  Based on the Floquet scattering theory, we analytically investigate the
topological spin pumping for an exactly solvable model. Floquet spin Chern
numbers are introduced to characterize the periodically time-dependent system.
The topological spin pumping remains robust both in the presence and in the
absence of the time-reversal symmetry, as long as the pumping frequency is
smaller than the band gap, where the electron transport involves only the
Floquet evanescent modes in the pump. For the pumping frequency greater than
the band gap, where the propagating modes in the pump participate in the
electron transport, the spin pumping rate decays rapidly, marking the end of
the topological pumping regime.
