Kondo cloud and spin-spin correlations around a partially screened magnetic impurity

We consider the spatial spin correlations around a partially screened spin-1 magnetic moment in a metal exhibiting the underscreened Kondo effect. We find that the underscreening of the impurity spin results in spatial spin correlations that are more pronounced as compared to the fully screened Kondo effect; their power-law decay is weaker but with characteristic logarithmic corrections at large distances. The spin correlator also changes sign as a function of distance to the impurity allowing for ferromagnetic correlations between conduction-electron spin density and the local moment. The numerical findings are shown to be in agreement with the predictions deriving from an effective ferromagnetic Kondo Hamiltonian.

Figure 1

(Color online) Panel a: the equal-time spin-spin correlation function $\chi \left(x,T\right)=⟨{\vec{S}}_{\text{imp}}\vec{s}\left(x\right)⟩$ as a function of the distance $x$ measured from the $S=1$ impurity for different values of the Kondo coupling $j=\nu J$. Unlike the situation for the completely screened Kondo impurity, $\chi$ changes sign as a function of distance, and the electron-spin density close to $x=\left(n+\frac{1}{2}\right)\frac{\pi }{k_F}$ is aligned, rather than being anticorrelated, with the impurity spin. In panel b, we show the amplitude of the $2k_F$ oscillating and uniform part of $\chi$ at $T=0$ that are universal functions of $x/{\xi }_K$ [see Eq. (5)]. While for short distances both functions decay as $\sim x^{-1}$, they both obtain logarithmic corrections for $x>{\xi }_K$. As shown in panel c, the envelope of the oscillating part crosses over from $\sim x^{-1}$ (dashed line) to $\sim 1/\left[x\text{ln}\left(x/{\xi }_K\right)\right]$ (dotted-dashed line) at around the Kondo coherence length ${\xi }_K$. Any finite temperature introduces another length scale ${\xi }_T=\hslash v_F/\left(k_{B}T\right)$ at which the envelope function crosses over to an exponential decay. The uniform part decays faster beyond the Kondo coherence length: it tends to zero as $\sim 1/\left[x{\text{ln}}^2\left(x/{\xi }_K\right)\right]$. For better visualization, we plot $1/\left[x\chi \left(x\right)\right]$ as a function of $x/{\xi }_K$ in panel d.

Figure 2

(Color online) Temperature correction $\chi \left(T\right)-\chi \left(0\right)$ to the integrated spatial spin correlator (4). At low temperatures ${\xi }_T⪡{\xi }_K$, it follows Eq. (8).