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arXiv:1607.04799 · Physical Review A

Shear viscosity in an anisotropic unitary Fermi gas

Rickmoy Samanta, Rishi Sharma, Sandip P. Trivedi

Published Nov 1, 2017• 17 citations
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Abstract

We consider a system consisting of a strongly interacting, ultracold unitary Fermi gas under harmonic confinement. Our analysis suggests the possibility of experimentally studying, in this system, an anisotropic shear viscosity tensor driven by the anisotropy in the trapping potential. In particular, we suggest that this experimental setup could mimic some features of anisotropic geometries that have recently been studied for strongly coupled field theories which have a dual gravitational description. Results using the AdS/CFT (anti-de Sitter/conformal field theory correspondence) in these theories show that in systems with a background linear potential, certain viscosity components can be made much smaller than the entropy density, parametrically violating the bound proposed by Kovtun, Son, and Starinets (KSS). This intuition, along with results from a Boltzmann analysis that we perform, suggests that a violation of the KSS bound can perhaps occur in the unitary Fermi gas system when it is subjected to a suitable anisotropic trapping potential which may be approximated to be linear in a suitable range of parameters. We give a concrete proposal for an experimental setup where an anisotropic shear viscosity tensor may arise. In such situations, it may also be possible to observe a reduction in the spin-1 component of the shear viscosity from its lowest value observed so far in ultracold Fermi gases. In extreme anisotropic situations, the reduction may be enough to reduce the shear viscosity to entropy ratio below the proposed KSS bound, although this regime is difficult to analyze in a theoretically controlled manner.

