Symmetry aspects of Yu-Shiba-Rusinov bands in magnetic atomic chains on a superconductor

Majorana zero modes (MZMs) have been proposed as a promising basis for Majorana qubits offering great potential for topological quantum computation. Such modes may form at the ends of a magnetic atomic chain on a superconductor. Here, we study the topological properties of Yu-Shiba-Rusinov (YSR) bands of excitations in Mn chains constructed on a Nb(110) and on a Ta(110) substrate using first-principles calculations and scanning tunneling microscopy and spectroscopy experiments. We demonstrate that even and odd YSR states with respect to mirroring on the symmetry plane containing the chain have different dispersions. On the Nb substrate, a wide but topologically trivial minigap is opened in the even states by the spin--orbit coupling, while the odd states display signatures of precursors of MZMs with a smaller minigap. On the Ta substrate, the stronger spin--orbit coupling increases the hybridization between the bands, but no clear minigap can be observed in either of them. These findings highlight the influence of symmetries on interpreting the spectroscopic signatures of candidates for MZMs.