Collective Modes of a Helical Liquid

We study low energy collective modes and transport properties of the “helical metal” on the surface of a topological insulator. At low energies, electrical transport and spin dynamics at the surface are exactly related by an operator identity equating the electric current to the in-plane components of the spin degrees of freedom. From this relation it follows that an undamped spin wave always accompanies the sound mode in the helical metal—thus it is possible to “hear” the sound of spins. In the presence of long range Coulomb interactions, the surface plasmon mode is also coupled to the spin wave, giving rise to a hybridized “spin-plasmon” mode. We make quantitative predictions for the spin-plasmon in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, and discuss its detection in a spin-grating experiment.

Figure 1

Dispersion of the collective modes in the long wavelength regime. The plasmon (solid red line) is shown for $\alpha =0.6$ and the zero-sound (dashed blue line) mode is shown for $U=0.3U_c$. The shaded region denotes the particle-hole continuum where the imaginary parts of the density and spin susceptibilities are nonzero. The collective modes are long-lived in the unshaded region $v_{f}q<\omega <2\mu -v_{f}q$.

Figure 2

The spin-plasmon collective mode: a density fluctuation (green surface) induces a transverse spin-fluctuation (red arrows) or vice versa. To detect the spin plasmon, a spin-density wave is generated by a spin grating. Two orthogonally polarized noncollinear incident beams (relative angle of $2\theta$) induce a spin polarization wave. The transverse component of the photon helicity is $P\cos \theta \cos \phi$, where $P$ is the total photon helicity amplitude and $\phi$ is the beam plane makes with the surface. The induced plasmon charge oscillation can be detected by conventional means.