Underscreened Kondo Compensation in a Superconductor

TL;DR

This paper studies how a magnetic impurity with spin-1 interacts with a superconductor, revealing a quantum phase transition affecting the Kondo screening and impurity compensation, with detailed analysis of correlation functions and spectral properties.

Contribution

It provides a detailed analysis of the quantum phase transition in underscreened Kondo impurities within superconductors, including universal compensation jumps and correlation behaviors, using advanced numerical methods.

Findings

Universal jump in Kondo compensation at phase transition

Oscillatory spin-spin correlations with 1/x decay

Exponential decay of correlations governed by superconducting length

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 $\Delta$ and the Kondo temperature $T_K$ induces a quantum phase transition from an underscreened doublet Kondo to an unscreened triplet phase, typically occurring when $\Delta/T_K\approx 1$. 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, $\Delta/T_K\gg 1$, the compensation vanishes, signifying complete decoupling of the impurity spin from the environment, while in the partially screened doublet phase, $\Delta/T_K\ll 1$, 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 $\sim 1/x$ at short distances. At larger distances, the superconducting gap induces an exponentially decaying behavior $\sim \exp(-x/\xi_\Delta)$ governed by the superconducting correlation length $\xi_\Delta$, 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.