Time-dependent Ginzburg-Landau theory of the vortex spin Hall effect

We develop a time-dependent Ginzburg-Landau theory of the vortex spin Hall effect, i.e., a spin Hall effect that is driven by the motion of superconducting vortices. For the direct vortex spin Hall effect in which an input charge current drives the transverse spin current accompanying the vortex motion, we start from the well-known Schmid-Caroli-Maki solution for the time-dependent Ginzburg-Landau equation under the applied electric field, and find out the expression of the induced spin current. For the inverse vortex spin Hall effect in which an input spin current drives the longitudinal vortex motion and produces the transverse charge current, we microscopically construct the time-dependent Ginzburg-Landau equation under the applied spin accumulation gradient, and calculate the induced transverse charge current as well as the open circuit voltage. The time-dependent Ginzburg-Landau equation and its analytical solution developed here can be a basis for more quantitative numerical simulations of the vortex spin Hall effect.