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Physical Review Letters

Bulk Viscosity and Conformal Symmetry Breaking in the Dilute Fermi Gas near Unitarity

Kevin Dusling, Thomas Schäfer

Published Sep 19, 2013• 42 citations
Publisher

Abstract

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 $P$ is the pressure and $\mathcal{E}$ is the energy density. At high temperature this ratio scales as $z\ensuremath{\lambda}/a$, where $z$ is the fugacity, $\ensuremath{\lambda}$ is the thermal wavelength, and $a$ is the scattering length. We show that the bulk viscosity $\ensuremath{\zeta}$ scales as the second power of this parameter, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.

DOI
10.1103/physrevlett.111.120603

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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−2/3E)/P(P\ensuremath{-}2/3\mathcal{E})/P(P\ensuremath−2/3E)/P, where PPP is the pressure and E\mathcal{E}E is the energy density. At high temperature this ratio scales as z\ensuremathλ/az\ensuremath{\lambda}/az\ensuremathλ/a, where zzz is the fugacity, \ensuremathλ\ensuremath{\lambda}\ensuremathλ is the thermal wavelength, and aaa is the scattering length. We show that the bulk viscosity \ensuremathζ\ensuremath{\zeta}\ensuremathζ scales as the second power of this parameter, \ensuremathζ\ensuremath∼(z\ensuremathλ/a)2\ensuremathλ\ensuremath−3\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}\ensuremathζ\ensuremath∼(z\ensuremathλ/a)2\ensuremathλ\ensuremath−3.
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.
We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, nnn is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.
We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity η(ω)\eta(\omega)η(ω) exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of 0.5TF0.5 T_F0.5TF​, thus invalidating a quasiparticle description. The ratio η/s\eta/sη/s between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound ℏ/(4πkB)\hbar/(4\pi k_B)ℏ/(4πkB​) by a factor of about seven.
We study the small frequency behavior of the bulk viscosity spectral function using stochastic fluid dynamics. We obtain a number of model independent results, including the long-time tail of the bulk stress correlation function and the leading nonanalyticity of the spectral function at small frequency. We also establish a lower bound on the bulk viscosity which is weakly dependent on assumptions regarding the range of applicability of fluid dynamics. The bound on the bulk viscosity $\ensuremath{\zeta}$ scales as ${\ensuremath{\zeta}}_{\mathrm{min}}\ensuremath{\sim}{(P\ensuremath{-}\frac{2}{3}\mathcal{E})}^{2}{\ensuremath{\sum}}_{i}{D}_{i}^{\ensuremath{-}2}$, where Di{D}_{i}Di​ are the diffusion constants for energy and momentum and $P\ensuremath{-}\frac{2}{3}\mathcal{E}$, where PPP is the pressure and E\mathcal{E}E is the energy density, is a measure of scale breaking. Applied to the cold Fermi gas near unitarity, $|\ensuremath{\lambda}/{a}_{s}|\ensuremath{\gtrsim}1$, where $\ensuremath{\lambda}$ is the thermal de Broglie wavelength and as{a}_{s}as​ is the sss-wave scattering length, this bound implies that the ratio of bulk viscosity to entropy density satisfies $\ensuremath{\zeta}/s\ensuremath{\gtrsim}0.1\ensuremath{\hbar}/{k}_{B}$. Here, $\ensuremath{\hbar}$ is Planck's constant and kB{k}_{B}kB​ is Boltzmann's constant.
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 $\ensuremath{\Gamma}/{\ensuremath{\omega}}_{0}\ensuremath{\sim}5%$.
We compute the shear viscosity of a superfluid atomic Fermi gas in the unitarity limit. The unitarity limit is characterized by a divergent scattering length between the atoms, and it has been argued that this will result in a very small viscosity. We show that in the low temperature TTT limit the shear viscosity scales as ${\ensuremath{\xi}}^{5}∕{T}^{5}$, where the universal parameter $\ensuremath{\xi}$ relates the chemical potential and the Fermi energy, $\ensuremath{\mu}=\ensuremath{\xi}{\ensuremath{\epsilon}}_{F}$. Combined with the high temperature expansions of the viscosity our results suggest that the viscosity has a minimum near the critical temperature Tc{T}_{c}Tc​. A na\"{\i}ve extrapolation indicates that the minimum value of the ratio of viscosity over entropy density is within a factor of $\ensuremath{\sim}5$ of the proposed bound $\ensuremath{\eta}∕s\ensuremath{\ge}\ensuremath{\hbar}∕(4\ensuremath{\pi}{k}_{B})$.