LAUSR

Recent experiments have shown the signatures of Majorana bound states at the ends of magnetic chains deposited on a superconducting substrate. Here, we employ first-principles calculations to directly investigate the topological properties of $3d$ transition metal nanochains (i.e., Mn, Cr, Fe and Co). In contrast to the previous studies [Nadj-Perge et al., Science 346, 602 (2014) and Ruby et al., Nano Lett. 17, 4473 (2017)], we found the exact tight-binding models in the Wannier orbital basis for the isolated chains as well as for the surface--deposited wires. Based on these models, we calculate the topological invariant of ${\mathbb{Z}}_{2}$ phases for all systems. Our results for the isolated chains demonstrate the existence of the topological phase only in Mn and Co systems. We also considered a noncollinear magnetic order as a source of the nontrivial topological phase and found that this type of magnetic order is not a stable ground state in the Fe and Co isolated chains. Further studies showed that a coupling between the chain and substrate leads to strong modification of the band structure. Moreover, the analysis of the topological invariant indicates the possibility of emergence of the topological phase in all studied nanochains deposited on the Pb surface. Therefore, our results demonstrate an important role of the coupling between deposited atoms and a substrate for topological properties of nanosystems.