Temperature dependent energy gap for Yu-Shiba-Rusinov states at the quantum phase transition

Motivated by recent experiments, which allow for fine-tuning of the effective magnetic interaction between the impurity and the superconductor, we investigate the regime around the quantum phase transition where the ground state of the system changes from a weakly coupled free spin to a screened spin regime. At this transition, we find that the Yu-Shiba-Rusinov (YSR) states remain at finite energies at low temperatures, thereby generating a gap in the spectrum, which is inconsistent with predictions of the original YSR theory. We investigate various gap-generating scenarios and determine that the local suppression of the order parameter, only captured by self-consistent calculations, generates the gap.