Current Correlations in Quantum Spin Hall Insulators

We consider a four-terminal setup of a two-dimensional topological insulator (quantum spin Hall insulator) with local tunneling between the upper and lower edges. The edge modes are modeled as helical Luttinger liquids and the electron-electron interactions are taken into account exactly. Using perturbation theory in the tunneling, we derive the cumulant generating function for the interedge current. We show that different possible transport channels give rise to different signatures in the current noise and current cross correlations, which could be exploited in experiments to elucidate the interplay between electron-electron interactions and the helical nature of the edge states.

Figure 1

The setup under investigation consists of a two-dimensional topological insulator with two helical edge modes. The four corners are connected to electron reservoirs held at bias voltages ${\mu }_{A,B,C,D}$. Using top gates, the upper and lower channels can be coupled locally, leading to interedge tunneling at $x=0$.

Figure 2

Graphical depiction of charged-pair tunneling ($T_c$) and spinful-pair tunneling ($T_s$) according to Eqs. (9, 10). Because of the helicity, a single $T_c$ event turns, e.g., two right movers into two left movers and thus changes the charge transferred from left to right by two. $T_s$ affects the spin sector in an analogous way. Both processes give rise to cross correlations between spin-up and spin-down tunnel currents $⟨⟨\delta N_{\uparrow }\delta N_{\downarrow }⟩⟩$. As $T_c$ transports two particles in opposite directions, whereas $T_s$ transports two particles in the same direction, the sign of the cross correlations is different for both processes.