Topological classification of interaction-driven spin pumps

When adiabatically varied in time, certain one-dimensional band insulators allow for the quantized noiseless pumping of spin even in the presence of strong spin-orbit scattering. These spin pumps are closely related to the quantum spin Hall system, and their properties are protected by a time-reversal restriction on the pumping cycle. In this paper we study pumps formed of one-dimensional insulators with a time-reversal restriction on the pumping cycle and a bulk energy gap which arises due to interactions. We find that the correlated gapped phase can lead to novel pumping properties. In particular, systems with $d$ different ground states can give rise to $d+1$ different classes of spin pumps, including a trivial class which does not pump quantized spin and $d$ nontrivial classes allowing for the pumping of quantized spin $\hslash /n$ on average per cycle, where $1\le n\le d$. We discuss an example of a spin pump that transfers an average of spin $\hslash /2$ without transferring charge.

Figure 1

The three different classes of pumps for systems with two ground states corresponding to (left to right) the trivial class with $z_3=n\times 0=0$, the integer pump with $z_3=1\times 1$, and the fractional pump with $z_3=2\times 1$. Here, the upper circles depict the extended cycle in parameter space, and for illustration purposes we restrict the reflection matrix to the two-sphere.

Figure 2

An interaction-driven pump described by the Hamiltonian in Eq. (8) characterized by a $z_3=2$ index. The two ground states at $t=0$ and $t=T$ can be obtained from one another by creating a single spin flip in the bulk which then propagates to the edges of the wire. The system returns to its original state after two pumping cycles. After a single pumping cycle the spin-density waves for the the two orientations have shifted by half a wavelength, corresponding to the transfer of $1/2[\hslash /2-(-\hslash /2\left)\right]$ spin from the left to the right edge of the system, without transferring charge.