Viscosity of strongly interacting quantum fluids: spectral functions and sum rules

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 \emph{non-relativistic} quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\zeta(\omega)$ and $\eta(\omega)$ respectively, to derive exact, non-perturbative results. Our results include: a microscopic connection between the shear viscosity $\eta$ and the normal fluid density $\rho_n$; sum rules for $\zeta(\omega)$ and $\eta(\omega)$ and their evolution through the BCS-BEC crossover; universal high-frequency tails for $\eta(\omega)$ and the dynamic structure factor $S({\bf q}, \omega)$. We use our sum rules to show that, at unitarity, $\zeta(\omega)$ is identically zero and thus relate $\eta(\omega)$ to density-density correlations. We predict that frequency-dependent shear viscosity $\eta(\omega)$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.