arXiv:cond-mat/0511721 · Physical Review Letters

Vanishing Bulk Viscosities and Conformal Invariance of the Unitary Fermi Gas

Dam Thanh Son

Published Jan 11, 2007• 141 citations

Abstract

By requiring general-coordinate and conformal invariance of the hydrodynamic equations, we show that the unitary Fermi gas has zero bulk viscosity, $\ensuremath{\zeta}=0$, in the normal phase. In the superfluid phase, two of the bulk viscosities have to vanish, ${\ensuremath{\zeta}}_{1}={\ensuremath{\zeta}}_{2}=0$, while the third one ${\ensuremath{\zeta}}_{3}$ is allowed to be nonzero.

DOI: 10.1103/physrevlett.98.020604

Related papers

Bulk viscosities for cold Fermi superfluids close to the unitary limit

Miguel Ángel Escobedo, Massimo Mannarelli, Cristina Manuel

arXiv:0904.3023Physical Review AJun 22, 200941 citations

Bulk Viscosity and Contact Correlations in Attractive Fermi Gases

Tilman Enss

arXiv:1904.12772Physical Review LettersNov 15, 201943 citations

Bulk viscosity and conformal symmetry breaking in the dilute Fermi gas near unitarity

Thomas Schäfer, Kevin Dusling

arXiv:1305.4688May 21, 20137 citations

Viscosity and scale invariance in the unitary Fermi gas

Tilman Enss, R. Haussmann, W. Zwerger

Jan 20, 2016164 citations

Determination of the Density and Temperature Dependence of the Shear Viscosity of a Unitary Fermi Gas Based on Hydrodynamic Flow

Marcus Bluhm, Jiaxun Hou, Thomas Schäfer

arXiv:1704.03720Physical Review LettersAug 10, 201736 citations

Bulk Viscosity and Conformal Symmetry Breaking in the Dilute Fermi Gas near Unitarity

Kevin Dusling, Thomas Schäfer

Physical Review LettersSep 19, 201342 citations

arXiv:cond-mat/0511721 · Physical Review Letters

Vanishing Bulk Viscosities and Conformal Invariance of the Unitary Fermi Gas

Dam Thanh Son

Published Jan 11, 2007• 141 citations

arXiv HTML PDF

Abstract

By requiring general-coordinate and conformal invariance of the hydrodynamic equations, we show that the unitary Fermi gas has zero bulk viscosity,

\ensuremath{\zeta}=0

, in the normal phase. In the superfluid phase, two of the bulk viscosities have to vanish,

{\ensuremath{\zeta}}_{1}={\ensuremath{\zeta}}_{2}=0

, while the third one

{\ensuremath{\zeta}}_{3}

is allowed to be nonzero.

DOI: 10.1103/physrevlett.98.020604

Related papers

Bulk viscosities for cold Fermi superfluids close to the unitary limit

Miguel Ángel Escobedo, Massimo Mannarelli, Cristina Manuel

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 ${\ensuremath{\zeta}}_{2}\ensuremath{\propto}1/{a}^{2}$.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 ${\ensuremath{\zeta}}_{2}\ensuremath{\propto}1/{a}^{2}$.

arXiv:0904.3023Physical Review AJun 22, 200941 citations

Bulk Viscosity and Contact Correlations in Attractive Fermi Gases

Tilman Enss

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.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.

arXiv:1904.12772Physical Review LettersNov 15, 201943 citations

Bulk viscosity and conformal symmetry breaking in the dilute Fermi gas near unitarity

Thomas Schäfer, Kevin Dusling

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 $(P-\frac{2}{3}{\cal E})/P$ , 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, $ζ\sim (\frac{zλ}{a})^2 λ^{-3}$ .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 $(P-\frac{2}{3}{\cal E})/P$, 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, $ζ\sim (\frac{zλ}{a})^2 λ^{-3}$.

arXiv:1305.4688May 21, 20137 citations

Viscosity and scale invariance in the unitary Fermi gas

Tilman Enss, R. Haussmann, W. Zwerger

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 $\eta(\omega)$ 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 $0.5 T_F$ , thus invalidating a quasiparticle description. The ratio $\eta/s$ 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 $\hbar/(4\pi k_B)$ by a factor of about seven.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 $\eta(\omega)$ 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 $0.5 T_F$, thus invalidating a quasiparticle description. The ratio $\eta/s$ 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 $\hbar/(4\pi k_B)$ by a factor of about seven.

