Prevalence of trivial zero-energy subgap states in nonuniform helical spin chains on the surface of superconductors

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. The existence of trivial zero-energy bound states can prevent the conclusive identification of MBSs and, given the limited tunability of atomic spin-chain systems, could present a major experimental roadblock. First, we show that the Hamiltonian of a helical spin chain with varying nonuniform rotation rate between neighboring magnetic moments on a superconductor can be mapped to an effective Hamiltonian reminiscent of a ferromagnetic chain with strong Rashba spin-orbit coupling and with smooth nonuniform chemical potential, reminding a Rashba nanowire setups. Previously it has been found that trivial zero-energy states are abundant in nanowire systems with smoothly changing 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 nonuniformities. In particular, utilizing 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 nonuniform helical spin chains compared with equivalent semiconductor nanowires, as well as the ability to measure states locally via scanning tunneling microscopy, should reduce the experimental barrier to identifying MBSs in such systems.