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

TL;DR

This paper investigates how electric field-induced Rashba spin-orbit coupling and electron-electron interactions influence the RKKY interaction in helical edge states of quantum spin Hall insulators, revealing new interaction terms and controllable quantum phase transitions.

Contribution

It introduces the impact of Rashba coupling and electron interactions on RKKY interactions, including new nematic terms and phase transition conditions, in helical liquids.

Findings

Rashba coupling introduces nematic-type RKKY terms.

Electron interactions modify the decay of RKKY range functions.

Quantum phase transition occurs at a specific Luttinger parameter value.

Abstract

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 originally, with the range functions depending 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 $1/|x|$ 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.