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arXiv:1409.5820 · Physical Review B

Nearly perfect fluidity in a high-temperature superconductor

J. D. Rameau, Theodore James Reber, H. B. Yang, Shimul Akhanjee, Genda D. Gu, P. D. Johnson, S. Campbell

Published Oct 13, 2014• 36 citations
arXivHTMLPDF

Abstract

Perfect fluids are characterized as having the smallest ratio of shear viscosity to entropy density, $\ensuremath{\eta}/s$, consistent with quantum uncertainty and causality. So far, nearly perfect fluids have only been observed in the quark-gluon plasma and in unitary atomic Fermi gases, exotic systems that are amongst the hottest and coldest objects in the known universe, respectively. We use angle resolved photoemission spectroscopy to measure the temperature dependence of an electronic analog of $\ensuremath{\eta}/s$ in an optimally doped cuprate high-temperature superconductor, finding it too is a nearly perfect fluid around, and above, its superconducting transition temperature ${T}_{c}$.

DOI
10.1103/physrevb.90.134509

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Perfect fluids are characterized as having the smallest ratio of shear viscosity to entropy density, \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/s, consistent with quantum uncertainty and causality. So far, nearly perfect fluids have only been observed in the quark-gluon plasma and in unitary atomic Fermi gases, exotic systems that are amongst the hottest and coldest objects in the known universe, respectively. We use angle resolved photoemission spectroscopy to measure the temperature dependence of an electronic analog of \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/s in an optimally doped cuprate high-temperature superconductor, finding it too is a nearly perfect fluid around, and above, its superconducting transition temperature Tc{T}_{c}Tc​.
In this paper, we examine in a unified fashion dissipative transport in strongly correlated systems. We thereby demonstrate the connection between "bad metals" (such as the high temperature superconductors) and "perfect fluids" (such as the ultracold Fermi gases, near unitarity). One motivation of this work is to communicate to the high energy physics community some of the central unsolved problems in high TcT_cTc​ superconductors. Because of interest in the nearly perfect fluidity of the cold gases and because of new tools such as the AdS/CFT correspondence, this better communication may lead to important progress in a variety of different fields. A second motivation is to draw attention to the great power of transport measurements which more directly reflect the excitation spectrum than, say, thermodynamics and thus strongly constrain microscopic theories of correlated fermionic superfluids. Our calculations show that bad metal and perfect fluid behavior is associated with the presence of a normal state excitation gap which suppresses the effective number of carriers leading to anomalously low conductivity and viscosity above the transition temperature TcT_cTc​. Below TcT_cTc​ we demonstrate that the condensate collective modes ("phonons") do not couple to transverse probes such as the shear viscosity. As a result, our calculated shear viscosity at low TTT becomes arbitrarily small as observed in experiments. In both homogeneous and trap calculations we do not find the upturn in ηηη or η/sη/sη/s (where sss is the entropy density) found in most theories. In the process of these studies we demonstrate compatibility with the transverse sum rule and find reasonable agreement with both viscosity and cuprate conductivity experiments.
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.
Experimental results obtained at the Relativistic Heavy Ion Collider (RHIC) have been interpreted in terms of a strongly interacting quark gluon plasma. The strongly interacting plasma is characterized by ``perfect fluidity'', i.e. a ratio of shear viscosity to entropy density that saturates a proposed lower bound. In this contribution we explore the possibility that a similar phenomenon takes place in a strongly coupled non-relativistic Fermi liquid in which the scattering length between the Fermions is infinitely large.
We present an ab initio calculation of the shear viscosity as a function of interaction strength in a two-component unpolarized Fermi gas near the unitary limit, within a finite temperature quantum Monte Carlo (QMC) framework and using the Kubo linear-response formalism. The shear viscosity decreases as we tune the interaction strength 1akF\frac{1}{a{k}_{F}}akF​1​ from the Bardeen-Cooper-Schrieffer side of the Feshbach resonance towards Bose-Einstein condensation limit and it acquires the smallest value for $\frac{1}{a{k}_{F}}\ensuremath{\approx}0.4$, with a minimum value of ${\frac{\ensuremath{\eta}}{s}|}_{\text{min}}\ensuremath{\approx}0.2\frac{\ensuremath{\hbar}}{{k}_{B}}$, which is about twice as small as the value reported for experiments in quark-gluon plasma ${\frac{\ensuremath{\eta}}{s}|}_{\text{QGP}}\ensuremath{\lesssim}0.4\frac{\ensuremath{\hbar}}{{k}_{B}}$. The Fermi gas near unitarity thus emerges as the most ``perfect fluid'' observed so far in nature. The clouds of dilute Fermi gas near unitarity exhibit the unusual attribute that, for the sizes realized so far in the laboratory, or larger (less than 109{10}^{9}109 atoms), can sustain quantum turbulence below the critical temperature, but at the same time the classical turbulence is suppressed in the normal phase.
In this paper we address the ratio of the shear viscosity to entropy density $\ensuremath{\eta}/s$ in bosonic and fermionic superfluids. A small $\ensuremath{\eta}/s$ is associated with nearly perfect fluidity, and more general measures of the fluidity perfection/imperfection are of wide interest to a number of communities. We use a Kubo approach to concretely address this ratio via low-temperature transport associated with the quasiparticles. Our analysis for bosonic superfluids utilizes the framework of the one-loop Bogoliubov approximation, whereas for fermionic superfluids we apply BCS theory and its BCS-BEC extension. Interestingly, we find that the transport properties of strict BCS and Bogoliubov superfluids have very similar structures, albeit with different quasiparticle dispersion relations. While there is a dramatic contrast between the power law and exponential temperature dependence for $\ensuremath{\eta}$ alone, the ratio $\ensuremath{\eta}/s$ for both systems is more similar. Specifically, we find the same linear dependence (on the ratio of temperature TTT to inverse lifetime $\ensuremath{\gamma}(T))$ with $\ensuremath{\eta}/s\ensuremath{\propto}T/\ensuremath{\gamma}(T)$, corresponding to imperfect fluidity. By contrast, near the unitary limit of BCS-BEC superfluids a very different behavior results, which is more consistent with near-perfect fluidity.
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.