We apply mean-field theory and the Hirsch-Fye quantum Monte Carlo method to study the spin-spin interaction in the bulk of three-dimensional topological insulators. We find that the spin-spin interaction has three different components: longitudinal, transverse, and transverse Dzyaloshinskii-Moriya-like terms. When the Fermi energy is located in the bulk gap of a topological insulator, the spin-spin interaction decays exponentially due to the Bloembergen-Rowland interaction. The longitudinal correlation is antiferromagnetic and the transverse correlations are ferromagnetic. When the chemical potential is in the conduction or valence band, the spin-spin interaction follows a power-law decay, and the isotropic ferromagnetic interaction dominates in the short separation limit.

• Received 12 November 2013
• Revised 19 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.115101