Optimal topological spin pump

We study the recently introduced ${\mathbb{Z}}_2$ pump consisting of a family of one-dimensional bulk insulators with time-reversal restriction on the pumping cycle. We find that the scattering matrices of these pumps are dichotomized by a topological index. We show that the class of pumps characterized by a nontrivial topological index allows, in contrast to its topologically trivial counterpart, for the noiseless pumping of quantized spin, even in the absence of spin conservation. This distinction sheds light on the ${\mathbb{Z}}_2$ classification of two-dimensional time-reversal invariant insulators.

Figure 1

(Color online) Time evolution of $\widehat{r}\left(t\right)$ of model, Eq. (10), in the absence of spin orbit (black curve), and for $\delta e/\delta \tau =0.3$ and $\Gamma /{\tau }_0=0.1$ (red/light gray curve). Squares indicate equally spaced time intervals and illustrate the time duration of the transition to $\widehat{r}=-\mathbb{1}$. The loops cross points $\pm \mathbb{1}$ at the TRIM 0 and $T/2$, respectively, corresponding to a nontrivial ${\mathbb{Z}}_2$ index $\nu =1$. Inset shows the time dependence of the $y$ component of ${\vec{e}}_{\varphi }\left(t\right)$ (black curve) and of $\varphi \left(t\right)$ for $\delta e/\delta \tau =0.3$ and $\Gamma /{\tau }_0=1$, 0.1, and 0.01 (dashed-dotted blue, dashed red, and dotted green lines, respectively). At $\Gamma /{\tau }_0\to 0$, the transition becomes sharper and $\widehat{r}\left(t\right)$ describes a rotation around a fixed axis, $e_{\varphi }\left(\vec{T}/2\right)$, implying that the pump is optimal.

Figure 2

(Color online) Time evolution of the $\widehat{r}\left(t\right)$ of model, Eq. (13), with ($\delta e/\delta \tau =0.1$, red/light gray curve) and without (black curve) spin-orbit coupling, for $\Gamma /{\tau }_0=0.1$. The loops cross two resonances during a cycle and thus belong to the trivial class of pumps. At finite spin-orbit coupling, $\widehat{r}\left(t\right)$ follows two different geodesics, with nonparallel tangents at the TRIM, indicated by the red arrows. The inset shows the time dependence of the $y$ component of ${\vec{e}}_{\varphi }\left(t\right)$ (black curve) and of $\varphi \left(t\right)$ for $\delta e/\delta \tau =0.1$ and $\Gamma /{\tau }_0=0.1$ (red/light gray curve). The different values of ${\vec{e}}_{\varphi }\left(t_i\right)$ at the resonances, $\varphi =\pi$, are marked by the blue circles.