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arXiv:0904.3023 · Physical Review A

Bulk viscosities for cold Fermi superfluids close to the unitary limit

Miguel Ángel Escobedo, Massimo Mannarelli, Cristina Manuel

Published Jun 22, 2009• 41 citations
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Abstract

We compute the coefficients of bulk viscosity for a nonrelativistic superfluid corresponding to a fermionic system close to the unitarity limit. We consider the low temperature regime assuming that the transport properties of the system are dominated by phonons. To compute the coefficients of bulk viscosity we use kinetic theory in the relaxation time approximation and the low energy effective field theory of the corresponding system. We show that the three independent bulk viscosity coefficients, ${\ensuremath{\zeta}}_{1},{\ensuremath{\zeta}}_{2},{\ensuremath{\zeta}}_{3}$, associated with irreversible flows vanish for phonons with a linear dispersion law. Considering a phonon dispersion law with a cubic term in momentum we find that in the conformal limit ${\ensuremath{\zeta}}_{1}={\ensuremath{\zeta}}_{2}=0$, while ${\ensuremath{\zeta}}_{3}$ is nonzero. Including a conformal breaking term that arises for a large but finite $s$-wave scattering length, $a$, at the leading order in $1/a$ we obtain that ${\ensuremath{\zeta}}_{1}\ensuremath{\propto}1/a$ and ${\ensuremath{\zeta}}_{2}\ensuremath{\propto}1/{a}^{2}$.

DOI
10.1103/physreva.79.063623

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Some results about dissipative processes in superfluids are presented. We focus on fermionic superfluidity and restrict our analysis to the contribution of phonons to bulk viscosity, shear viscosity and thermal conductivity. At sufficiently low temperatures phonons give the dominant contribution to the transport coefficients if all the other low energy excitation of the system are gapped. We first consider a system of cold fermionic atoms close to the unitarity limit. Then we turn to the superfluid phase of quark matter that may be realized at high baryonic density.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio (P−23E)/P(P-\frac{2}{3}{\cal E})/P(P−32​E)/P, where PPP is the pressure and E{\cal E}E is the energy density. In the high temperature limit this ratio scales as zλa\frac{zλ}{a}azλ​, where zzz is the fugacity, λλλ is the thermal wave length, and aaa is the scattering length. We show that the bulk viscosity ζζζ scales as the second power of this parameter, ζ∼(zλa)2λ−3ζ\sim (\frac{zλ}{a})^2 λ^{-3}ζ∼(azλ​)2λ−3.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute, in the high temperature limit, the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio $(P\ensuremath{-}2/3\mathcal{E})/P$, where PPP is the pressure and E\mathcal{E}E is the energy density. At high temperature this ratio scales as $z\ensuremath{\lambda}/a$, where zzz is the fugacity, $\ensuremath{\lambda}$ is the thermal wavelength, and aaa is the scattering length. We show that the bulk viscosity $\ensuremath{\zeta}$ scales as the second power of this parameter, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.
We study the recently proposed effective-field theory for the phonon of an arbitrary nonrelativistic superfluid. After computing the one-loop phonon self-energy, we obtain the low-temperature TTT contributions to the phonon dispersion law at low momentum and see that the real part of those can be parametrized as a thermal correction to the phonon velocity. Because the phonons are the quanta of the sound waves, at low momentum their velocity should agree with the speed of sound. We find that our results match at order T4lnT{T}^{4}\mathrm{ln}TT4lnT with those predicted by Andreev and Khalatnikov for the speed of sound, derived from the superfluid hydrodynamical equations and the phonon kinetic theory. We get also higher-order corrections of order T4{T}^{4}T4, which are not reproduced pushing naively the kinetic theory computation. Finally, as an application, we consider the cold Fermi gas in the unitarity limit and find a universal expression for the low-TTT relative correction to the speed of sound for these systems.
We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity η(ω)\eta(\omega)η(ω) exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of 0.5TF0.5 T_F0.5TF​, thus invalidating a quasiparticle description. The ratio η/s\eta/sη/s between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound ℏ/(4πkB)\hbar/(4\pi k_B)ℏ/(4πkB​) by a factor of about seven.
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.