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

TL;DR

This paper develops a method to classify the topological phases of subgap states induced by magnetic chains in 2D superconductors, accounting for their complex spatial structure and broken translational symmetry.

Contribution

It introduces spatially varying topological Hamiltonians that accurately capture the topological properties of subgap bands in the presence of magnetic chains.

Findings

Spatially extended wave functions hinder straightforward Hamiltonian extraction.

Constructed topological Hamiltonians adapt to broken translational symmetry.

Transition to trivial phase characterized by zeros and poles of Green's functions.

Abstract

We investigate the topological classification of the subgap bands induced in a two-dimensional superconductor by a densely packed chain of magnetic moments with ferromagnetic or spiral alignments. The wave functions for these bands are composites of Yu-Shiba-Rusinov-type states and magnetic scattering states and have a significant spatial extension away from the magnetic moments. We show that this spatial structure prohibits a straightforward extraction of a Hamiltonian useful for the topological classification. To address the latter correctly we construct a family of spatially varying topological Hamiltonians for the subgap bands adapted for the broken translational symmetry caused by the chain. The spatial dependence in particular captures the transition to the topologically trivial bulk phase when moving away from the chain by showing how this, necessarily discontinuous, transition can be understood from an alignment of zeros with poles of Green's functions. Through the latter the topological Hamiltonians reflect a characteristic found otherwise primarily in strongly interacting systems.