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We theoretically obtain the phase diagram of localized magnetic impurityspins arranged in a one-dimensional chain on top of a one- or two-dimensionalelectron gas with Rashba spin-orbit coupling. The interactions between thespins are mediated by the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanismthrough the electron gas. Recent work predicts that such a system mayintrinsically support topological superconductivity when a helical spin-densitywave is formed in the spins, and superconductivity is induced in the electrongas. We analyze, using both analytical and numerical techniques, the conditionsunder which such a helical spin state is stable in a realistic situation in thepresence of disorder. We show that it becomes unstable towards the formation of(anti) ferromagnetic domains if the disorder in the impurity spin positions$\delta R$ becomes comparable with the Fermi wave length. We also examine thestability of the helical state against Gaussian potential disorder in theelectronic system using a diagrammatic approach. Our results suggest that inorder to stabilize the helical spin state, and thus the emergent topologicalsuperconductivity, a sufficiently strong Rashba spin-orbit coupling, givingrise to Dzyaloshinskii-Moriya interactions, is required.

Helical order in one-dimensional magnetic atom chains and possible emergence of Majorana bound states