  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.
