Numerical study of the Kondo cloud using finite-U slave bosons

In this work, using the finite-U slave boson mean-field approximation to solve the single-impurity Anderson model, the authors apply two different strategies to study the Kondo cloud, by analyzing quantities that are dependent on the distance to the magnetic impurity and then finding a universal distance scale ${\xi }_K$ through the collapse of the results into an universal function. The first method is based on the analysis of the local density of states of the conduction electrons (denoted as ${\xi }_K^L)$, while the second relies on the analysis of spin correlations $\left({\xi }_K^{\mathrm{\Sigma }}\right)$. Our calculations show that there is exact quantitative agreement, in the way ${\xi }_K$ depends on $U/\mathrm{\Gamma }$, between ${\xi }_K^{\mathrm{\Sigma }}$ and the results obtained through the heuristic expression ${\xi }_K\propto v_F/T_K$, while there is very close quantitative agreement between ${\xi }_K^{\mathrm{\Sigma }}$ and ${\xi }_K^L$. The use of the slave boson technique to calculate the spin correlations, which eliminates finite size effects, allowed us to study very large Kondo clouds, something that is very difficult using other techniques, like the density matrix renormalization Group method, for example. In addition, the very smooth curves obtained for the spin correlations allowed us to qualitatively identify a region in the Kondo cloud, adjacent to the impurity, that had been connected to the free orbital fixed point in previous numerical renormalization group calculations [Phys. Rev. B 84, 115120 (2011)].

Figure 1

Anderson impurity, with Coulomb interaction $U$, coupled by a hopping $t^{\prime }$ to a semi-infinite tight-binding chain with half-bandwidth $D=2t$. Top panel illustrates the situation when there is no Kondo effect $\left(t^{\prime }=0\right)$. The black curve under the chain is the LDOS of site $n=4$. Bottom panel illustrates the situation when the Kondo state is formed (finite $t^{\prime })$. Notice the change in the LDOS of site $n=4$, mainly around the Fermi energy, when comparing upper and lower panel.

Figure 2

LDOS at sites $n=4$ and 52 inside the tight-binding chain, in (a) and (b), respectively. Results are shown when the impurity is coupled to the chain, for $U=1.25$ and $U/\mathrm{\Gamma }=10.0$ [(black) solid line], and when it is decoupled from it $\mathrm{[}\mathrm{\Gamma }=0.0$, (red) dashed line]. The inset in each panel shows a zoom of the main-panel data around the Fermi energy. A finite $\mathrm{\Gamma }$ results in a Kondo state, while a vanishing $\mathrm{\Gamma }$ suppresses it, and the difference between the two corresponding LDOS is used to calculate the Kondo cloud extension function $L\left(n\right)$ [see Eq. (7)].

Figure 3

The (red) solid curves show $\left|{\rho }_n^{\mathrm{K}}\left(\omega \right)-{\rho }_n^{\mathrm{NK}}\left(\omega \right)\right|$ results for $U/\mathrm{\Gamma }=10.0$ [(a) and (b)] and $U/\mathrm{\Gamma }=5.0$ [(c) and (d)] at sites $n=12$ [(a) and (c)] and $n=52$ [(b) and (d)]. The black dashed curves are the LDOS at the impurity (the Kondo peak), equivalent to the $L_{\mathrm{\Delta }}\left(\omega \right)$ used in our previous method [6] [see Eq. (6)]. It is clear that a convolution with $L_{\mathrm{\Delta }}\left(\omega \right)$ suppresses spectral weight at higher energies. In addition, by comparing the left-side panels with the right-side panels it is easy to see that, for fixed $\mathrm{\Gamma }$, the envelope function of $\left|{\rho }_n^{\mathrm{K}}\left(\omega \right)-{\rho }_n^{\mathrm{NK}}\left(\omega \right)\right|$ depends very weakly on $n$. All results obtained for $U=1.25$.

