Optically Engineering the Topological Properties of a Spin Hall Insulator

Time-periodic perturbations can be used to engineer topological properties of matter by altering the Floquet band structure. This is demonstrated for the helical edge state of a spin Hall insulator in the presence of monochromatic circularly polarized light. The inherent spin structure of the edge state is influenced by the Zeeman coupling and not by the orbital effect. The photocurrent (and the magnetization along the edge) develops a finite, helicity-dependent expectation value and turns from dissipationless to dissipative with increasing radiation frequency, signalling a change in the topological properties. The connection with Thouless’ charge pumping and nonequilibrium zitterbewegung is discussed, together with possible experiments.

Figure 1

The quantum spin Hall insulator (light yellow rectangle) with its helical edge state (counterpropagating red and blue arrows) in a circularly polarized electromagnetic field with frequency $\omega$ and wave vector $k$. In the plane $z=0$, the rotating magnetic field $\mathbf{B}(t)=B_0(\cos \omega t,\sin \omega t)$ is perpendicular to the ${\sigma }^z$ direction (vertical green arrows). A small tilting from the $z$ axis does not influence our results.

Figure 2

The induced photocurrent (blue solid line) and the ${\mathcal{C}}_{-}$ Chern number (black dashed line) are shown as a function of the frequency $\omega$. The photocurrent roughly behaves as $⟨j⟩\approx e\mathrm{sgn}(\omega )\min (g,|\omega |/2)/\pi$, and it is maximal at the transition $|\omega |=4g$ between the low- and the high-frequency regimes. The Chern number becomes nonquantized when band touching occurs at $4g=|\omega |$.

Figure 3

The induced dc (by the Zeeman term, solid line) $\pm$ the ac (by the vector potential, dashed lines) currents are plotted as a function of the vector potential, for several values of $g$; $j_{\mathrm{ac}}$ is always smaller than $j_{\mathrm{dc}}$ for physically relevant parameters. The current behaves as $⟨j⟩\approx j_{\mathrm{dc}}+j_{\mathrm{ac}}\cos (\omega t)$.