  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.
