A Large-$N$ Approach to Magnetic Impurities in Superconductors

TL;DR

This paper develops a large-N mean-field theory for magnetic impurities in superconductors, accurately describing strong coupling regimes and subgap states, complementing existing methods like NRG.

Contribution

The authors introduce a large-N mean-field approach for quantum impurities in superconductors, effective in strong coupling regimes and extendable to complex models beyond NRG capabilities.

Findings

Accurately describes Yu-Shiba-Rusinov states in strong coupling

Performs well when Kondo temperature exceeds superconducting gap

Provides analytical insights complementing numerical methods

Abstract

Quantum spin impurities coupled to superconductors are under intense investigation for their relevance to fundamental research as well as the prospects to engineer novel quantum phases of matter. Here we develop a large-$N$ mean-field theory of a strongly coupled spin-$\tfrac{1}{2}$ quantum impurity in a conventional $s$-wave superconductor. The approach is benchmarked against Wilson's numerical renormalization group (NRG). While the large-$N$ method is not applicable in the weak-coupling regime where the Kondo temperature $T_K$ is smaller than the superconducting gap $\Delta$, it performs very well in the strong coupling regime where $T_K \gtrsim \Delta$, thus allowing to obtain a reasonably accurate description of experimentally relevant quantities. The latter includes the energy of the Yu-Shiba-Rusinov subgap states, their spectral weight, as well as the local density of continuum states. The method provides a reliable analytical tool that complements other perturbative and non-perturbative methods, and can be extended to more complex impurity models for which NRG may be not easily applicable.