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 $\delta \eta \left(T\right)\propto ln(T-T_c)$. 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.

Figure 1

These diagrams define the main contributions to viscosity discussed and calculated in the main text. (a) This “Drude-like diagram” defines viscosity of a Fermi liquid with short-range interactions. The interactions give rise to a finite relaxation rate encoded in the Green's functions (the solid lines). The short double wavy lines correspond to the viscosity vertices $p_x{p}_y/m$. (b) The long wavy line is the superconducting fluctuation propagator [see Eq. (5)]. It diverges at the transition point for $Q=0$. (c) The Aslamazov-Larkin (AL) diagram for viscosity, which corresponds to the viscosity of the fluid of fluctuating pairs. (d) The Maki-Thomson (MT) diagram for viscosity, which corresponds to scattering of electrons off of the fluctuating pairs. (e) and (f) The density of states (DOS) diagrams for viscosity, which describe the deficit of single-electron excitations contributing to $\eta$, because some electrons participate in fluctuating pairing.

Figure 2

Single-electron Green's function renormalized by pairing fluctuations. The self-energy (12) gives rise to a strongly enhanced relaxation term calculated in Eq. (13).