Exchange and collective behavior of magnetic impurities in a disordered helical metal

We study the exchange interaction and the subsequent collective behavior of magnetic impurities embedded in a disordered two-dimensional helical metal. The exchange coupling follows a statistical distribution whose moments are calculated to the lowest order in ${\left(p_F\ell \right)}^{-1}$, where $p_F$ is the Fermi momentum of itinerant electrons and $\ell$ is the mean free path. We find that (i) the first moment of the distribution decays exponentially, and (ii) the variance of the interaction is long range, however, it becomes independent of the orientation of the localized magnetic moments due to the locking between spin and momentum of the electrons that mediate the interaction. As a consequence, long-range magnetic order tends to be suppressed, and a spin glass phase emerges. The formalism is applied to the surface states of a three-dimensional topological insulator. The lack of a net magnetic moment in the glassy phase and the full randomization of spin polarization at distances larger than $\ell$ excludes a spectral gap for surface states. Hence, nonmagnetic disorder may explain the dispersion in results for photoemission experiments in magnetically doped topological insulators.

Figure 1

Sketch of the RKKY interaction mediated by itinerant electrons in a disordered helical metal.

Figure 2

Diffuson and cooperon ladder contributions to ${\chi }_{\alpha \beta }{\chi }_{{\alpha }^{\prime }{\beta }^{\prime }}$. The latin indices refer to spin components of the diffuson/cooperon kernel, and the greek indices label the spin operators at the vertex.

Figure 3

Phase diagram of magnetic adatoms randomly distributed over the surface of a 3D topological insulator, showing paramagnetic (PM), ferromagnetic (FM), modified ferromagnetic $({\mathrm{FM}}^{\prime }$) and spin glass (SG) behavior. The results are based on the SK model for $\delta \ll 1$.