  Recently, it has been suggested that topological superconductivity and
Majorana end states can be realized in a chain of magnetic impurities on the
surface of an s-wave superconductor when the magnetic moments form a spin helix
as a result of the RKKY interaction mediated by the superconducting substrate.
Here, we investigate this scenario theoretically by developing a tight-binding
Bogoliubov-de Gennes description starting from the Shiba bound states induced
by the individual magnetic impurities. While the resulting model Hamiltonian
has similarities with the Kitaev model for one-dimensional spinless p-wave
superconductors, there are also important differences, most notably the
long-range nature of hopping and pairing as well as the complex hopping
amplitudes. We use both analytical and numerical approaches to explore the
consequences of these differences for the phase diagram and the localization
properties of the Majorana end states when the Shiba chain is in a topological
superconducting phase.
