Critical viscosity of a fluctuating superconductor

We consider a fluctuating superconductor in the vicinity of the transition temperature, ${T}_c$. The fluctuation shear viscosity is calculated. In two dimensions, the leading correction to viscosity is negative and scales as $\ensuremath{\delta}\ensuremath{\eta}(T)\ensuremath{\propto}ln(T\ensuremath{-}{T}_c)$. Critical hydrodynamics of the fluctuating superconductor involves two fluids: a fluid of fluctuating pairs and a quasiparticle fluid of single-electron excitations. The pair viscosity (Aslamazov-Larkin) term is shown to be zero. The (density of states) correction to viscosity of single-electron excitations is negative, which is due to fluctuating pairing that results in a reduction of electron density. Scattering of electrons off of the fluctuations gives rise to an enhanced quasiparticle scattering and another (Maki-Thomson) negative correction to viscosity. Our results suggest that fluctuating superconductors provide a promising platform to investigate low-viscosity electronic media and may potentially host fermionic/electronic turbulence. Some experimental probes of two-fluid critical hydrodynamics are proposed such as time-of-flight measurement of turbulent energy cascades in critical cold atom superfluids and magnetic dynamos in three-dimensional fluctuating superconductors.