Tunable and direction-dependent group velocities in topologically protected edge states

Topological effects continue to fascinate physicists since more than three decades. One of their main applications are high-precision measurements of the resistivity. We propose to make also use of the spatially separated edge states. It is possible to realize strongly direction-dependent group velocities. They can also be tuned over orders of magnitude so that robust and tunable delay lines and interference devices are within reach.

Figure 1

Sketch of a strip of the honeycomb lattice with NN hopping (black bonds) and NNN hopping (colored arrows). Note that the color of the arrows is swapped starting from the other sublattice. The strip height is $N$ units. The upper boundary is decorated by weakly coupled $(\lambda \le 1$) additional sites with local potential $\delta$. The lattice constant $a$ is set to unity.

Figure 2

Dispersion of the edge modes: in green (dark gray), the left mover at the upper edge and in orange (light gray) the right mover at the lower edge for $\lambda =0.2$. The shaded areas show the continua from the Chern bands. The inset shows the Fermi velocity $v_{\mathrm{F}}=\partial \omega /\partial k_x{|}_{{\epsilon }_{\mathrm{F}}}$ relative to the one at the undecorated edge vs the relative coupling $\lambda$ at $t_2=0.2t$ and $\varphi =\pi /2$.

Figure 3

(Top) Dispersion of the edge modes; in green (dark gray) the left mover at the upper edge and in orange (light gray) the right mover at the lower edge for $\lambda =0.2$ and $\delta =0.1t$. The shaded areas show the continua from Chern bands. (Bottom) Fermi velocity $v_{\mathrm{F}}=\partial \omega /\partial k_x{|}_{{\epsilon }_{\mathrm{F}}}$ relative to the one at the undecorated edge vs the gate voltage $\delta$ for various values of the relative coupling $\lambda$ at $t_2=0.2t$ and $\varphi =\pi /2$.

Figure 4

(Left) Sketch of a circuit to use the tunable delay line in an interference measurement for the determination of the delay of signal transmission in a sample; almost destructive interference is shown. (Right) Sketch of a decorated Hall sample to realize tunable and direction-dependent Fermi velocities. The black hatched area is impenetrable for the electrons, while the potential of the colored areas (blue (dark gray) and red (light gray) can be tuned independently by gate voltages ${\delta }_{\mathrm{low}}$ and ${\delta }_{\mathrm{up}}$, while the bulk of the sample (green) remains unchanged. A magnetic field is applied perpendicular to the plane shown to generate the edge states. It should be approximately so large that circular Landau orbits fit into the bays.