Proposal for a topological spin Chern pump

A one-dimensional electron model with parameters modulated adiabatically in closed cycles is proposed which can continuously pump spin to leads. By defining the spin-polarized Wannier functions, we reveal that the spin pumping is protected by the spin Chern numbers, so that it is stable to perturbations violating the time-reversal symmetry and spin conservation. Our work demonstrates the principle for realizing robust topological spin pumps independent of the time-reversal symmetry. A possible way to directly observe the bulk topological invariants is also suggested.

Figure 1

A zigzag chain of atoms is affixed on a substrate. Oscillatory shear deformation of the substrate is stimulated, resulting in a periodic modulation of hopping integrals between nearest-neighboring atomic sites. A nonuniform oscillating magnetic field is applied in such a manner that electrons on the $A$ and $B$ sites experience opposite Zeeman fields at any time.

Figure 2

Evolution of centers of the Wannier functions for the spin-up and spin-down sectors. For clarity, only ten unit cells of a long chain with periodic boundary condition are displayed, in which $10\%$ of the atoms are assumed to be replaced with magnetic impurities. The other parameters are taken as $t_1=g_0=0.1$, and $V_0=0$ (a), $V_0=2.5$ (b), and $V_0=4.5$ (c), with $t_0$ as the unit of energy.

Figure 3

Trajectories of $r_{\uparrow \uparrow }$ on the complex plane for three different strengths of impurity scattering potential.

Figure 4

Experimental detection of the spin Chern pumping effect. (a) When the spin polarization of the half-metallic lead is parallel to the $z$ axis, a quantized charge $(-e)$ will be pumped into the lead per cycle. (b) If the spin polarization is flipped, a quantized charge $(-e)$ will be pumped in the opposite direction.