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

TL;DR

This paper investigates how magnetic impurities interact and behave collectively in a disordered 2D helical metal, revealing long-range interactions that lead to spin glass phases instead of magnetic order, with implications for topological insulators.

Contribution

It provides a statistical analysis of exchange interactions among magnetic impurities in disordered helical metals, highlighting the emergence of spin glass behavior and its effects on surface states of topological insulators.

Findings

First moment of interaction decays exponentially

Variance of interaction is long-range and orientation-independent

Spin glass phase suppresses magnetic order without opening a spectral gap

Abstract

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.