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arXiv:1003.2601 · Physical Review A

Thermal conductivity and sound attenuation in dilute atomic Fermi gases

Matt Braby, Jingyi Chao, Thomas Schäfer

Published Sep 27, 2010• 51 citations
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Abstract

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.

DOI
10.1103/physreva.82.033619

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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, Tc{T}_{c}Tc​, the quasiparticles are fermions, whereas below Tc{T}_{c}Tc​, the dominant excitations are phonons. We calculate the thermal conductivity in both cases. We find that at unitarity the thermal conductivity \ensuremathκ\ensuremath{\kappa}\ensuremathκ in the normal phase scales as \ensuremathκ\ensuremath∝T3/2\ensuremath{\kappa}\ensuremath{\propto}{T}^{3/2}\ensuremathκ\ensuremath∝T3/2. In the superfluid phase we find \ensuremathκ\ensuremath∝T2\ensuremath{\kappa}\ensuremath{\propto}{T}^{2}\ensuremathκ\ensuremath∝T2. 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 Tc{T}_{c}Tc​ 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.
Recent experiments on sound waves in a unitary Fermi gas reveal many transport properties about strongly interacting fermions. Sound propagates through the coupling of momentum and heat transport, and attenuates strongly with the presence of a phase transition. In this work, focusing on the temperature regimes near and below the superfluid critical temperature [Formula: see text] in the BCS-BEC crossover, we present a Kubo-based microscopic calculation of thermal conductivity [Formula: see text], which has not attracted much attention compared to the shear viscosity. Our approach primarily addresses the contributions of the fermionic quasiparticles to thermal transport and our results return to the kinetic descriptions at high temperatures. [Formula: see text] drops upon crossing the pseudogap temperature [Formula: see text], and its temperature dependence changes below [Formula: see text]. The drops become more pronounced on the weakly coupled BCS side, where the Pauli blocking causes the upturn of [Formula: see text] above [Formula: see text]. Our calculations fit well with the sound measurement on the damping rate.
We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length aaa, including the unitarity limit a→∞a\to\inftya→∞. 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 ω0≃(2ϵ)/(3η)\omega_0\simeq (2\epsilon)/(3\eta)ω0​≃(2ϵ)/(3η) where ϵ\epsilonϵ is the energy density and η\etaη is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of order ω0∼k(T/m)1/2\omega_0\sim k(T/m)^{1/2}ω0​∼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 .
We compute the shear viscosity of a superfluid atomic Fermi gas in the unitarity limit. The unitarity limit is characterized by a divergent scattering length between the atoms, and it has been argued that this will result in a very small viscosity. We show that in the low temperature TTT 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 Tc{T}_{c}Tc​. 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 $\ensuremath{\eta}∕s\ensuremath{\ge}\ensuremath{\hbar}∕(4\ensuremath{\pi}{k}_{B})$.
We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, nnn 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.
Transport of strongly interacting fermions is crucial for the properties of modern materials, nuclear fission, the merging of neutron stars, and the expansion of the early Universe. Here, we observe a universal quantum limit of diffusivity in a homogeneous, strongly interacting atomic Fermi gas by studying sound propagation and its attenuation through the coupled transport of momentum and heat. In the normal state, the sound diffusivity D monotonically decreases upon lowering the temperature, in contrast to the diverging behavior of weakly interacting Fermi liquids. Below the superfluid transition temperature, D attains a universal value set by the ratio of Planck's constant and the particle mass. Our findings inform theories of fermion transport, with relevance for hydrodynamic flow of electrons, neutrons, and quarks.