Viscosity spectral functions of the dilute Fermi gas in kinetic theory

Jingyi Chao, Thomas Schäfer, Matt Braby

Published Jul 2, 2021• 4 citations

Abstract

We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length $a$, including the unitarity limit $a\to\infty$. We perform the calculation in kinetic theory by studying the response to a non-trivial background metric. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is $\omega_0\simeq (2\epsilon)/(3\eta)$ where $\epsilon$ is the energy density and $\eta$ is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of order $\omega_0\sim k(T/m)^{1/2}$.

DOI: 10.17615/h5kq-d915

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Viscosity spectral functions of the dilute Fermi gas in kinetic theory

Jingyi Chao, Thomas Schäfer, Matt Braby

Published Jul 2, 2021• 4 citations

Publisher

Abstract

We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length

a

, including the unitarity limit

a\to\infty

. We perform the calculation in kinetic theory by studying the response to a non-trivial background metric. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is

\omega_0\simeq (2\epsilon)/(3\eta)

where

\epsilon

is the energy density and

\eta

is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of order

\omega_0\sim k(T/m)^{1/2}

DOI: 10.17615/h5kq-d915

Related papers

Viscosity spectral functions of the dilute Fermi gas in kinetic theory

Matt Braby, Jingyi Chao, Thomas Schäfer

We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length a , including the unitarity limit a → ∞ . We perform the calculation in kinetic theory by studying the response to a non-trivial background metric. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is ω 0 ≃(2ε)/(3η) where ε is the energy density and η is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of the order of ω 0 ∼ k ( T / m ) 1/2 .We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length a , including the unitarity limit a → ∞ . We perform the calculation in kinetic theory by studying the response to a non-trivial background metric. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is ω 0 ≃(2ε)/(3η) where ε is the energy density and η is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of the order of ω 0 ∼ k ( T / m ) 1/2 .

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We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length

a

, including the unitarity limit

a\to\infty

\omega_0\simeq (2\epsilon)/(3\eta)

where

\epsilon

is the energy density and

\eta

is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of order

\omega_0\sim k(T/m)^{1/2}

n

is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.

We compute the thermal conductivity and sound attenuation length of a dilute atomic Fermi gas in the framework of kinetic theory. Above the critical temperature for superfluidity,

{T}_{c}

, the quasiparticles are fermions, whereas below

{T}_{c}

, the dominant excitations are phonons. We calculate the thermal conductivity in both cases. We find that at unitarity the thermal conductivity $\ensuremath{\kappa}$ in the normal phase scales as $\ensuremath{\kappa}\ensuremath{\propto}{T}^{3/2}$. In the superfluid phase we find $\ensuremath{\kappa}\ensuremath{\propto}{T}^{2}$. At high temperature the Prandtl number, the ratio of the momentum and thermal diffusion constants, is 2/3. The ratio increases as the temperature is lowered. As a consequence we expect sound attenuation in the normal phase just above

{T}_{c}

to be dominated by shear viscosity. We comment on the possibility of extracting the shear viscosity of the dilute Fermi gas at unitarity using measurements of the sound absorption length.

(P-\frac{2}{3}{\cal E})/P

, where

P

is the pressure and

{\cal E}

is the energy density. In the high temperature limit this ratio scales as

\frac{zλ}{a}

, where

z

is the fugacity,

λ

is the thermal wave length, and

a

is the scattering length. We show that the bulk viscosity

ζ

scales as the second power of this parameter,

ζ\sim (\frac{zλ}{a})^2 λ^{-3}

T

limit the shear viscosity scales as ${\ensuremath{\xi}}^{5}∕{T}^{5}$, where the universal parameter $\ensuremath{\xi}$ relates the chemical potential and the Fermi energy, $\ensuremath{\mu}=\ensuremath{\xi}{\ensuremath{\epsilon}}_{F}$. Combined with the high temperature expansions of the viscosity our results suggest that the viscosity has a minimum near the critical temperature

{T}_{c}

. A na\"{\i}ve extrapolation indicates that the minimum value of the ratio of viscosity over entropy density is within a factor of $\ensuremath{\sim}5$ of the proposed bound $\ensuremath{\eta}∕s\ensuremath{\ge}\ensuremath{\hbar}∕(4\ensuremath{\pi}{k}_{B})$.

P

is the pressure and

\mathcal{E}

is the energy density. At high temperature this ratio scales as $z\ensuremath{\lambda}/a$, where

z

is the fugacity, $\ensuremath{\lambda}$ is the thermal wavelength, and

a

is the scattering length. We show that the bulk viscosity $\ensuremath{\zeta}$ scales as the second power of this parameter, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.