Shear viscosity of strongly interacting fermionic quantum fluids

Abstract: Eighty years ago Eyring proposed that the shear viscosity of a liquid, $\eta$, has a quantum limit $\eta \gtrsim n\hbar$ where $n$ is the density of the fluid. Using holographic duality and the AdS/CFT correspondence in string theory Kovtun, Son, and Starinets (KSS) conjectured a universal bound $\frac{\eta}{s}\geq \frac{\hbar}{4\pi k_{B}}$ for the ratio between the shear viscosity and the entropy density, $s$. Using Dynamical Mean-Field Theory (DMFT) we calculate the shear viscosity and entropy density for a fermionic fluid described by a single band Hubbard model at half filling. Our calculated shear viscosity as a function of temperature is compared with experimental data for liquid $^{3}$He. At low temperature the shear viscosity is found to be well above the quantum limit and is proportional to the characteristic Fermi liquid $1/T^{2}$ dependence, where $T$ is the temperature. With increasing temperature and interaction strength $U$ there is significant deviation from the Fermi liquid form. Also, the shear viscosity violates the quantum limit near the crossover from coherent quasi-particle based transport to incoherent transport (the bad metal regime). Finally, the ratio of the shear viscosity to the entropy density is found to be comparable to the KSS bound for parameters appropriate to liquid $^{3}$He. However, this bound is found to be strongly violated in the bad metal regime for parameters appropriate to lattice electronic systems such as organic charge transfer salts.