Kondo screening cloud in the single-impurity Anderson model: A density matrix renormalization group study

A magnetic moment in a metal or in a quantum dot is, at low temperatures, screened by the conduction electrons through the mechanism of the Kondo effect. This gives rise to spin-spin correlations between the magnetic moment and the conduction electrons, which can have a substantial spatial extension. We study this phenomenon, the so-called Kondo cloud, by means of the density matrix renormalization group method for the case of the single-impurity Anderson model. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. For several mechanisms—the gate potential and a magnetic field—which destroy the Kondo effect, we investigate the behavior of the screening cloud induced by these perturbations.

Figure 1

(Color online) Integrated spin-spin correlations $\Sigma \left(x\right)$ [from Eq. (3)] for systems of different sizes at $U=1,\Gamma =0.20$ and ${ϵ}_d=-U/2$. As an example, the threshold of 0.1 that we use in Eq. (5) to extract ${\xi }_{0.9}$ is indicated by the dashed horizontal line. As an illustration of the typical raw data, we show the absolute value of the spin-spin correlations $\left|⟨{\vec{S}}_d\cdot {\vec{S}}_i⟩\right|$ for $L=300$ in the inset.

Figure 2

(Color online) (a) Rescaled integrated spin-spin correlations $\Sigma \left(x\right)$, collapsed onto a universal curve via suitable choices of ${\xi }_K$. (b) Comparison of the $U$ and $\Gamma$ dependence of ${\xi }_K$ and ${\xi }_K^0$ [from Eq. (4)]: $\sqrt{U\Gamma }{\xi }_K$ (symbols) and $\sqrt{U\Gamma }p{T}_K^{-1}$ (lines) plotted vs $U/\Gamma$, using $p$ as the fitting parameter (resulting in $p=6.8$).

Figure 3

(Color online) System size dependence of ${\xi }_{0.9}\left(L\right)$ for ${ϵ}_d=-U/2$. Points represent numerical data; lines serve as guides to the eyes.

Figure 4

(Color online) Results of a ${\xi }_a\left(L\right)$ scaling analysis for ${ϵ}_d=-U/2$. (a) Scaling collapse of ${\xi }_{0.9}\left(L\right)/{\xi }_{0.9}$ vs $L/{\xi }_{0.9}$, obtained by the two-step scaling strategy described in the text in Sec. 3B. (b) Comparison of the $U$ and $\Gamma$ dependence of ${\xi }_a$ and ${\xi }_K^0$ [from Eq. (4)] for several values of $a$: $\sqrt{U\Gamma }{\xi }_a$ (symbols) and $\sqrt{U\Gamma }p\left(a\right)T_K^{-1}$ (lines) plotted vs $U/\Gamma$, using the fit parameters $p\left(a\right)$ shown in the inset (squares). The dotted line in the inset indicates the prefactor $p=6.8$ obtained from the $\Sigma \left(x\right)$ scaling analysis of Fig. 2b.

Figure 5

(Color online) (a) Kondo screening length ${\xi }_{0.9}$ vs gate potential ${ϵ}_d/U$ for several $U/\Gamma$ and $L=500$. (b) Dot occupation $⟨n_d⟩$ vs gate potential.

Figure 6

(Color online) (a) Kondo screening length ${\xi }_{0.9}$ as a function of the magnetic field applied to the dot for $L=500$. In all panels, $T_K$ is given by $T_K=\hslash v_F/{\xi }_K^0$ with ${\xi }_K^0$ from Eq. (4). (b) Scaling collapse of ${\xi }_{0.9}\left(B\right)/{\xi }_{0.9}\left(B\to 0\right)$ vs $B/T_K$ (c) Magnetic moment $\mu =⟨{\left(S_d^z\right)}^2⟩-{⟨S_d^z⟩}^2$ vs $B/T_K$. The inset shows the rescaled data $\mu \left(B\right)/\mu \left(B\to 0\right)$.

Figure 7

(Color online) Convergence of ${\xi }_{0.9}$ vs the residual norm per site $\delta r/L$ for ${ϵ}_d=-U/2$, extracted from ground-state calculations using the SU(2) symmetry. For each combination of $U/\Gamma$ and $L$, the number of states kept increases for data points from right to left as $m=200,400,600,800,1000,1200,1500,2000,2500,3000$, except for the case $U/\Gamma =5.6$, $L=500$, where no point with $m=3000$ is shown.

Figure 8

(Color online) Convergence of ${\xi }_a\left(B\right)$ and the magnetic moment $\mu$ vs the residual norm per site $\delta r/L$ for a finite magnetic field of $B/T_K=3\cdot {10}^{-3}$ and ${ϵ}_d=-U/2$. For comparison, we add the $B=0$ data, represented by the open triangles, from calculations exploiting the SU(2) symmetry ($m=1500$ states kept). The calculation with a magnetic field (symbols connected with lines) uses the U(1) symmetry only ($m=100,200,400,600,800,1000,1500$ states kept from right to left). The results for the dot’s magnetic moment $\mu$ are also included for comparison (solid diamonds).