Non-adiabatic topological spin pumping
  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. 
W. Y. Deng, W. Luo, H. Geng, M. N. Chen, L. Sheng and D. Y. Xing
cond-mat.mes-hall