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arXiv:1206.0059

$η/s$ of the Normal Phase of Unitary Fermi Gas from $\varepsilon$ Expansion

Andrei B. Kryjevski

Published Jun 1, 2012• 0 citations
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Abstract

Using $\varepsilon$-expansion technique we compute $η/s$, where $η$ is the shear viscosity, $s$ is the entropy density, of the normal phase of unitary Fermi gas in $d=4-\varepsilon$ dimensions to LO in $\varepsilon$. We use kinetic theory approach and solve transport equations for medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to $\varepsilon^2$ which is LO. The LO result is temperature independent with $η/\rm s\simeq (0.11/\varepsilon^2)(\hbar/k_B).$ The $d=3$ prediction for $η/\rm s$ exceeds the $\hbar/4 πk_B$ bound by a factor of about $1.4.$

DOI
10.48550/arxiv.1206.0059

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η/sη/sη/s of the Normal Phase of Unitary Fermi Gas from ε\varepsilonε Expansion
Using ε\varepsilonε-expansion technique we compute η/sη/sη/s, where ηηη is the shear viscosity, sss is the entropy density, of the normal phase of unitary Fermi gas in d=4−εd=4-\varepsilond=4−ε dimensions to LO in ε\varepsilonε. We use kinetic theory approach and solve transport equations for medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to ε2\varepsilon^2ε2 which is LO. The LO result is temperature independent with η/s≃(0.11/ε2)(ℏ/kB).η/\rm s\simeq (0.11/\varepsilon^2)(\hbar/k_B).η/s≃(0.11/ε2)(ℏ/kB​). The d=3d=3d=3 prediction for η/sη/\rm sη/s exceeds the ℏ/4πkB\hbar/4 πk_Bℏ/4πkB​ bound by a factor of about 1.4.1.4.1.4.
Using the $\ensuremath{\varepsilon}$-expansion technique, we compute $\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and sss is the entropy density of the normal phase of unitary Fermi gas in $d=4\ensuremath{-}\ensuremath{\varepsilon}$ dimensions to leading order (LO) in $\ensuremath{\varepsilon}$. We use the kinetic theory approach and solve transport equations for a medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to ${\ensuremath{\varepsilon}}^{2}$, which is LO. The LO result is temperature independent with $\ensuremath{\eta}/s\ensuremath{\simeq}(0.11/{\ensuremath{\varepsilon}}^{2})(\ensuremath{\hbar}/{k}_{B})$. The d=3d=3d=3 prediction for $\ensuremath{\eta}/s$ exceeds the $\ensuremath{\hbar}/4\ensuremath{\pi}{k}_{B}$ bound by a factor of about 1.41.41.4.
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 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 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.