Kondo Cloud in a Superconductor

Magnetic impurities embedded in a metal are screened by the Kondo effect, signaled by the formation of an extended correlation cloud, the so-called Kondo or screening cloud. In a superconductor, the Kondo state turns into subgap Yu-Shiba-Rusinov states, and a quantum phase transition occurs between screened and unscreened phases once the superconducting energy gap $\mathrm{\Delta }$ exceeds sufficiently the Kondo temperature, $T_K$. Here we show that, although the Kondo state does not form in the unscreened phase, the Kondo cloud does exist in both quantum phases. However, while screening is complete in the screened phase, it is only partial in the unscreened phase. Compensation, a quantity introduced to characterize the integrity of the cloud, is universal, and shown to be related to the magnetic impurities’ $g$ factor, monitored experimentally by bias spectroscopy.

Figure 1

Schematic “phase diagram” of the model at zero temperature. When $\mathrm{\Delta }>T_K$, the ground state is a doublet with an asymptotically free spin decoupled from the superconductor, while in the opposite limit, when $\mathrm{\Delta }<T_K$, the ground state is a many-body singlet.

Figure 2

Compensation $\kappa$ across the quantum phase transition as a function of $\mathrm{\Delta }/T_K$, for $j_0=0.05$. In the fully screened regime, $\mathrm{\Delta }/T_K\lesssim 1.1$, $\kappa$ remains 1, and it displays a universal jump of size $\mathrm{\Delta }\kappa \simeq 0.719$ at the quantum phase transition, followed by a monotonous decrease in the partially screened regime. Blue and yellow squares represent DMRG and NRG results, while the solid line presents the theoretical result, Eqs. (10) and (11).

Figure 3

Envelope for the equal-time spin-spin correlation function $C(x)$ defined in Eq. (1) as a function of the distance from the impurity spin in a one-dimensional chain, as computed by DMRG. For $\mathrm{\Delta }=0$, the envelope function shows the expected universal scaling in the near and far regions. For $\mathrm{\Delta }\ne 0$, the algebraic behavior turns into an exponential decay, $C(x)\propto \exp (-2x/\xi )$, with $\xi =v_F/\mathrm{\Delta }$ the coherence length. We used a chain length $L=200$, the Kondo temperature was fixed to $T_K=0.1t$, with $t$ the hopping along the chain, which corresponds to a dimensionless coupling, $j_0=0.286$. The associated Kondo correlation length is ${\xi }_K\approx 20$ lattice sites.

Figure 4

Experimental setup for measuring the compensation $\kappa$. The quantum dot is coupled to a normal ($N$), a superconducting (SC), and a ferromagnetic (FM) lead.