Corner Junction as a Probe of Helical Edge States

We propose and analyze interedge tunneling in a quantum spin Hall corner junction as a means to probe the helical nature of the edge states. We show that electron-electron interactions in the one-dimensional helical edge states result in Luttinger parameters for spin and charge that are intertwined, and thus rather different from those for a quantum wire with spin rotation invariance. Consequently, we find that the four-terminal conductance in a corner junction has a distinctive form that could be used as evidence for the helical nature of the edge states.

Figure 1

Proposed geometry of a point contact device for probing the helical nature of QSH edge states. The painted (pale grey) area defines the bulk of the QSH insulator. The red (solid) line represents the up-spin right-movers while the blue (dashed) line represents the down-spin left-movers for both edges. Electrons can tunnel between the two edges at $x_{1,2}=0$, where the corners come into close proximity. There are four contact leads, where voltages $V_i$ can be applied, before and after the tunneling point.

Figure 2

(a) The phase diagram of a junction of two spinful LL quantum wires. The black curve indicates the trajectory that respects the HLL constraint $g_c\times g_s=1$. The label $C$ stands for conducting while the label I stands for insulating phase. The adjacent labeling, $AB$, indicates the phase $A$ and phase $B$ for the charge and spin degrees of freedom, respectively. Depending on the detailed structure of the point contact, both $CC$ and II fixed points can be stable at low energies, and this is represented in the dashed area. (b) shows the phase diagram of the point contact between two HLL edge states in terms of $g_c$, obtained by following the $g_c\times g_s=1$ trajectory in (a).

Figure 3

Proposed measurement for detecting the IC fixed point and the helical nature of the QSH edge states. A positive voltage $V_1$ and a negative voltage $V_3$ are applied to leads 1 and 3, such that $V_1>|V_3|$, while leads 2 and 4 are grounded. The arrows indicate the direction of the current flow and the quantized conductance $e^2/h=1$. Notice that currents flow out from lead 3, even though it has the lowest applied voltage.