  We theoretically obtain the phase diagram of localized magnetic impurity
spins arranged in a one-dimensional chain on top of a one- or two-dimensional
electron gas with Rashba spin-orbit coupling. The interactions between the
spins are mediated by the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism
through the electron gas. Recent work predicts that such a system may
intrinsically support topological superconductivity when a helical spin-density
wave is formed in the spins, and superconductivity is induced in the electron
gas. We analyze, using both analytical and numerical techniques, the conditions
under which such a helical spin state is stable in a realistic situation in the
presence 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 the
stability of the helical state against Gaussian potential disorder in the
electronic system using a diagrammatic approach. Our results suggest that in
order to stabilize the helical spin state, and thus the emergent topological
superconductivity, a sufficiently strong Rashba spin-orbit coupling, giving
rise to Dzyaloshinskii-Moriya interactions, is required.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons