  Theoretical descriptions of Yu-Shiba-Rusinov (YSR) states induced by magnetic
impurities inside the gap of a superconductor typically rely on a classical
spin model or are restricted to spin-1/2 quantum spins. These models fail to
account for important aspects of YSR states induced by transition-metal
impurities, including the effects of higher quantum spins coupled to several
conduction-electron channels, crystal or ligand-field effects, and magnetic
anisotropy. We introduce and explore a zero-bandwidth model, which incorporates
these aspects, is readily solved numerically, and analytically tractable in
several limiting cases. The principal simplification of the model is to neglect
Kondo renormalizations of the exchange couplings between impurity spin and
conduction electrons. Nevertheless, we find excellent correspondence in those
cases, in which we can compare our results to existing
numerical-renormalization-group calculations. We apply the model to obtain and
understand phase diagrams as a function of pairing strength and magnetic
anisotropy as well as subgap excitation spectra. The single-channel case is
most relevant for transition-metal impurities embedded into metallic
coordination complexes on superconducting substrates, while the multi-channel
case models transition-metal adatoms.