DOI
10.1103/physreva.96.053601

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We consider a system consisting of a strongly interacting, ultracold unitary Fermi gas under harmonic confinement. Our analysis suggests the possibility of experimentally studying, in this system, an anisotropic shear viscosity tensor driven by the anisotropy in the trapping potential. In particular, we suggest that this experimental setup could mimic some features of anisotropic geometries that have recently been studied for strongly coupled field theories which have a dual gravitational description. Results using the AdS/CFT (anti-de Sitter/conformal field theory correspondence) in these theories show that in systems with a background linear potential, certain viscosity components can be made much smaller than the entropy density, parametrically violating the bound proposed by Kovtun, Son, and Starinets (KSS). This intuition, along with results from a Boltzmann analysis that we perform, suggests that a violation of the KSS bound can perhaps occur in the unitary Fermi gas system when it is subjected to a suitable anisotropic trapping potential which may be approximated to be linear in a suitable range of parameters. We give a concrete proposal for an experimental setup where an anisotropic shear viscosity tensor may arise. In such situations, it may also be possible to observe a reduction in the spin-1 component of the shear viscosity from its lowest value observed so far in ultracold Fermi gases. In extreme anisotropic situations, the reduction may be enough to reduce the shear viscosity to entropy ratio below the proposed KSS bound, although this regime is difficult to analyze in a theoretically controlled manner.
We present a proposal to measure anisotropic shear viscosity in a strongly interacting, ultra-cold, unitary Fermi gas confined in a harmonic trap. We introduce anisotropy in this setup by strongly confining the gas in one of the directions with relatively weak confinement in the remaining directions. This system has a close resemblance to anisotropic strongly coupled field theories studied recently in the context of gauge-gravity duality. Computations in such theories (which have gravity duals) revealed that some of the viscosity components of the anisotropic shear viscosity tensor can be made much smaller than the entropy density, thus parametrically violating the bound proposed by Kovtun, Son and Starinets (KSS): ηs≥14π\frac η {s} \geq \frac{1}{4 π}sη​≥4π1​. A Boltzmann analysis performed in a system of weakly interacting particles in a linear potential also shows that components of the viscosity tensor can be reduced. Motivated by these exciting results, we propose two hydrodynamic modes in the unitary Fermi gas whose damping is governed by the component of shear viscosity expected to violate the KSS bound. One of these modes is the well known scissor mode. We estimate trap parameters for which the reduction in the shear viscosity is of order unity and find that the trap geometry, the damping timescales, and mode amplitudes are within the range of existing experimental setups on ultra-cold Fermi gases.
Eighty years ago Eyring proposed that the shear viscosity of a liquid, ηηη, has a quantum limit η≳nℏη\gtrsim n\hbarη≳nℏ where nnn is the density of the fluid. Using holographic duality and the AdS/CFT correspondence in string theory Kovtun, Son, and Starinets (KSS) conjectured a universal bound ηs≥ℏ4πkB\fracη{s}\geq \frac{\hbar}{4πk_{B}}sη​≥4πkB​ℏ​ for the ratio between the shear viscosity and the entropy density, sss. Using Dynamical Mean-Field Theory (DMFT) we calculate the shear viscosity and entropy density for a fermionic fluid described by a single band Hubbard model at half filling. Our calculated shear viscosity as a function of temperature is compared with experimental data for liquid 3^{3}3He. At low temperature the shear viscosity is found to be well above the quantum limit and is proportional to the characteristic Fermi liquid 1/T21/T^{2}1/T2 dependence, where TTT is the temperature. With increasing temperature and interaction strength UUU there is significant deviation from the Fermi liquid form. Also, the shear viscosity violates the quantum limit near the crossover from coherent quasi-particle based transport to incoherent transport (the bad metal regime). Finally, the ratio of the shear viscosity to the entropy density is found to be comparable to the KSS bound for parameters appropriate to liquid 3^{3}3He. However, this bound is found to be strongly violated in the bad metal regime for parameters appropriate to lattice electronic systems such as organic charge transfer salts.
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 anti-de Sitter/conformal field theory correspondence (AdS/CFT) has been used to determine a lower bound on the ratio of shear viscosity to entropy density ( s ) for strongly-coupled field theories with a gravity dual. The conjectured universal lower bound, given as , is a measure of interaction strength in a quantum fluid where equality indicates a perfect quantum fluid. In this paper we study η / s in a Fermi gas in the unitary limit. We show that in addition to a local minimum for η / s at which obeys the lower bound, a more interesting result exists in the violation of the η / s lower bound due to the superfluid fluctuations above T c . To conclude, we examine the viscoelastic properties of the unitary Fermi gas. Previous work brought to light the connection between violation of the η / s bound and a viscoelastic response in the context of holographic solids. We ultimately find that, in addition to holographic solids, all Fermi liquids with a viscoelastic response produced by superfluid fluctuations can violate the universal η / s lower bound.
This thesis considers out-of-equilibrium dynamics of strongly interacting non-relativistic Fermi gases in several two and three dimensional geometries. The tools of second-order hydrodynamics and gauge-gravity duality will be utilized to address this system. Many of the themes of this work are motivated by the observed similarities in transport properties between strongly interacting Fermi gases and other very different strongly interacting quantum fluids such as the quark-gluon plasma, high temperature superconductors, and quantum field theories described by gauge-gravity duality. In particular, these systems all nearly saturate the conjectured lower bound on the ratio of shear viscosity to entropy density η/s≥ℏ/(4πkB)η/s \geq \hbar/(4 πk_B)η/s≥ℏ/(4πkB​) coming from the AdS/CFT correspondence. Among other things, this observation, in conjunction with current experiment and data analysis in atomic, condensed matter, and nuclear physics lends itself to the following questions: How perfect of a fluid is the strongly interacting Fermi gas, and can one find a more stringent constraint on η/sη/sη/s in Fermi gases? Do the similarities in transport properties among strongly interacting quantum systems extend beyond dynamics controlled by the hydrodynamical shear viscosity? In regards to the first question, by utilizing second-order hydrodynamics, it will be demonstrated that higher-order collective modes of a harmonically trapped Fermi gas may serve as a more sensitive probe of the shear viscosity. For the second question, both second-order hydrodynamics and a gravity dual theory are used to make predictions about dynamics occurring on short timescales where hydrodynamics is expected to break down. In particular the appearance of a class of "non-hydrodynamic" collective modes not contained within a Navier-Stokes description of the strongly interacting Fermi gas will be discussed.