Non-Linear Spin Susceptibility in Topological Insulators

We theoretically study the effect of impurity resonances on the indirect exchange interaction between magnetic impurities in the surface states of a three-dimensional topological insulator. The interaction is composed of an isotropic Heisenberg, and anisotropic Ising and Dzyaloshinskii-Moriya contributions. We find that all three contributions are finite at the Dirac point, which is in stark contrast to the linear response theory which predicts a vanishing Dzyaloshinskii-Moriya contribution. We show that the spin-independent component of the impurity scattering can generate large values of the DM term in comparison with the Heisenberg and Ising terms, while these latter contributions drastically reduce in magnitude and undergo sign changes. As a result, both collinear and non-collinear configurations are allowed magnetic configurations of the impurities.