Point contacts and localization in generic helical liquids

We consider two helical liquids on opposite edges of a two-dimensional topological insulator, which are connected by one or several local tunnel junctions. In the presence of spatially inhomogeneous Rashba spin-orbit coupling, the spin of the helical edge states is momentum dependent, and this spin texture can be different on opposite edges. We demonstrate that this has a strong impact on the electron transport between the edges. In particular, in the case of many random tunnel contacts, the localization length depends strongly on the spin textures of the edge states.

Figure 1

For a single point contact, the angle $\theta \left(\mu \right)$ [see Eq. (11)] determines the branching ratio of right movers on the upper edge into right- and left-movers on the lower edge. The right panel shows the spectra of the upper and lower edges for $V=0$. $\theta \left(\mu \right)$ is given by the difference in spin axis rotation at the Fermi energy. For clarity, we set ${\vartheta }_1=0$, which corresponds to spin-conserving tunneling.

Figure 2

Conductance (in units of $e^2/h$) of a 2D topological insulator with two tunnel junctions as a function of the angle $\theta \left(\mu \right)$ and distance between tunnel junctions $\Delta x_2$, where ${\varphi }_2=\Delta x_2\mu /v_F$. We use ${\vartheta }_1={\vartheta }_2=0$, which corresponds to spin-conserving tunneling.

Figure 3

Charge puddles in a narrow 2D topological insulator between two edges can form tunnel paths at random positions $x_j$ and with random tunnel amplitudes ${\gamma }_j$. Tunneling leads to localization, with a localization length which depends strongly on the spin rotation angle $\theta \left(\mu \right)$.