Viscosity in dilute Fermi gases: spectral functions and sum rules

Recently there has been considerable interest in the viscosity of strongly interacting systems, especially in regimes where the mean free path is of the order of the interparticle spacing and a quasiparticle description breaks down. We derive exact results for the spectral functions and sum rules for the shear and bulk viscosities, ${\rm Re}\ \eta(\omega)$ and ${\rm Re}\ \zeta(\omega)$, of strongly interacting Fermi and Bose systems. The zero-frequency limits of these functions give the viscosities measured in hydrodynamic damping experiments. For a two-component Fermi gas, we find the exact sum rules $\int^{\infty}_{0}d\omega\; {\rm Re}\eta(\omega)/{\pi}={\varepsilon}/{3}-{2\langle V\rangle}/{5}$ and $\int^{\infty}_{0}d\omega\; {\rm Re}\ \zeta(\omega)/{\pi} ={(\varepsilon +P)}/{3}-{\rho c^2_{{s}}}/{2}$, where $\varepsilon$ is the internal energy density, $\langle V\rangle$ the potential energy density, $P$ the pressure, $\rho$ the mass density, and $c_{{s}}$ the speed of sound. These results are valid at all temperatures and for all values of $1/(k_F a)$ through the BCS-BEC crossover. We will discuss the implications of these sum rules including universal high-frequency tails of the spectral functions.