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We investigate the topological classification of the subgap bands induced ina two-dimensional superconductor by a densely packed chain of magnetic momentswith ferromagnetic or spiral alignments. The wave functions for these bands arecomposites of Yu-Shiba-Rusinov-type states and magnetic scattering states andhave a significant spatial extension away from the magnetic moments. We showthat this spatial structure prohibits a straightforward extraction of aHamiltonian useful for the topological classification. To address the lattercorrectly we construct a family of spatially varying topological Hamiltoniansfor the subgap bands adapted for the broken translational symmetry caused bythe chain. The spatial dependence in particular captures the transition to thetopologically trivial bulk phase when moving away from the chain by showing howthis, necessarily discontinuous, transition can be understood from an alignmentof zeros with poles of Green's functions. Through the latter the topologicalHamiltonians reflect a characteristic found otherwise primarily in stronglyinteracting systems.

Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification