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Viscosity and scale invariance in the unitary Fermi gas

Tilman Enss, R. Haussmann, W. Zwerger

Published Jan 20, 2016• 164 citations
OpenAlex

Abstract

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 $\eta(\omega)$ 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 $0.5 T_F$, thus invalidating a quasiparticle description. The ratio $\eta/s$ 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 $\hbar/(4\pi k_B)$ by a factor of about seven.

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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 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 $\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$.
The thermodynamic and transport properties of the unitary Fermi gas at finite temperature TTT are governed by a quantum critical point at T=0T=0T=0 and zero density. We compute the universal shear viscosity to entropy ratio $\ensuremath{\eta}/s$ in the high-temperature quantum critical regime $T\ensuremath{\gg}|\ensuremath{\mu}|$ and find that this strongly coupled quantum fluid comes close to perfect fluidity $\ensuremath{\eta}/s=\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$. Using a controlled large-NNN expansion, we show that already at the first nontrivial order the equation of state and the Tan contact density CCC agree well with the most recent experimental measurements and theoretical Luttinger-Ward and bold diagrammatic Monte Carlo calculations.
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 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.
The viscosity of strongly interacting systems is a topic of great interest in diverse fields. We focus here on the bulk and shear viscosities of nonrelativistic quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$, respectively, to derive exact, nonperturbative results. Our results include a microscopic connection between the shear viscosity $\ensuremath{\eta}$ and the normal-fluid density ${\ensuremath{\rho}}_{n}$; sum rules for $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$ and their evolution through the BCS-BEC crossover (where BEC denotes Bose-Einstein condensate); and universal high-frequency tails for $\ensuremath{\eta}(\ensuremath{\omega})$ and the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$. We use our sum rules to show that, at unitarity, $\ensuremath{\zeta}(\ensuremath{\omega})$ is identically zero and thus relate $\ensuremath{\eta}(\ensuremath{\omega})$ to density-density correlations. We predict that frequency-dependent shear viscosity $\ensuremath{\eta}(\ensuremath{\omega})$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.