Topological properties of spin block magnetic ladders in proximity of a superconductor: application to BaFe$_{2}$S$_{3}$

Monoatomic chains with magnetic order in proximity of a s-wave superconductor can host Majorana edge modes. In this paper, we extend this idea to more complex spin-block chains such as the BaFe$_{2}$S$_{3}$ magnetic material that has a spin-ladder like structure. We investigate the topological phase diagram of such a system as function of the system parameters. We show that the coupling between chains within the ladder leads to the topological phase with a winding number larger than the sum of two single magnetic chains. Furthermore, strong coupling between chains leads to fractal-like substructure in the topological phase diagram. By investigating the real space properties of such a system and particularly its edge modes, we find that the system spectrum contains several in-gap states that we analyze in detail.