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) 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
0.5TF, thus invalidating a quasiparticle description. The ratio
η/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) by a factor of about seven.
The viscosity of strongly interacting systems is a topic of great interest in diverse fields. We focus here on the bulk and shear viscosities of nonrelativistic quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$, respectively, to derive exact, nonperturbative results. Our results include a microscopic connection between the shear viscosity $\ensuremath{\eta}$ and the normal-fluid density ${\ensuremath{\rho}}_{n}$; sum rules for $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$ and their evolution through the BCS-BEC crossover (where BEC denotes Bose-Einstein condensate); and universal high-frequency tails for $\ensuremath{\eta}(\ensuremath{\omega})$ and the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$. We use our sum rules to show that, at unitarity, $\ensuremath{\zeta}(\ensuremath{\omega})$ is identically zero and thus relate $\ensuremath{\eta}(\ensuremath{\omega})$ to density-density correlations. We predict that frequency-dependent shear viscosity $\ensuremath{\eta}(\ensuremath{\omega})$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.
This dissertation describes the first experimental measurement of the energy and interaction dependent shear viscosity
eta and bulk viscosity
zeta in the hydrodynamic expansion of a two-component Fermi gas near a broad collisional (Feshbach) resonance. This expansion also provides a precise test of scale invariance and an examination of local thermal equilibrium as a function of interaction strength. After release from an anisotropic optical trap, we observe that a resonantly interacting gas obeys scale-invariant hydrodynamics, where the mean square cloud size
langlemathbfr2rangle=langlex2+y2+z2rangle expands ballistically (like a noninteracting gas) and the energy-averaged bulk viscosity is consistent with zero,
0.00(0.04),hbar,n, with
n the density. In contrast, the aspect ratios of the cloud exhibit anisotropic ``elliptic" flow with an energy-dependent shear viscosity. Tuning away from resonance, we observe conformal symmetry breaking, where
langlemathbfr2rangle deviates from ballistic flow. We find that
eta has both a quadratic and a linear dependence on the interaction strength
1/(kFIa), where
a is the s-wave scattering length and
kFI is the Fermi wave vector for an ideal gas at the trap center. At low energy, the minimum is less than the resonant value and is significantly shifted toward the BEC side of resonance, to
1/(kFIa)=0.2.
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.