Ruderman-Kittel-Kasuya-Yosida interaction in silicene

We investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities in a monolayer silicene. Due to spin-orbit (SO) coupling, the RKKY interaction can be further divided into various types according to the polarization directions of the magnetic impurities. We demonstrate that the spatial behaviors of the RKKY interaction closely relate to the external electric field, which together with the SO coupling gives rise to various phases in undoped silicene. Consequently, by probing the RKKY interaction, which contains all the information in the momentum space, differences between the topological phase and the trivial phase can be identified explicitly. Moreover, the change of chemical potential induced by the doping of silicene can also have a profound influence on the spatial behaviors of the RKKY interaction