Magnetic correlation between two local spins in a quantum spin Hall insulator

Two spins located at the edge of a quantum spin Hall insulator may interact with each other via indirect spin-exchange interaction mediated by the helical edge states, namely, the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which can be measured by the magnetic correlation between the two spins. By means of the newly developed natural orbitals renormalization group (NORG) method, we investigated the magnetic correlation between two Kondo impurities interacting with the helical edge states, based on the Kane-Mele model defined in a finite zigzag graphene nanoribbon (ZGNR) with spin-orbital coupling (SOC). We find that the SOC effect breaks the symmetry in spatial distribution of the magnetic correlation, leading to anisotropy in the RKKY interaction. Specifically, the total correlation is always ferromagnetic (FM) when the two impurities are located at the same sublattice, while it is always antiferromagnetic (AFM) when at the different sublattices. Meanwhile, the behavior of the in-plane correlation is consistent with that of the total correlation. However, the out-of-plane correlation can be tuned from FM to AFM by manipulating either the Kondo coupling or the interimpurity distance. Furthermore, the magnetic correlation is tunable by the SOC, especially that the out-of-plane correlation can be adjusted from FM to AFM by increasing the strength of SOC. Dynamic properties of the system, represented by the spin-staggered excitation spectrum and the spin-staggered susceptibility at the two impurity sites, are finally explored. It is shown that the spin-staggered susceptibility is larger when the two impurities are located at the different sublattices than at the same sublattice, which is consistent with the behavior of the out-of-plane correlation. On the other hand, our study further demonstrates that the NORG is an effective numerical method for studying the quantum impurity systems.

Automated summary

The interaction between two spins at the edge of a quantum spin Hall insulator was studied, along with the magnetic correlation between two Kondo impurities in a graphene nanoribbon with spin-orbital coupling. The study revealed how the spatial symmetry and anisotropy in the magnetic correlation were affected by the strength of SOC and the position of the impurities, demonstrating the effectiveness of NORG in studying quantum impurity systems.