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arXiv:1111.1173 · Physical Review Letters

Scale Invariance and Viscosity of a Two-Dimensional Fermi Gas

Enrico Vogt, Michael Feld, Bernd Fröhlich, Daniel Pertot, Marco Koschorreck, Michael Köhl

Published Feb 17, 2012• 157 citations
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Abstract

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.

DOI
10.1103/physrevlett.108.070404

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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.
We examine the breathing mode of a strongly interacting two-dimensional Fermi gas and the role of temperature on the anomalous breaking of scale invariance. By calculating the equation of state with different many-body TTT-matrix theories and the virial expansion, we obtain a hydrodynamic equation of the harmonically trapped Fermi gas (with trapping frequency ${\ensuremath{\omega}}_{0}$) through the local density approximation. By solving the hydrodynamic equations, we determine the breathing mode frequencies as a function of interaction strength and temperature. We find that the breathing mode anomaly depends sensitively on both interaction strength and temperature. In particular, in the strongly interacting regime, we predict a significant downshift of the breathing mode frequency, below the scale invariant value of $2{\ensuremath{\omega}}_{0}$, for temperatures of the order of the Fermi temperature.
We derive analytical expressions for the frequency and damping of the lowest collective modes of a two-dimensional Fermi gas using kinetic theory. For strong coupling, we furthermore show that pairing correlations overcompensate the effects of Pauli blocking on the collision rate for a large range of temperatures, resulting in a rate which is larger than that of a classical gas. Our results agree well with experimental data, and they recover the observed crossover from collisionless to hydrodynamic behavior with increasing coupling for the quadruple mode. Finally, we show that a trap anisotropy within the experimental bounds results in a damping of the breathing mode which is comparable to what is observed, even for a scale-invariant system.
The collective excitations of a zero-temperature, spin-polarized, harmonically trapped, two-dimensional dipolar Fermi gas are examined within the Thomas-Fermi--von Weizs\"acker hydrodynamic theory. We focus on repulsive interactions and investigate the dependence of the excitation frequencies on the strength of the dipolar interaction and particle number. We find that the mode spectrum can be classified according to bulk modes, whose frequencies are shifted upward as the interaction strength is increased, and an infinite ladder of surface modes, whose frequencies are independent of the interactions in the large particle limit. We argue quite generally that it is the local character of the two-dimensional energy density that is responsible for the insensitivity of surface excitations to the dipolar interaction strength and not the precise form of the equation of state. This property will not be found for the collective excitations of harmonically trapped, dipolar Fermi gases in one and three dimensions, where the energy density is manifestly nonlocal.
Recent experiments on a 2D Fermi gas find an undamped breathing mode at twice the trap frequency over a wide range of parameters. To understand this seemingly scale-invariant behavior in a system with a scale, we derive two exact results valid across the entire Bardeen-Cooper-Schrieffer--Bose-Einstein condensation (BCS--BEC) crossover at all temperatures. First, we relate the shift of the mode frequency from its scale-invariant value to ${\ensuremath{\gamma}}_{d}\ensuremath{\equiv}(1+2/d)P\ensuremath{-}\ensuremath{\rho}(\ensuremath{\partial}P/\ensuremath{\partial}\ensuremath{\rho}{)}_{s}$ in ddd dimensions. Next, we relate ${\ensuremath{\gamma}}_{d}$ to dissipation via a new low-energy bulk viscosity sum rule. We argue that 2D is special, with its logarithmic dependence of the interaction on density, and thus ${\ensuremath{\gamma}}_{2}$ is small in both the BCS--BEC regimes, even though $P\ensuremath{-}2\ensuremath{\epsilon}/d$, sensitive to the dimer binding energy that breaks scale invariance, is not.
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, PPP is the pressure, cs2{c}_{s}^{2}cs2​ is the speed of sound, $\ensuremath{\eta}$ is the shear viscosity, zzz is the fugacity, and EB{E}_{B}EB​ 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 $\ensuremath{\Gamma}/{\ensuremath{\omega}}_{0}\ensuremath{\sim}5%$.
For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently, the frequency of the breathing mode ${\ensuremath{\omega}}_{B}$ of the 2D Fermi gas can be different from $2{\ensuremath{\omega}}_{0}$, with ${\ensuremath{\omega}}_{0}$ the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for ${\ensuremath{\omega}}_{B}$. We further calculate ${\ensuremath{\omega}}_{B}$ through the Euler equations in the hydrodynamic regime. The obtained value of ${\ensuremath{\omega}}_{B}$ satisfies the upper bounds and shows deviation from $2{\ensuremath{\omega}}_{0}$, which can be as large as about 88%8.