In-gap states induced by magnetic impurities on wide-band s -wave superconductors: Self-consistent calculations

In this study, we examine the behavior of the self-consistent order parameter in the presence of magnetic impurities and nanostructures on a superconducting surface. Using the Bogoliubov-de Gennes equations in a self-consistent lattice Green's function implementation, we reproduce the single-impurity behavior, where a strong interaction with the substrate can induce a $0\ensuremath{-}\ensuremath{\pi}$ transition, characterized by a local reversal in the sign of the order parameter. The presence of multiple impurities, forming structures such as spin chains, complicates the behavior of the order parameter. These impurities trigger quantum phase transitions, evident from changes in the ground-state quantum numbers, such as fermion parity. Overall, the order parameter serves as a valuable tool for confirming or refuting potential transitions, which can be challenging to analyze through numerical evaluations of the topological invariants alone.