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arXiv:1504.07131 · Physical Review B

Shear viscosity of strongly interacting fermionic quantum fluids

Nandan Pakhira, Ross H. McKenzie

Published Sep 1, 2015• 9 citations
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Abstract

Eighty years ago Eyring proposed that the shear viscosity of a liquid, $η$, has a quantum limit $η\gtrsim n\hbar$ where $n$ 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 $\fracη{s}\geq \frac{\hbar}{4πk_{B}}$ for the ratio between the shear viscosity and the entropy density, $s$. 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 $^{3}$He. At low temperature the shear viscosity is found to be well above the quantum limit and is proportional to the characteristic Fermi liquid $1/T^{2}$ dependence, where $T$ is the temperature. With increasing temperature and interaction strength $U$ 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 $^{3}$He. 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.

DOI
10.1103/physrevb.92.125103

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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.
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.
Shear viscosity is a measure of the amount of dissipation in a simple fluid. In kinetic theory shear viscosity is related to the rate of momentum transport by quasi-particles, and the uncertainty relation suggests that the ratio of shear viscosity η to entropy density s in units of ℏ/ k B is bounded by a constant. Here, ℏ is Planck's constant and k B is Boltzmann's constant. A specific bound has been proposed on the basis of string theory where, for a large class of theories, one can show that η/ s ⩾ ℏ/(4π k B ). We will refer to a fluid that saturates the string theory bound as a perfect fluid. In this review we summarize theoretical and experimental information on the properties of the three main classes of quantum fluids that are known to have values of η/ s that are smaller than ℏ/ k B . These fluids are strongly coupled Bose fluids, in particular liquid helium, strongly correlated ultracold Fermi gases and the quark gluon plasma. We discuss the main theoretical approaches to transport properties of these fluids: kinetic theory, numerical simulations based on linear response theory and holographic dualities. We also summarize the experimental situation, in particular with regard to the observation of hydrodynamic behavior in ultracold Fermi gases and the quark gluon plasma.
In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $({k}_{B}/\ensuremath{\hbar})\phantom{\rule{0.16em}{0ex}}\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and sss is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension d=2d=2d=2 by an expansion below d=5/2d=5/2d=5/2. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\ensuremath{\eta}$ scales in the same manner as a chiral conductivity, and the ratio $\ensuremath{\eta}/s$ diverges at low temperature (TTT) as ${T}^{\ensuremath{-}2/z}$, where zzz is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.
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 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.