Helical Quantum Edge Gears in 2D Topological Insulators

We show that two-terminal transport can measure the Luttinger liquid (LL) parameter $K$, in helical LLs at the edges of two-dimensional topological insulators (TIs) with Rashba spin-orbit coupling. We consider a Coulomb drag geometry with two coplanar TIs and short-ranged spin-flip interedge scattering. Current injected into one edge loop induces circulation in the second, which floats without leads. In the low-temperature ($T\to 0$) perfect drag regime, the conductance is $(e^2/h)(2K+1)/(K+1)$. At higher $T$, we predict a conductivity $\sim T^{-4K+3}$. The conductivity for a single edge is also computed.

Figure 1

Using helical quantum edge gears to measure the Luttinger parameter. We consider ${\mathbb{Z}}_2$ TI edge states in two adjacent topological regions. The blue and red arrows indicate the propagation directions of edge electrons with opposite helicities. The left TI is connected to external leads; $I_1$ and ${I_1}^{\prime }$ denote the currents of the edges connected to these. The right TI edge floats as an electrically isolated closed loop. Rashba spin-orbit coupling [1] enables Coulomb drag due to short-ranged spin-flip scattering [23] between the adjacent edges. This induces a current $I_2$ that circulates in the right edge. In the case of identical TIs with an interacting edge region of size $L\to \infty$, at zero temperature strong backscattering “locks” the currents $I_1=I_2$, associated to perfect drag [10, 12]. We then predict that the zero-temperature conductance is $G=(I_1+{I_1}^{\prime })/V=(e^2/h)(2K+1)/(K+1)$, where $K$ is the Luttinger parameter. In a real system of finite length $L\gg \xi$ and at temperatures $T$ satisfying $\hslash v/L\lesssim k_{B}T\ll \mathrm{\Delta }$ [11] with $\xi =\hslash v/\mathrm{\Delta }$ and $\mathrm{\Delta }$ the Mott gap of the antisymmetric mode, the result for $G$ holds up to terms exponentially small in $L/\xi$ and $\mathrm{\Delta }/k_{B}T$ [10, 11, 12]. Here $v$ is the charge velocity.