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arXiv:1801.08627 · Journal of Physics Communications

Viscosity bound violation in viscoelastic Fermi liquids

M P Gochan, Hua Li, K S Bedell

Published Jun 1, 2019• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2399-6528/ab292b

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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.
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 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 calculate the ratio of the viscosity to the entropy density for both Bose and Fermi gases in the unitary limit using a new approach to the quantum statistical mechanics of gases based on the S -matrix. In the unitary limit the scattering length diverges and the S -matrix equals − 1. For the fermion case, we obtain η/ s >4.7 times the proposed lower bound of ℏ/4π k B , which came from the anti-Desitter space/conformal field theory correspondence (AdS/CFT) for gauge theories, consistent with the most recent experiments. For the bosonic case, we present evidence that the gas undergoes a phase transition to a strongly interacting Bose–Einstein condensate and is a more perfect fluid, with η/ s <1.3 times the bound.
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.
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 $\mathrm{ln}(1/\ensuremath{\omega})$.