Topological quantized edge pumping spin flip in the Rice-Mele model with spin-orbit coupling

The quantized Thouless pumping charge in a spinless Rice-Mele model originates from a degeneracy point in the parameter space and cannot be detected when open boundary conditions are applied. In this work, we investigate the topological features of a spinful Rice-Mele model. We demonstrate that spin-orbit coupling facilitates the transition of a single degenerate point into a degenerate loop, which is anticipated to be the source of the topological characteristics. When periodic boundary conditions are considered, we find that the pumping spin is zero for an adiabatic loop within the nodal loop and is 2 (in units of $\ensuremath{\hbar}/2$) for an adiabatic passage enclosing the nodal loop. When open boundary conditions are considered, the boundary-bulk correspondence is demonstrated by quantized pumping spin flips at the edges, which can be obtained by completing double periods of a closed passage, rather than a single cycle. Our findings reveal an alternative dynamic manifestation of the boundary-bulk correspondence.