Viscosity sum rules at large scattering lengths

Abstract

We use the operator product expansion (OPE) and dispersion relations to obtain model-independent ``Borel-resummed'' sum rules for both shear and bulk viscosity of many-body systems of spin-1/2 fermions with predominantly short-range $S$-wave interactions. These sum rules relate Gaussian weights of the frequency-dependent viscosities to the Tan contact parameter $\mathcal{C}(a)$. Our results are valid for arbitrary values of the scattering length $a$, but receive small corrections from operators of dimension $\ensuremath{\Delta}>5$ in the OPE and can be used to study transport properties in the vicinity of the $a\ensuremath{\rightarrow}\ensuremath{\infty}$ fixed point. In particular, we find that the exact dependence of the shear viscosity sum rule on scattering length is controlled by the function $\mathcal{C}(a)$. The sum rules that we obtain depend on a frequency scale ${\ensuremath{\omega}}_{0}$ that can be optimized to maximize their overlap with low-energy data.