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arXiv:1804.10209

Strongly Interacting Fermi Gases: Hydrodynamics and Beyond

William E. Lewis

Published Apr 26, 2018• 0 citations
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Abstract

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 $η/s \geq \hbar/(4 πk_B)$ 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 $η/s$ 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.

DOI
10.48550/arxiv.1804.10209

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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.
Unitary fermi gases have been widely studied as they provide a tabletop archetype for research on strongly coupled many body systems and perfect fluids. Research into unitary fermi gases can provide insight into may other strongly interacting systems including high temperature superconductor, quark-gluon plasmas, and neutron stars. Within the unitary regime, the equilibrium transport coefficients and thermodynamic properties are universal functions of density and temperature. Thus, unitary fermi gases provide a archetype to study nonperturbative many-body physics, which is of fundamental significance and crosses several fields.\nThis thesis reports on two topics regarding unitary fermi gases. A recent string theory conjecture gives a lower bound for the dimensionless ratio of shear viscosity of entropy, η=s ≥ 4πℏ =kb. Unitary fermi gases are a candidate for prefect fluids, yet η=s is well above the string theory bound. Using a stochastic formulation of hydrodynamics, we calculate a lower bound for this ratio accounting for the momentum dissipation from fluctuations. This lower bound is in good agreement with both theoretical and experimental results.\nThe second question addressed is the simulation of elliptic flow. Elliptic flow, first observed in 2002, is a characteristic of strongly coupled systems and has been studied in both quark-gluon plasmas and unitary fermi gases. As such, simulations of these systems are of interest. We test a variety of lattice Boltzmann models and compare the simulation results to the theoretical and experimental findings.
Very different strongly interacting quantum systems such as Fermi gases, quark-gluon plasmas formed in high-energy ion collisions, and black holes studied theoretically in string theory are known to exhibit quantitatively similar damping of hydrodynamic modes. It is not known if such similarities extend beyond the hydrodynamic limit. Do nonhydrodynamic collective modes in Fermi gases with strong interactions also match those from string theory calculations? In order to answer this question, we use calculations based on string theory to make predictions for modes outside the hydrodynamic regime in trapped Fermi gases. These predictions are amenable to direct testing with current state-of-the-art cold atom experiments.
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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.
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.