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arXiv:1905.05133 · Physical Review A

High-temperature expansion of the viscosity in interacting quantum gases

Johannes Hofmann

Published Jan 16, 2020• 42 citations
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Abstract

The frequency-dependent shear and bulk viscosity spectral functions of an interacting Fermi gas at high temperature are computed using a systematic virial expansion of the stress tensor correlation function. The field-theoretical methods employed have interesting relations with many-body techniques, such as the Keldysh formalism and can be compared with numerical calculations.

DOI
10.1103/physreva.101.013620

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The frequency-dependent shear and bulk viscosity spectral functions of an interacting Fermi gas at high temperature are computed using a systematic virial expansion of the stress tensor correlation function. The field-theoretical methods employed have interesting relations with many-body techniques, such as the Keldysh formalism and can be compared with numerical calculations.
The viscosity of strongly interacting systems is a topic of great interest in diverse fields. We focus here on the bulk and shear viscosities of nonrelativistic quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$, respectively, to derive exact, nonperturbative results. Our results include a microscopic connection between the shear viscosity $\ensuremath{\eta}$ and the normal-fluid density ${\ensuremath{\rho}}_{n}$; sum rules for $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$ and their evolution through the BCS-BEC crossover (where BEC denotes Bose-Einstein condensate); and universal high-frequency tails for $\ensuremath{\eta}(\ensuremath{\omega})$ and the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$. We use our sum rules to show that, at unitarity, $\ensuremath{\zeta}(\ensuremath{\omega})$ is identically zero and thus relate $\ensuremath{\eta}(\ensuremath{\omega})$ to density-density correlations. We predict that frequency-dependent shear viscosity $\ensuremath{\eta}(\ensuremath{\omega})$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.
The bulk viscosity determines dissipation during hydrodynamic expansion. It vanishes in scale invariant fluids, while a nonzero value quantifies the deviation from scale invariance. For the dilute Fermi gas the bulk viscosity is given exactly by the correlation function of the contact density of local pairs. As a consequence, scale invariance is broken purely by pair fluctuations. These fluctuations give rise also to logarithmic terms in the bulk viscosity of the high-temperature nondegenerate gas. For the quantum degenerate regime I report numerical Luttinger-Ward results for the contact correlator and the dynamical bulk viscosity throughout the BEC-BCS crossover. The ratio of bulk to shear viscosity ζ/η is found to exceed the kinetic theory prediction in the quantum degenerate regime. Near the superfluid phase transition the bulk viscosity is enhanced by critical fluctuations and has observable effects on dissipative heating, expansion dynamics, and sound attenuation.
We briefly review current research progress in the field of strongly correlated atomic Fermi gases. A deep understanding of strongly correlated quantum gases is plagued by the lack of either exact solutions or a controllable small interaction parameter. In this work, we present a systematic virial expansion method to study the high-temperature thermodynamics of a strongly correlated Fermi gas,where the controllable parameter is given by the fugacity,exp (μkBT),with μ being the chemical potential. A practical way is proposed to obtain the expansion coefficients for both homogeneous and harmonically trapped Fermi gases. We calculate, for the first time accurately, the third-order virial coefficient. The resulting equation of states at high temperatures is compared with a recent thermodynamic measurement and with Monte Carlo simulations.
One-dimensional Bose and Fermi gases with contact interactions are known to exhibit the weak-strong duality, where the equilibrium thermodynamic properties of one system at weak coupling are identical to those of the other system at strong coupling. Here, we show that such duality extends beyond the thermodynamics to the frequency-dependent complex bulk viscosity, which is provided by the contact-contact response function. In particular, we confirm that the bulk viscosities of the Bose and Fermi gases agree in the high-temperature limit, where the systematic expansion in terms of fugacity is available at arbitrary coupling. We also compute their bulk viscosities perturbatively in the weak-coupling limit at arbitrary temperature, which via the duality serve as those of the Fermi and Bose gases in the strong-coupling limit.
In this thesis, cold gases as well as the electron gas are analyzed with two expansion techniques that have, in essence, the same aim: To describe parameter regimes where the few-body aspect dominates over the collective many-body behavior.The Operator Product Expansion (OPE) of quantum field theory is used to extract high-frequency properties of the correlation function that determines the radio-frequency (RF) response of a two-dimensional Fermi gas with zero-range interactions.Exact sum rules that relate the zeroth and first moment of the radio-frequency spectrum to the particle number and the contact of the gas, are derived from these properties.A quantum cluster (virial-) expansion is employed to study the properties of the spectral function of the two-dimensional Fermi gas at small to moderate degeneracy.From the spectral function, observables like the density of states and the momentum distribution are calculated and are found to be in excellent agreement with Tan's universal relations.In the second part of the thesis, the cluster expansion is applied to both bosonic and fermionic three-dimensional quantum gases.Virial coefficients, contacts and momentum distribution are calculated up to third order in the fugacity.This includes three-particle correlations exactly.The results are are in excellent agreement with known universal relations.In the third part of this thesis, an OPE is applied to the problem of the interacting electron gas in both two and three dimensions.A power-law decay of the momentum distribution and the static structure factor at high momenta is found, compatible with earlier findings by J. Kimball.
In interaction-dominated two-dimensional electron gases at intermediate temperatures, electron transport is not diffusive as in the conventional Drude picture but instead hydrodynamic. The relevant transport coefficient in this regime is the shear viscosity. Here, we develop a numerically exact basis expansion to solve the Fermi liquid equation, and apply it to compute the shear viscosity of the electron gas with screened Coulomb interactions. Our calculations are valid at all temperatures and in particular describe the response beyond the asymptotic low-temperature limit, where perturbative approaches exist. We show that even in this low-temperature limit, there is a nonanalytic exchange contribution to the shear viscosity, highlighting the need for a full nonperturbative solution of the Fermi liquid equation. We hope that the techniques developed in this work will serve as a platform to determine the response of interacting Fermi liquids.