Here, $P$ is the pressure, ${c}_{s}^{2}$ is the speed of sound, $\ensuremath{\eta}$ 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, Feld, Fr\"ohlich, Pertot, Koschorreck, and K\"ohl [Phys. Rev. Lett. 108, 070404 (2012)]. Vogt et al. reported a frequency shift $\ensuremath{\delta}\ensuremath{\omega}/\ensuremath{\omega}$ of the order of a few percent and a damping rate smaller than the background rate $\ensuremath{\Gamma}/{\ensuremath{\omega}}_{0}\ensuremath{\sim}5%$.

DOI
10.1103/physreva.88.043636

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The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute, in the high temperature limit, the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio $(P\ensuremath{-}2/3\mathcal{E})/P$, where PPP is the pressure and E\mathcal{E}E is the energy density. At high temperature this ratio scales as $z\ensuremath{\lambda}/a$, where zzz is the fugacity, $\ensuremath{\lambda}$ is the thermal wavelength, and aaa is the scattering length. We show that the bulk viscosity $\ensuremath{\zeta}$ scales as the second power of this parameter, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.
We investigate collective excitations of a harmonically trapped two-dimensional Fermi gas from the collisionless (zero sound) to the hydrodynamic (first sound) regime. The breathing mode, which is sensitive to the equation of state, is observed with an undamped amplitude at a frequency 2 times the dipole mode frequency for a large range of interaction strengths and different temperatures. This provides evidence for a dynamical SO(2,1) scaling symmetry of the two-dimensional Fermi gas. Moreover, we investigate the quadrupole mode to measure the shear viscosity of the two-dimensional gas and study its temperature dependence.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio (P−23E)/P(P-\frac{2}{3}{\cal E})/P(P−32​E)/P, where PPP is the pressure and E{\cal E}E is the energy density. In the high temperature limit this ratio scales as zλa\frac{zλ}{a}azλ​, where zzz is the fugacity, λλλ is the thermal wave length, and aaa is the scattering length. We show that the bulk viscosity ζζζ scales as the second power of this parameter, ζ∼(zλa)2λ−3ζ\sim (\frac{zλ}{a})^2 λ^{-3}ζ∼(azλ​)2λ−3.
The shear viscosity of a two-dimensional Fermi gas interacting via a short-range potential with scattering length a2d{a}_{2\mathrm{d}}a2d​ in kinetic theory is computed. It is found that classical kinetic theory predicts that the shear viscosity to entropy density ratio of a strongly interacting two-dimensional gas is comparable to that of the three-dimensional unitary gas. Results are applied to the damping of collective modes of a trapped Fermi gas, and compared to experimental data recently obtained by Vogt et al. [Phys. Rev. Lett. 108, 070404 (2012)].
The frequency of the breathing mode of a two-dimensional Fermi gas with zero-range interactions in a harmonic confinement is fixed by the scale invariance of the Hamiltonian. Scale invariance is broken in the quantized theory by introducing the two-dimensional scattering length as a regulator. This is an example of a quantum anomaly in the field of ultracold atoms and leads to a shift of the frequency of the collective breathing mode of the cloud. In this work, we study this anomalous frequency shift for a two-component Fermi gas in the strongly interacting regime. We measure significant upwards shifts away from the scale-invariant result that show a strong interaction dependence. This observation implies that scale invariance is broken anomalously in the strongly interacting two-dimensional Fermi gas.
Strongly interacting two-dimensional Fermi systems are one of the great remaining challenges in many-body physics due to the interplay of strong local correlations and enhanced long-range fluctuations. Here, we probe the thermodynamic and transport properties of a 2D Fermi gas across the BEC-BCS crossover by studying the propagation and damping of sound modes. We excite particle currents by imprinting a phase step onto homogeneous Fermi gases trapped in a box potential and extract the speed of sound from the frequency of the resulting density oscillations. We measure the speed of sound across the BEC-BCS crossover and compare the resulting dynamic measurement of the equation of state both to a static measurement based on recording density profiles and to quantum Monte Carlo calculations and find reasonable agreement between all three. We also measure the damping of the sound mode, which is determined by the shear and bulk viscosities as well as the thermal conductivity of the gas. We find that the damping is minimal in the strongly interacting regime and the diffusivity approaches the universal quantum bound ℏ/m of a perfect fluid.