Electronic states associated with a chain of magnetic adatoms on the surface of an ordinary $s$-wave superconductor have been shown theoretically to form a one-dimensional (1D) topological phase with unpaired Majorana fermions bound to its ends. In a simple 1D effective model the system exhibits an interesting self-organization property: The pitch of the spiral formed by the adatom magnetic moments tends to adjust itself so that electronically the chain remains in the topological phase whenever such a state is physically accessible. Here we examine the physics underlying this self-organization property in the framework of a more realistic two-dimensional model of a superconducting surface coupled to a 1D chain of magnetic adatoms. Treating both the superconducting order and the magnetic moments self-consistently, we find that the system retains its self-organization property, even if the topological phase extends over a somewhat smaller portion of the phase diagram compared to the 1D model. We also study the effect of imperfections and find that, once established, the topological phase survives moderate levels of disorder.

• Received 19 June 2014

DOI:https://doi.org/10.1103/PhysRevB.90.085124