Figure 4

(a) Function $L\left(n\right)$, obtained through Eq. (7), calculated for $4.17\le U/\mathrm{\Gamma }\le 16.7$, as indicated in the legend in (b), and $U=1.25$. The scale ${\xi }_K^L$ (marked by a black star) corresponds to the value of $n$ for which the normalized $L\left(n\right)/L_{\max }$ vs $n/{\xi }_K^L$ curves collapse into a universal curve [see (b)]. The factors indicated on the right side of panel (a) were used to narrow the window in the vertical axis and improve the visualization of the data. (b) Semilogarithmic plot of $L\left(n\right)/L_{\max }$, the normalized $L\left(n\right)$ curves from (a), as a function of $n/{\xi }_K^L$, showing their collapse into a universal curve $f(n/{\xi }_K^L)$ (black solid and dotted lines).

Figure 5

(a) Spin-spin correlations $\langle {\vec{S}}_d·{\vec{S}}_n\rangle$ between the impurity and the conduction electron in the $n^{\mathrm{th}}$ site inside the chain for $U=1.25$ and different values of $U/\mathrm{\Gamma }$, as indicated in the inset legend (same values as in Fig. 4). The inset shows a zoom for $n<30$. Note that only results for odd values of $n$ are shown, as $\langle {\vec{S}}_d·{\vec{S}}_n\rangle$ vanishes for even $n$. (b) Semilogarithmic plot of $1-\mathrm{\Sigma }\left(N\right)/\mathrm{\Sigma }\left(0\right)$ as a function of $N/{\xi }_K^{\mathrm{\Sigma }}$, for $U=1.25$ and same values of $U/\mathrm{\Gamma }$ as in (a). Note the very good collapse of the curves (aside from a region around the impurity), indicating their universality. The inset shows how the ${\xi }_K^{\mathrm{\Sigma }}$ values (indicated as black stars, except for the (red) squares curve, which falls outside the range shown) used to collapse the curves in the main panel were obtained: ${\xi }_K^{\mathrm{\Sigma }}$ is the value of $N$ for which the sum $\mathrm{\Sigma }\left(N\right)$ reaches $f_s=95\%$ of $\mathrm{\Sigma }\left(0\right)=-\langle {\vec{S}}_d^2\rangle$, i.e., ${\xi }_K^{\mathrm{\Sigma }}$ is the distance from the impurity for which $95\%$ of the impurity's magnetic moment is screened by the conduction electrons that form a singlet with the impurity.

Figure 6

(a) and (b) show a zoom of Figs. 4 and 5, respectively, at the region adjacent to the impurity up to the point where the data collapse into a universal curve. The arrows in (b) approximately indicate the point where each curve, starting with the one for $U/\mathrm{\Gamma }=4.17$ [(magenta) diamonds], starts to deviate from the others (i.e., the point where the free orbital region starts, see text). Note that in (a) we are not showing the data for even sites to facilitate the comparison with data in (b).

Figure 7

Log-log plot of $\left|{\vec{S}}_d·{\vec{S}}_n\right|$ (dashed black curves) as a function of $n/{\xi }_K^{\mathrm{\Sigma }}$ for $U/\mathrm{\Gamma }=4.17$, 10.0, and 16.7 in (a) to (c). All results for $U=1.25$. The gray (red) solid curves are $\propto 1/r_n^2$, with a proportionality constant chosen so that it matches the spin correlation calculated for the largest value of $n$, and $r_n$ is the distance to the impurity. The comparison of the two curves clearly indicates that the spin correlations decay as the second power of distance to the impurity outside of the cloud, as defined by ${\xi }_K^{\mathrm{\Sigma }}$.

Figure 8

Semilogarithmic plot of the Kondo screening length ${\xi }_K$, as a function of $U/\mathrm{\Gamma }$, obtained through $v_F/T_K$ [(blue) solid line], the $L\left(n\right)$ function [(green) triangles], and through the $\mathrm{\Sigma }\left(N\right)$ spin correlations method [(red) plus signs]. The first two were multiplied by a scaling factor (as indicated in the legend) to facilitate the comparison. Note that $v_F=2.0$ and $T_K$ was obtained through the finite-U SBMFA.