Yu-Shiba-Rusinov states, the BCS-BEC crossover, and the exact solution in the flat-band limit

TL;DR

This paper investigates Yu-Shiba-Rusinov states across the BCS-BEC crossover in a superconductor with a magnetic impurity, providing an exact solution in the flat-band limit and showing the weak impact of pairing strength on the YSR spectrum.

Contribution

It offers an analytical solution in the deep BEC limit of Richardson's model, connecting it to flat-band superconductivity and demonstrating the effectiveness of zero-bandwidth approximations.

Findings

YSR states are weakly affected by pairing strength when properly scaled.

Exact solution in the flat-band limit relates to a zero-bandwidth BCS model.

Zero-bandwidth models can quantitatively predict the full problem with proper rescaling.

Abstract

We study the subgap Yu-Shiba-Rusinov (YSR) states in the Richardson's model of a superconductor with a magnetic impurity for different electron pairing strengths from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) regime to the strong-coupling Bose-Einstein-condensate (BEC) regime. We observe that the effect of the increasing pairing strength on the YSR excitation spectrum as a function of hybridisation strength and impurity on-site potential is only quantitative and, in fact, rather weak when the results are appropriately rescaled. We furthermore show that the problem is analytically solvable in the deep BEC limit which is equivalent to flat-band superconductivity. This exact solution can be related to a zero-bandwidth (ZBW) effective BCS mean field Hamiltonian where the superconductor is described by a single electron level with onsite pairing. The small difference between the BCS and BEC regimes of the Richardson's model explains the success of the simple ZBW calculations for BCS mean-field Hamiltonians. A ZBW model requires only a suitable parameter rescaling to become useful as a quantitative predictive tool for the full problem.