Fully compensated Kondo effect for a two-channel spin $S=1$ impurity

We study the low-temperature properties of the generalized Anderson impurity model in which two localized configurations, one with two doublets and the other with a triplet, are mixed by two degenerate conduction channels. By using the numerical renormalization group and the noncrossing approximation, we analyze the impurity entropy, its spectral density, and the equilibrium conductance for several values of the model parameters. Marked differences with respect to the conventional one-channel spin $s=1/2$ Anderson model, which can be traced as hallmarks of an impurity spin $S=1$, are found in the Kondo temperature, the width and position of the charge-transfer peak, and the temperature dependence of the equilibrium conductance. Furthermore, we analyze the rich effects of a single-ion magnetic anisotropy $D$ on the Kondo behavior. In particular, as shown before, for large enough positive $D$ the system behaves as a “non-Landau” Fermi liquid that cannot be adiabatically connected to a noninteracting system turning off the interactions. For negative $D$ the Kondo effect is strongly suppressed. While the model is suitable for the description of a single Ni impurity embedded in an O-doped Au chain, it is a generic one for $S=1$ and two channels and might be realized in other nanoscopic systems.

Figure 1

Scheme of the system proposed in Ref. [42]. The Ni ion has two holes, with total spin $S=1$, that can jump to conduction bands of the same symmetry either to the left or to the right.

Figure 2

Top panel: NRG impurity contribution to the entropy as a function of temperature, $S_{\mathrm{imp}}\left(T\right)$, for the model of Eq. (2). Bottom panel: Same data in units of $T/T_K,$ with $T_K=6.7\times {10}^{-7},1.7\times {10}^{-7},4.0\times {10}^{-8},9.2\times {10}^{-9},2.1\times {10}^{-9}$ for $-{\epsilon }_d/\mathrm{\Delta }=2,2.5,3,3.5,4$, respectively.

Figure 3

NRG total impurity occupancy, $n_{\mathrm{imp}}-(M-1),$ as a function of ${\epsilon }_d$, $M$ being the number of channels. Squares indicate the results for the model of Eq. (2), $M=2$. The circles correspond to the impurity population for the spin $s=1/2$ infinite-$U$-limit one-channel Anderson model where $M=1$.

Figure 4

NRG impurity spectral density as a function of frequency for the same set of values of ${\epsilon }_d$ and other parameters as in Fig. 2.

Figure 5

Same as Fig. 4 calculated with NCA. The inset shows the charge-transfer peaks shifted by ${\epsilon }_d^{*}$ [see Eq. (13)].

Figure 6

Renormalized position of the charge-transfer peak, ${\epsilon }_d^{*}$, as a function of ${\epsilon }_d$. Other parameters as in Fig. 2.

Figure 7

Top panel: The squares indicate the NRG Kondo temperature as a function of $|{\epsilon }_d|/\mathrm{\Delta }$ calculated for $|J_H|=1000W$. Dashed line: Linear fitting giving $\ln \left(T_K\right)=-8.70-2.80\left|{\epsilon }_d\right|/\mathrm{\Delta }$ with a correlation coefficient of 0.999. Bottom panel: $T^{*}=T_K\left(J_H\right)/\sqrt{\mathrm{\Delta }/W^3}{\left(T_K^{s=1/2}\right)}^2$ as a function of $J_H$ for $|{\epsilon }_d|/\mathrm{\Delta }=3$.

Figure 8

Top panel: Equilibrium conductance $G\left(T\right)$ in units of $2e^2/h$ as a function of temperature. Bottom panel: The same data as in the top panel in units of $T/T_K$ (see caption of Fig. 2 for the values of $T_K$) and normalized by the saturated conductance $G_s=G(T\to 0)$.

Figure 9

Squares: NRG data for $G_{\alpha }\left(T\right)={\sum }_{\sigma }{G}_{\alpha \sigma }\left(T\right)$ in units of its maximum $G_s$ as a function of $T/T_K,$ for ${\epsilon }_d=-4\mathrm{\Delta }$. Dashed black line: Fitting of numerical data with expression Eq. (16) with $s=0.15$. Solid red line: Eq. (16) with $s=0.22$.

Figure 10

NRG conductance $G_{\alpha }\left(T\right)/G_0$ of the $S=1$ Kondo impurity model for several positive magnetic anisotropies $D$ across the topological quantum phase transition. The black (red) curves correspond to $D<D_c$ ($D>D_c$). From top to bottom: $D=0.0,1\times {10}^{-4}$, $1.15\times {10}^{-4},1.3\times {10}^{-4},1.35\times {10}^{-4},1.3505\times {10}^{-4}$, $1.351\times {10}^{-4},1.315\times {10}^{-4},1.352\times {10}^{-4},1.3525\times {10}^{-4}$, $1.353\times {10}^{-4},1.3535\times {10}^{-4},1.354\times {10}^{-4},1.345\times {10}^{-4}$, $1.355\times {10}^{-4},1.4\times {10}^{-4},1.6\times {10}^{-4}.$ The parameters are $J=0.2$ and $W=1$. We take $\mathrm{\Lambda }=3$ and keep 4000 NRG states. For these parameters, $T_K^0=4.2\times {10}^{-5}$ and $D_c=1.352\times {10}^{-4}$.

Figure 11

Normalized NRG spectral function ${\rho }_{\alpha }\left(\omega \right)$ of the $S=1$ Kondo impurity model as a function of $T/T_K^0,$ for several positive $D$ across the topological quantum phase transition. The black (red) curves correspond to $D<D_c$ ($D>D_c$). From top to bottom: $D=0.0,5\times {10}^{-5},8\times {10}^{-5}$, $1\times {10}^{-4},1.15\times {10}^{-4}$, $1.2\times {10}^{-4},1.21\times {10}^{-4},1.22\times {10}^{-4}$, $1.23\times {10}^{-4},1.25\times {10}^{-4}$, $1.3\times {10}^{-4},1.35\times {10}^{-4}$, $1.6\times {10}^{-4},1.8\times {10}^{-4},2\times {10}^{-4},5\times {10}^{-4}.$ The parameters are $W=1,J=0.2$. We take $\mathrm{\Lambda }=2$ and keep 3000 NRG states. For these parameters, $D_c=1.225\times {10}^{-4}$ and $T_K^0=3.8\times {10}^{-4}$.

Figure 12

A fourth-order hybridization process that leads to an effective spin flip between the $S_z=\pm 1$ projections for negative $D$.

Figure 13

Left: NRG impurity entropy $S_{\mathrm{imp}}$ as a function of $T/T_K^0$. Right: NRG impurity spectral function ${\rho }_{\alpha \sigma }\left(\omega \right)$ as a function of $\omega /T_K^0$. The model parameters are $\mathrm{\Delta }=0.1,{\epsilon }_d=-0.2\mathrm{\Delta }$, with a corresponding $T_K^0\simeq 1.245\times {10}^{-3}$, and $D=-16T_K^0$.

Figure 14

NRG conductance $G_{\alpha }\left(T\right)$ as a function of $T/T_K^0$ for three different negative magnetic anisotropies $D$. The other parameters are the same as in Fig. 13.