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arXiv:1209.1604 · Physical Review A

Implications of hydrodynamic fluctuations for the minimum shear viscosity of the dilute Fermi gas at unitarity

Paul Romatschke, Ryan E. Young

Published May 10, 2013• 29 citations
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Abstract

We confirm and expand on work by Chafin and Sch\"afer [Phys. Rev. A 87, 023629 (2013)] on hydrodynamic fluctuations in the unitary Fermi gas. Using the result for the equation of state from a recent MIT experiment, we derive lower bounds for $\ensuremath{\eta}/n$ and $\ensuremath{\eta}/s$ as a function of temperature. Reanalyzing recent quantum Monte Carlo data for the shear-viscosity spectral function, we point out a possible resolution for the tension between the viscosity bound $\ensuremath{\eta}/n\ensuremath{\gtrsim}0.3$ from Chafin and Sch\"afer and the quantum Monte Carlo results $\ensuremath{\eta}/n\ensuremath{\lesssim}0.2$ from Wlazlowski, Magierski, and Drut [Phys. Rev. Lett. 109, 020406 (2012)] near the critical temperature.

DOI
10.1103/physreva.87.053606

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We confirm and expand on work by Chafin and Sch"afer [Phys. Rev. A 87, 023629 (2013)] on hydrodynamic fluctuations in the unitary Fermi gas. Using the result for the equation of state from a recent MIT experiment, we derive lower bounds for \ensuremathη/n\ensuremath{\eta}/n\ensuremathη/n and \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/s as a function of temperature. Reanalyzing recent quantum Monte Carlo data for the shear-viscosity spectral function, we point out a possible resolution for the tension between the viscosity bound \ensuremathη/n\ensuremath≳0.3\ensuremath{\eta}/n\ensuremath{\gtrsim}0.3\ensuremathη/n\ensuremath≳0.3 from Chafin and Sch"afer and the quantum Monte Carlo results \ensuremathη/n\ensuremath≲0.2\ensuremath{\eta}/n\ensuremath{\lesssim}0.2\ensuremathη/n\ensuremath≲0.2 from Wlazlowski, Magierski, and Drut [Phys. Rev. Lett. 109, 020406 (2012)] near the critical temperature.
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 $\mathrm{ln}(1/\ensuremath{\omega})$.
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 present an ab initio determination of the shear viscosity for the unitary Fermi gas based on finite temperature quantum Monte Carlo (QMC) calculations and the Kubo linear-response formalism. The results are confronted with the bound for the shear viscosity originating from hydrodynamic fluctuations. Assuming smoothness of the frequency dependent shear viscosity $\ensuremath{\eta}(\ensuremath{\omega})$, we show that the bound is violated in the low temperature regime and the violation occurs simultaneously with the onset of the Cooper pairing in the system. In order to preserve the hydrodynamic bound in QMC $\ensuremath{\eta}(\ensuremath{\omega})$ has to possess a sharp structure located in the vicinity of zero frequency which is not resolved by an analytic continuation procedure.
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 $\ensuremath{\eta}∕s\ensuremath{\ge}\ensuremath{\hbar}∕(4\ensuremath{\pi}{k}_{B})$.
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 present an ab initio calculation of the shear viscosity as a function of interaction strength in a two-component unpolarized Fermi gas near the unitary limit, within a finite temperature quantum Monte Carlo (QMC) framework and using the Kubo linear-response formalism. The shear viscosity decreases as we tune the interaction strength 1akF\frac{1}{a{k}_{F}}akF​1​ from the Bardeen-Cooper-Schrieffer side of the Feshbach resonance towards Bose-Einstein condensation limit and it acquires the smallest value for $\frac{1}{a{k}_{F}}\ensuremath{\approx}0.4$, with a minimum value of ${\frac{\ensuremath{\eta}}{s}|}_{\text{min}}\ensuremath{\approx}0.2\frac{\ensuremath{\hbar}}{{k}_{B}}$, which is about twice as small as the value reported for experiments in quark-gluon plasma ${\frac{\ensuremath{\eta}}{s}|}_{\text{QGP}}\ensuremath{\lesssim}0.4\frac{\ensuremath{\hbar}}{{k}_{B}}$. The Fermi gas near unitarity thus emerges as the most ``perfect fluid'' observed so far in nature. The clouds of dilute Fermi gas near unitarity exhibit the unusual attribute that, for the sizes realized so far in the laboratory, or larger (less than 109{10}^{9}109 atoms), can sustain quantum turbulence below the critical temperature, but at the same time the classical turbulence is suppressed in the normal phase.