  Sequential plateau transitions of quantum spin chains ($S$=1,3/2,2 and 3) are
demonstrated by a spin pump using dimerization and staggered magnetic field as
synthetic dimensions. The bulk is characterized by the Chern number associated
with the boundary twist and the pump protocol as a time. It counts the number
of critical points in the loop that is specified by the $Z_2$ Berry phases.
With open boundary condition, discontinuity of the spin weighted center of mass
due to emergent effective edge spins also characterizes the pump as the bulk
edge correspondence. It requires extra level crossings in the pump as a
super-selection rule that is consistent with the Valence Bond Solid (VBS)
picture.
