Electrical control and interaction effects of the RKKY interaction in helical liquids

We study the RKKY interaction mediated by the helical edge states of a quantum spin Hall insulator in the presence of the Rashba spin-orbital coupling induced by an external electric field and the electron-electron interaction. We show that in the presence of the Rashba coupling, the RKKY interaction induced by the helical edge states contains not only the Heisenberg-like and the Dzyaloshinskii-Moria terms, but also the nematic-type term that is not present in the absence of the Rashba coupling. Moreover, the range functions depend on the strength of the Rashba coupling. We also show that the electron-electron interaction changes the strength of the RKKY interaction by modifying the power of the $1/\left|x\right|$ dependence of the range functions. In particular, by varying the strength of the interaction or the Rashba coupling, there is an (impurity) quantum phase transition involving the sign change of the RKKY interaction at the value of the Luttinger liquid parameter $K=1/2$. Since both the strength of the Rashba coupling and the chemical potential of the helical edge states are electrically controllable by external gate voltages, our results not only shed light on the nature of magnetic impurity correlations in the edge of a two-dimensional topological insulator, but also pave a way to manipulate the qubits in quantum computing.

Figure 1

Schematic setup of two local spins ${\mathit{S}}_1$ and ${\mathit{S}}_2$ located near the helical edge states of a QSHI that occupies the $xy$ plane with $y<0$. The local spins can be realized by quantum dots with strong on-site interactions. Both are exchange coupled to the spin density of the helical edge states, denoted by the dashed lines. The solid lines show the directions of the momenta of the edge states, with the associated spins indicated by the arrows.

Figure 2

The occupation probability $P_{+-}$ of the configuration $|+-\rangle$ as a function of $\alpha /v_F$. We have set $\left|\mu x\right|/v_F=0.1\pi$. The solid and dashed lines correspond to $K>1/2$ and $1/4<K<1/2$, respectively.