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arXiv:1612.07729 · Physical Review A

Relation connecting thermodynamics and transport of atomic unitary Fermi superfluids

Hao Guo, Weimin Cai, Yan He, Chih-Chun Chien

Published Mar 31, 2017• 3 citations
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Abstract

The shear viscosity has been shown to be equal to the product of the pressure and relaxation time in normal scale-invariant fluids, but the presence of superfluidity at low temperatures can alter the relation. Using the mean-field BCS-Leggett theory with a gauge-invariant linear response theory for unitary Fermi superfluids, we present an explicit relation between thermodynamic quantities, including the pressure and chemical potential, and transport coefficients, including the shear viscosity, superfluid density, and anomalous shear viscosity from momentum transfer via Cooper pairs. The relation is modified when pairing fluctuations associated with noncondensed Cooper pairs are considered. Within a pairing fluctuation theory consistent with the BCS-Leggett ground state, we found an approximate relation for unitary Fermi superfluids. The exact mean-field relation and the approximate one with pairing fluctuations advance our understanding of relations between equilibrium and transport quantities in superfluids, and they help determine or constrain quantities which otherwise can be difficult to measure.

DOI
10.1103/physreva.95.033638

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The shear viscosity has been shown to be equal to the product of the pressure and relaxation time in normal scale-invariant fluids, but the presence of superfluidity at low temperatures can alter the relation. Using the mean-field BCS-Leggett theory with a gauge-invariant linear response theory for unitary Fermi superfluids, we present an explicit relation between thermodynamic quantities, including the pressure and chemical potential, and transport coefficients, including the shear viscosity, superfluid density, and anomalous shear viscosity from momentum transfer via Cooper pairs. The relation is modified when pairing fluctuations associated with noncondensed Cooper pairs are considered. Within a pairing fluctuation theory consistent with the BCS-Leggett ground state, we found an approximate relation for unitary Fermi superfluids. The exact mean-field relation and the approximate one with pairing fluctuations advance our understanding of relations between equilibrium and transport quantities in superfluids, and they help determine or constrain quantities which otherwise can be difficult to measure.
The shear viscosity has been an important topic in ultracold Fermi gases, and it has served as a diagnostic of various theories. Due to the complicated phase structures of population-imbalanced (polarized) Fermi gases with tunable attraction, past works on the shear viscosity mainly focused on unpolarized Fermi gases. Here we investigate the shear viscosity of homogeneous, population-imbalanced Fermi superfluid at finite temperatures by a pairing fluctuation theory for thermodynamical quantities and a gauge-invariant linear response theory for transport coefficients. The Cooper pairs lead to the anomalous shear viscosity analogous to the shear viscosity. We derive an exact relation connecting certain thermodynamic quantities and transport coefficients at the mean-field level for polarized unitary Fermi superfluids. An approximate relation beyond mean-field is proposed and only exhibits mild deviations from our numerical results. In the unitary and Bose-Einstein condensation (BEC) regimes, the total shear viscosity increases with the polarization because the excess majority fermions cause gapless excitations acting like a normal fluid. Moreover, competition among the excess fermions, noncondensed pairs, and fermionic quasiparticles may lead to non-monotonic behavior of the ratio between the shear viscosity and relaxation time as the polarization increases.
We theoretically investigate the shear viscosity ηηη in the BCS-BEC crossover regime of an ultracold Fermi gas with a Feshbach resonance. Within the framework of the strong-coupling self-consistent TTT-matrix approximation, we examine how a strong pairing interaction associated with a Feshbach resonance affects this transport coefficient, in the normal state above the superfluid phase transition temperature TcT_{\rm c}Tc​. We show that, while ηηη diverges in both the weak-coupling BCS and strong-coupling BEC limits, it becomes small in the unitary regime. The minimum of ηηη is obtained, not at the unitarity, but slightly in the strong-coupling BEC side. This deviation is consistent with the recent experiment on a 6^66Li Fermi gas. In the weak-coupling BCS regime, we also find that ηηη exhibits anomalous temperature dependence near TcT_{\rm c}Tc​, which is deeply related to the pseudogap phenomenon originating form strong pairing fluctuations.
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.
In this paper we address the ratio of the shear viscosity to entropy density $\ensuremath{\eta}/s$ in bosonic and fermionic superfluids. A small $\ensuremath{\eta}/s$ is associated with nearly perfect fluidity, and more general measures of the fluidity perfection/imperfection are of wide interest to a number of communities. We use a Kubo approach to concretely address this ratio via low-temperature transport associated with the quasiparticles. Our analysis for bosonic superfluids utilizes the framework of the one-loop Bogoliubov approximation, whereas for fermionic superfluids we apply BCS theory and its BCS-BEC extension. Interestingly, we find that the transport properties of strict BCS and Bogoliubov superfluids have very similar structures, albeit with different quasiparticle dispersion relations. While there is a dramatic contrast between the power law and exponential temperature dependence for $\ensuremath{\eta}$ alone, the ratio $\ensuremath{\eta}/s$ for both systems is more similar. Specifically, we find the same linear dependence (on the ratio of temperature TTT to inverse lifetime $\ensuremath{\gamma}(T))$ with $\ensuremath{\eta}/s\ensuremath{\propto}T/\ensuremath{\gamma}(T)$, corresponding to imperfect fluidity. By contrast, near the unitary limit of BCS-BEC superfluids a very different behavior results, which is more consistent with near-perfect fluidity.
We compute second-order transport coefficients of the dilute Fermi gas at unitarity. The calculation is based on kinetic theory and the Boltzmann equation at second order in the Knudsen expansion. The second-order transport coefficients describe the shear stress relaxation time, nonlinear terms in the strain-stress relation, and nonlinear couplings between vorticity and strain. An exact calculation in the dilute limit gives ${\ensuremath{\tau}}_{R}=\ensuremath{\eta}/P$, where ${\ensuremath{\tau}}_{R}$ is the shear stress relaxation time, $\ensuremath{\eta}$ is the shear viscosity, and PPP is pressure. This relation is identical to the result obtained using the Bhatnagar-Gross-Krook approximation to the collision term, but other transport coefficients are sensitive to the exact collision integral.
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.