Underscreened Kondo compensation in a superconductor

Abstract

A magnetic impurity with a larger S=1 spin remains partially screened by the Kondo effect when embedded in a metal. However, when placed within an s-wave superconductor, the interplay between the superconducting energy gap Δ and the Kondo temperature TK induces a quantum phase transition from an underscreened doublet Kondo to an unscreened triplet phase, typically occurring when Δ/TK≈1. We investigate the Kondo compensation of the impurity spin resulting from this partial screening across the quantum phase transition, which together with the spin-spin correlation function serves as a measure of the Kondo cloud's integrity. Deep within the unscreened triplet phase, Δ/TK≫1, the compensation vanishes, signifying complete decoupling of the impurity spin from the environment, while in the partially screened doublet phase, Δ/TK≪1, it asymptotically approaches 1/2, indicating that half of the spin is screened. Notably, there is a universal jump in the compensation precisely at the phase transition, which we accurately calculate. The spin-spin correlation function exhibits an oscillatory pattern with an envelope function decaying as ≈1/x at short distances. At larger distances, the superconducting gap induces an exponentially decaying behavior ≈exp(−x/ξΔ) governed by the superconducting correlation length ξΔ, irrespective of the phase, without any distinctive features across the transition. Furthermore, the spectral functions of some relevant operators are evaluated and discussed. In terms of the methods used, a consistent description is provided through the application of multiplicative, numerical, and density-matrix renormalization group techniques.

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