Dynamics of transport by helical edge states

Abstract

Topologically nontrivial band structure of a material may give rise to special states that are confined to the material's boundary and protected against disorder and scattering. Quantum spin Hall effect (QSHE) is a paradigmatic example of phenomenon in which such states appear in the presence of time-reversal symmetry in two dimensions. Whereas the spatial structure of these helical edge states has been largely studied, their dynamic properties are much less understood. We design a microwave experiment mimicking QSHE and explore the spatiotemporal dynamics of unidirectional transport of optical angular momentum (or pseudospin) by edge states. Pseudospin-polarized signal propagation is shown to be immune to scattering by defects introduced along the edge. Its velocity is 2 to 3 orders of magnitude slower than the speed of light in the free space, which may have important consequences for practical applications of topological edge states in modern optical and quantum-information technologies.

Figures (4)

• Figure 1: (a) Experimental setup. $N = 384$ dielectric cylinders of 3 mm in radius and 5 mm in height, dielectric constant $\epsilon = 45$, are grouped in six-cylinder hexagonal clusters of side $R$. In their turn, the clusters are arranged in a triangular lattice with $3a = 30$ mm lattice spacing and placed on the lower of the two reflecting aluminum plates separated by a distance of $h = 13$ mm and forming a Fabry-Perot cavity with no propagating TE modes. TE-polarized microwaves are emitted by the first of the two loop antennas introduced into the cavity via small holes in the plates. The second antenna can move together with the upper plate. (b) Band structure of an infinite ribbon (see inset for a sketch of its single unit cell repeated along dotted lines) divided in two halves with $R < a$ on the left and $R > a$ on the right, and having the same types of boundaries as our sample, in the tight-binding approximation. ${\mathbf k}_{\text{edge}}$ is parallel to ribbon edges. Color lines show bands corresponding to the edge states at the interface between the two halves of the sample (yellow) and at the sample boundary (blue). A (mini-)gap between edge states is due to the breakdown of the $C_6$ symmetry at an edge. Calculations done using PythTB Coh_2024. (c) Gray-scale plot of measured DOS compared to a calculation of band edges in the tight-binding model (red lines), as a function of frequency and deformation $R/a$. The apparent persistence of the gap in the measured DOS for $R/a = 1$ is due to finite frequency resolution of our measurements. (d) Spin Chern number $C_{\text{SC}}$ for the infinite lattice (circles) and spin Bott index $C_{\text{SB}}$ for lattices of two different sizes and open (OBC) or periodic (PBC) boundary conditions. Both $C_{\text{SC}}$ and $C_{\text{SB}}$ are calculated for the middle of the spectral gap in panel (c).
• Figure 2: Unidirectional propagation of microwaves along edges of a sample with a topologically nontrivial band gap. The left half of the sample is topologically trivial ($R/a = 0.94$) whereas the right half is topologically nontrivial ($R/a = 1.06$). The color code shows the time-integrated orbital angular momentum of light (pseudospin) in six-cylinder clusters $J_z({\mathbf r})$. The pulsed source position is indicated by an antenna pictogram near the top or bottom of the boundary between the two halves of the sample.
• Figure 3: Velocity of propagation of the orbital angular momentum $v_J$ (solid orange circles) and group velocity $v_g$ (open green squares) along the interface between topologically distinct parts of the sample compared to the theoretical predictions for the group velocity of the corresponding tight-binding model (black dashed line), and a 1D chain of coupled harmonic oscillators (blue dash-dotted line). Six-cylinder cluster size is $R$ and $2a-R$ for the two topologically distinct parts of the sample, respectively. Data are averaged over several central frequencies $f_c$ of the source within the band gap.
• Figure 4: Topological protection of edge states against disorder. (a) Color-scale plot of the time-integrated pseudospin $J_z( {\mathbf r})$ for excitation by a loop antenna located at the top of the sample. A defect consisting of two six-cylinder clusters of topologically trivial type ($R/a = 0.94$, same as in the topologically trivial left half of the sample) is protruding into the topologically nontrivial right half of the sample where $R/a = 1.06$. (b) Temporal profiles of $J_z( {\mathbf r},t)$ for clusters marked as 1 and 2 in panel (a). Panels (c) and (d) are the same as panels (a) and (b) except for the emitting antenna at the bottom of the sample.