Spin Correlations and Finite-Size Effects in the One-Dimensional Kondo Box

We analyze the Kondo effect of a magnetic impurity attached to an ultrasmall metallic wire using the density matrix renormalization group. The spatial spin correlation function and the impurity spectral density are computed for system sizes of up to $L=511$ sites, covering the crossover from $L<l_K$ to $L>l_K$, with $l_K$ the spin screening length. We establish a proportionality between the weight of the Kondo resonance and $l_K$ as a function of $L$. This suggests a spectroscopic way of detecting the Kondo cloud.

Figure 1

Finite-size and even/odd effects in the 1D Kondo box. Solid line: $T_K$ as a function of impurity position $x_0$ for $L=511$, $N=512$, $V=0.35$ and open boundary conditions, as obtained from Eq. (3). Open circles: Weight of the Kondo peak, $W_K$, as defined in the text and in the inset. The results shown are for even $x_0$ only (see text). The inset shows the Kondo peak for $V=0.35$, $x_0=4$ and for various successive values of $L$ and $N$.

Figure 2

The impurity spectral density $A_d(\omega )$ for $L=127$, $N=128$, $V=0.35$ and two different $x_0$; $\eta =0.05$. Comparison with the noninteracting spectrum ($U=0$) exhibits the Kondo enhancement of the peak near ${\epsilon }_F=0$. The inset shows the upper Hubbard peak near $\omega ={\epsilon }_d+U$.

Figure 3

The average $C(r)$ of the spin correlation function $K(r)$ is shown as function of $1/r$ for $L=511$, $N=512$ and various hybridization strengths $V$; $r=x-x_0$, $x_0=4$. Inset: $K(r)$ showing RKKY oscillations. The $V=0.55$ curve is offset by $-0.005$ for clarity.

Figure 4

The weight of the Kondo resonance $W_K$ and the numerically determined screening length $l_K^{*}$ as function of size $L$ for $x_0=4$ and $N=L+1$, $N$ odd.

Figure 5

$l_K^{*}$ as in Fig. 4, but for $N$ even.