Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. I. Continuum description

We consider subgap bands induced in a two-dimensional superconductor by a densely packed chain of magnetic moments with ferromagnetic or spiral alignments. We show that by contrast with sparsely packed chains a consistent description requires that all wavelengths are taken into account for the scattering at the magnetic moments. The resulting subgap states are a composition of Yu-Shiba-Rusinov-type states and magnetic scattering states, whose mixture becomes especially important to understand the nature and dimensional renormalization of gap closures for spiral magnetic alignments under increasing scattering strength, particularly as the spiral becomes commensurate with the Fermi wavelength. The results are fully analytic in the form of Green's functions and provide the tools for further analysis of the properties of the subgap states.