Kondo screening cloud in a one-dimensional wire: Numerical renormalization group study

We study the Kondo model—a magnetic impurity coupled to a one-dimensional wire via exchange coupling—by using Wilson’s numerical renormalization group technique. By applying an approach similar to which was used to compute the two-impurity problem we managed to improve the poor spatial resolution of the numerical renormalization group method. In this way we have calculated the impurity-spin–conduction-electron-spin correlation function which is a measure of the Kondo compensation cloud whose existence has been a long-standing problem in solid-state physics. We also present results on the temperature dependence of the Kondo correlations.

Figure 1

(a) Spherical shells in real space depicting the extent of the wave functions of states which are represented as on-site states on the Wilson chain. As shown, the NRG method has a good spatial resolution around the impurity only. (b) A straightforward extension of the NRG method tackles that problem providing a good spatial resolution at the impurity site as well as at another, freely chosen point by introducing a second electron field.

Figure 2

(Color online) The equal-time spin-spin correlation function ${\chi }_{\mathrm{t}=0}\left(x\right)=⟨\vec{S}\vec{\sigma }\left(x\right)⟩$ as a function of the distance $x$ measured from the impurity for different values of the Kondo coupling $j={\varrho }_{0}J$. ${\chi }_{\mathrm{t}=0}\left(x\right)$ oscillates as $\sim {\cos }^2\left(k_{F}x\right)$. As shown in the inset the envelope of the oscillating part [i.e., ${\chi }_{\mathrm{t}=0}(x=n\pi ∕k_F)$ where $n$ is an integer] crosses over from $\sim x^{-1}$ to $\sim x^{-2}$ at around the Kondo coherence length ${\xi }_K$. The envelope functions for different couplings nicely collapse into one universal curve.

Figure 3

(Color online) The envelope of ${\chi }_{t=0}\left(x\right)$ for different couplings and different temperatures: Any finite temperature introduces another energy scale ${\xi }_T=\hslash v_F∕k_{B}T$ at which the envelope function crosses over from an algebraic to an exponential decay.