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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 (2D) 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 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 (3D) 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, non-magnetic disorder may explain the dispersion in results for photoemission experiments in magnetically-doped topological insulators.