Self-consistent theory of ferromagnetism on the surface of a topological insulator

The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities, mediated by Dirac surface states on the surface of a topological insulator, leads to impurities' ferromagnetic ordering. We present a self-consistent theory of the ordering, which takes into account a gap in the surface spectrum induced by the exchange field of magnetic impurities. We show that the gap does not change the general structure of RKKY interaction but considerably influences its strength. This feedback can be both positive and negative, depending on the ratio between the chemical potential and the gap, and it qualitatively modifies the temperature dependence of the spin polarization of magnetic impurities. The resulting unusual temperature dependence can be directly measured in angle resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) experiments.