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arXiv:0707.1520 · Physical Review A

Shear viscosity of a superfluid Fermi gas in the unitarity limit

Gautam Rupak, Thomas Schäfer

Published Nov 13, 2007• 105 citations
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Abstract

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 $T$ 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 ${T}_{c}$. 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 $\ensuremath{\eta}∕s\ensuremath{\ge}\ensuremath{\hbar}∕(4\ensuremath{\pi}{k}_{B})$.

DOI
10.1103/physreva.76.053607

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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 \ensuremathξ5∕T5{\ensuremath{\xi}}^{5}∕{T}^{5}\ensuremathξ5∕T5, where the universal parameter \ensuremathξ\ensuremath{\xi}\ensuremathξ relates the chemical potential and the Fermi energy, \ensuremathμ=\ensuremathξ\ensuremathϵF\ensuremath{\mu}=\ensuremath{\xi}{\ensuremath{\epsilon}}_{F}\ensuremathμ=\ensuremathξ\ensuremathϵF​. 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∼5\ensuremath{\sim}5\ensuremath∼5 of the proposed bound \ensuremathη∕s\ensuremath≥\ensuremathℏ∕(4\ensuremathπkB)\ensuremath{\eta}∕s\ensuremath{\ge}\ensuremath{\hbar}∕(4\ensuremath{\pi}{k}_{B})\ensuremathη∕s\ensuremath≥\ensuremathℏ∕(4\ensuremathπkB​).
We determine the shear viscosity of the ultracold Fermi gas at unitarity in the normal phase using hydrodynamic expansion data. The analysis is based on a generalized fluid dynamic framework which ensures a smooth transition between the fluid dynamic core of the cloud and the ballistic corona. We use expansion data taken by Joseph, Elliott, and Thomas [Shear Viscosity of a Universal Fermi Gas Near the Superfluid Phase Transition, Phys. Rev. Lett. 115, 020401 (2015).PRLTAO0031-900710.1103/PhysRevLett.115.020401] and measurements of the equation of state by Ku et al. [Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 563 (2012).SCIEAS0036-807510.1126/science.1214987]. We find that the shear viscosity to particle density ratio just above the critical temperature is η/n|_{T_{c}}=0.41±0.11. We also obtain evidence that the shear viscosity to entropy density ratio has a minimum slightly above T_{c} with η/s|_{min}=0.50±0.10.
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 analyze the contributions of Nambu-Goldstone modes to the shear viscosity of a superfluid atomic Fermi gas close to unitarity. We show that the low temperature experimental values of the shear viscosity to entropy ratio, η/sη/sη/s, are reproduced considering an effective shear viscosity which takes into account the collisions among Nambu-Goldstone bosons and the finite size of the optical trap. We predict that, for T≲0.1TFT \lesssim 0.1 T_FT≲0.1TF​, η/sη/sη/s should correlate with the size of the trap and linearly decrease with decreasing temperature. For T≳0.1TFT \gtrsim 0.1 T_FT≳0.1TF​, we find agreement with the experimental data assuming that the Nambu-Goldstone modes have an anomalous dispersion law.
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 present an ab initio determination of the shear viscosity $\ensuremath{\eta}$ of the unitary Fermi gas, based on finite temperature quantum Monte Carlo calculations and the Kubo linear-response formalism. We determine the temperature dependence of the shear viscosity-to-entropy density ratio $\ensuremath{\eta}/s$. The minimum of $\ensuremath{\eta}/s$ appears to be located above the critical temperature for the superfluid-to-normal phase transition with the most probable value being $(\ensuremath{\eta}/s{)}_{\mathrm{min}}\ensuremath{\approx}0.2\ensuremath{\hbar}/{k}_{B}$, which is close the Kovtun-Son-Starinets universal value $\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$.
We study hydrodynamic fluctuations in a nonrelativistic fluid. We show that in three dimensions, fluctuations lead to a minimum in the shear viscosity to entropy density ratio $\ensuremath{\eta}/s$ as a function of the temperature. The minimum provides a bound on $\ensuremath{\eta}/s$ which is independent of the conjectured bound in string theory, $\ensuremath{\eta}/s\ensuremath{\ge}\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$, where sss is the entropy density. For the dilute Fermi gas at unitarity, we find $\ensuremath{\eta}/s\ensuremath{\gtrsim}0.2\ensuremath{\hbar}$. This bound is not universal---it depends on the thermodynamic properties of the unitary Fermi gas and on empirical information about the range of validity of hydrodynamics. We also find that the viscous relaxation time of a hydrodynamic mode with frequency $\ensuremath{\omega}$ diverges as $1/\sqrt{\ensuremath{\omega}}$, and that the shear viscosity in two dimensions diverges as $\mathrm{ln}(1/\ensuremath{\omega})$.