Prevalence of trivial zero-energy sub-gap states in non-uniform helical spin chains on the surface of superconductors

TL;DR

This paper investigates the occurrence of trivial zero-energy states in helical spin chains on superconductors, finding they are less common than in nanowire systems, which simplifies the search for Majorana bound states.

Contribution

The study maps helical spin chains to effective nanowire models and performs extensive searches, revealing a lower prevalence of zero-energy states in these chains compared to nanowires.

Findings

Zero-energy states are less prevalent in helical spin chains than in nanowire systems.

Varying rotation rates do not produce zero-energy states in 3D superconductor setups.

Reduced zero-energy states facilitate experimental detection of Majorana bound states.

Abstract

Helical spin chains, consisting of magnetic (ad-)atoms, on the surface of bulk superconductors are predicted to host Majorana bound states (MBSs) at the ends of the chain. Here, we investigate the prevalence of trivial zero-energy bound states in these helical spin chain systems. First, we show that the Hamiltonian of a helical spin chain on a superconductor can be mapped to an effective Hamiltonian reminiscent of a semiconductor nanowire with strong Rashba spin-orbit coupling. In particular, we show that a varying rotation rate between neighbouring magnetic moments maps to smooth non-uniform potentials in the effective nanowire Hamiltonian. Previously it has been found that trivial zero-energy states are abundant in nanowire systems with smooth potentials. Therefore, we perform an extensive search for zero-energy bound states in helical spin chain systems with varying rotation rates. Although bound states with near zero-energy do exist for certain dimensionalities and rotation profiles, we find that zero-energy bound states are far less prevalent than in semiconductor nanowire systems with equivalent non-uniformities. In particular, utilising varying rotation rates, we do not find zero-energy bound states in the most experimentally relevant setup consisting of a one-dimensional helical spin chain on the surface of a three-dimensional superconductor, even for profiles that produce near zero-energy states in equivalent one- and two- dimensional systems. Although our findings do not rule them out, the much reduced prevalence of zero-energy bound states in long non-uniform helical spin chains compared with equivalent semi-conductor nanowires, as well as the ability to measure states locally via STM, should reduce the experimental barrier to identifying MBSs in such systems.