Jan 20, 2016164 citations

Determination of the Density and Temperature Dependence of the Shear Viscosity of a Unitary Fermi Gas Based on Hydrodynamic Flow

Marcus Bluhm, Jiaxun Hou, Thomas Schäfer

We determine the shear viscosity of the ultracold Fermi gas at unitarity in the normal phase using hydrodynamic expansion data. The analysis is based on a generalized fluid dynamic framework which ensures a smooth transition between the fluid dynamic core of the cloud and the ballistic corona. We use expansion data taken by Joseph, Elliott, and Thomas [Shear Viscosity of a Universal Fermi Gas Near the Superfluid Phase Transition, Phys. Rev. Lett. 115, 020401 (2015).PRLTAO0031-900710.1103/PhysRevLett.115.020401] and measurements of the equation of state by Ku et al. [Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 563 (2012).SCIEAS0036-807510.1126/science.1214987]. We find that the shear viscosity to particle density ratio just above the critical temperature is η/n|_{T_{c}}=0.41±0.11. We also obtain evidence that the shear viscosity to entropy density ratio has a minimum slightly above T_{c} with η/s|_{min}=0.50±0.10.We determine the shear viscosity of the ultracold Fermi gas at unitarity in the normal phase using hydrodynamic expansion data. The analysis is based on a generalized fluid dynamic framework which ensures a smooth transition between the fluid dynamic core of the cloud and the ballistic corona. We use expansion data taken by Joseph, Elliott, and Thomas [Shear Viscosity of a Universal Fermi Gas Near the Superfluid Phase Transition, Phys. Rev. Lett. 115, 020401 (2015).PRLTAO0031-900710.1103/PhysRevLett.115.020401] and measurements of the equation of state by Ku et al. [Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 563 (2012).SCIEAS0036-807510.1126/science.1214987]. We find that the shear viscosity to particle density ratio just above the critical temperature is η/n|_{T_{c}}=0.41±0.11. We also obtain evidence that the shear viscosity to entropy density ratio has a minimum slightly above T_{c} with η/s|_{min}=0.50±0.10.

arXiv:1704.03720Physical Review LettersAug 10, 201736 citations

Bulk Viscosity and Conformal Symmetry Breaking in the Dilute Fermi Gas near Unitarity

Kevin Dusling, Thomas Schäfer

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 $P$ 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, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.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 $P$ 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, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.

Physical Review LettersSep 19, 201342 citations

By requiring general-coordinate and conformal invariance of the hydrodynamic equations, we show that the unitary Fermi gas has zero bulk viscosity,

\ensuremath{\zeta}=0

, in the normal phase. In the superfluid phase, two of the bulk viscosities have to vanish,

{\ensuremath{\zeta}}_{1}={\ensuremath{\zeta}}_{2}=0

, while the third one

{\ensuremath{\zeta}}_{3}

is allowed to be nonzero.

s

-wave scattering length,

a

, at the leading order in

1/a

we obtain that ${\ensuremath{\zeta}}_{1}\ensuremath{\propto}1/a$ and ${\ensuremath{\zeta}}_{2}\ensuremath{\propto}1/{a}^{2}$.

(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}

\eta(\omega)

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

0.5 T_F

, thus invalidating a quasiparticle description. The ratio

\eta/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

\hbar/(4\pi k_B)

by a factor of about seven.

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, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.