Yu-Shiba-Rusinov states in a superconductor with topological Z2 bands

A Yu-Shiba-Rusinov (YSR) state is a localized in-gap state induced by a magnetic impurity in a superconductor. Recent experiments used an STM tip to manipulate the exchange coupling between an Fe adatom and the ${\rm FeTe}_{0.55}{\rm Se}_{0.45}$ superconductor possessing a $Z_2$ nontrivial band structure with topological surface states. As the tip moves close to the single Fe adatom, the energy of the in-gap state is modulated and exhibits two unusual zero-energy crossings, which cannot be understood by coupling the magnetic impurity to the superconducting topological surface Dirac cone. Here, we numerically and analytically study the YSR states in superconductors with nontrivial $Z_2$ bands and explain the origin of the two zero-energy crossings as a function of the exchange coupling between the magnetic impurity and the bulk states. We analyze the role of the topological surface states and compare in-gap states to systems with trivial $Z_2$ bands. The spin-polarization of the YSR states is further studied for future experimental measurement.