Transport properties of fully screened Kondo models

We study the nonequilibrium transport properties of fully (exactly) screened Kondo quantum dots subject to a finite bias voltage or a finite temperature. First, we calculate the Fermi-liquid coefficients of the conductance for models with arbitrary spin, i.e., its leading behavior for small bias voltages or temperatures. Second, we determine the low-temperature behavior of the static susceptibility from the exactly known Bethe ansatz results for the magnetization. Third, we study the crossover from strong to weak coupling in the spin-1/2 and the spin-1 models coupled to one or two screening channels, respectively. Using a real-time renormalization group method we calculate the static and dynamical spin-spin correlation functions for the spin-1/2 model as well as the linear and differential conductance and the static susceptibility for the spin-1 model. We define various Kondo scales and discuss their relations. We assess the validity of the renormalization-group treatment by comparing with known results for the temperature dependence of the linear conductance and static susceptibility as well as the Fermi-liquid behavior at low energies.

Figure 1

Schematic picture of fully (exactly) screened Kondo models. (a) Spin-1/2 dot coupled to one screening channel ($N=2S=1$) which is divided in two leads held at chemical potentials ${\mu }_{L/R}=\pm V/2$. (b) Spin-1 dot coupled to two screening channels ($N=2$). The left (right) leads of these channels are both held at ${\mu }_L=V/2$ (${\mu }_R=-V/2$). We note that the exchange interaction does not mix the screening channels.

Figure 2

Second-order diagrams describing the inelastic processes, or type (iii), in the mean current calculation. The open circles represent a current vertex while filled black dots correspond to interaction vertices. The four diagrams correspond to the four terms in Eq. (12) with interaction vertices proportional to ${\varphi }_1/3$, ${\varphi }_1$, $2{\varphi }_1/3$, and ${\varphi }_1/3$.

Figure 3

RG flow of the renormalized exchange coupling $\tilde{J}$ governed by the scaling equation (54). The maximum given by Eq. (79) is reached at $\Lambda =0$. In contrast to the poor-man's scaling result the coupling does not diverge. Inset: RG flow of the effective relaxation rate $\Gamma$. We note that $\Gamma$ remains finite for $\Lambda \to 0$.

Figure 4

Static spin-susceptibility $\chi \left(T\right)T$ for the spin-1/2 model and $V=0$. We compare to the exact result from Bethe ansatz [data are taken from Table 3 of Ref. 10] and asymptotic results Eqs. (81) and (82), respectively. There is no fit parameter in the RTRG result. The dashed line at $\chi \left(T\right)T=0.25$ shows the susceptibility of the asymptotically free impurity.

Figure 5

Static spin susceptibility ${\chi }_S$ of the $S=1/2$ (blue) and $S=1$ (red) models. The solid lines show the temperature dependence at $V=0$, ${\chi }_S\left(T\right)$, while the dashed lines show the voltage dependence at $T=0$, ${\chi }_S\left(V\right)$. All curves are normalized to ${\chi }_S\left(0\right)$, and both temperature $T$ and voltage $V$ are rescaled with $T_{0,S}$ defined in Eq. (7). Inset: Comparison of ${\chi }_S\left(T\right)/{\chi }_S^{*}$ for $S=1/2$ and $S=1$. The approach to the free spin susceptibility ${\chi }_S^{*}=S(S+1)/\left(3T\right)$ is slower in the spin-1 model.

Figure 6

The spin-spin correlation function $S\left(\Omega \right)$ and the imaginary part of the spin susceptibility ${\chi }^{\prime \prime }\left(\Omega \right)$ for the spin-1/2 model and $V=0$. We observe the maximum of ${\chi }^{\prime \prime }\left(\Omega \right)$ at $\Omega \approx \Gamma$ and $S(\Omega =0)=1/\left(2\Gamma \right)$, where $\Gamma (T,V=0)$ is shown in the inset.

Figure 7

The spin-spin correlation function $S\left(\Omega \right)$ and the imaginary part of the spin susceptibility ${\chi }^{\prime \prime }\left(\Omega \right)$ for the spin-1/2 model and $T=0$. We observe that the maximum of ${\chi }^{\prime \prime }\left(\Omega \right)$ appears at lower frequencies as compared to the temperature-dependent susceptibility shown in Fig. 6, which was also observed in Ref. [85]. $\Gamma (T=0,V)$ is shown in the inset.

Figure 8

Linear and differential conductance $G\left(T\right)$ and $G\left(V\right)$ for $S=1$ scaled to the Kondo scale $T_K^{*}$ defined in Eq. (87). The dashed lines are a guide to the eye to extract the relation $G(V=T_K^{*})\approx 0.605G_0$. Inset: Second-order (crosses) and third-order (solid lines) RTRG results for $G\left(T\right)$ as a function of $T/T_K^{*}$ and $G\left(V\right)$ as a function of $V/T_K^{**}$. The results are almost independent of the truncation.

Figure 9

Comparison of the linear conductance $G\left(T\right)$ for the fully screened Kondo model with $S=1/2$ and $S=1$ between the RTRG (solid lines) and numerical RG [95] (dashed lines).

Figure 10

Linear and differential conductance $G\left(T\right)$ and $G\left(V\right)$ for the spin-1 model plotted in the form $\frac{G_0-G(T,V)}{G_0}/{\left(\frac{T,eV}{T_K^{*}}\right)}^2$ such that the FL coefficients $c_T^{*}$ and $c_V^{*}$ can be readily determined by a quadratic fit (dashed line). Similarly, for the spin-1/2 model we obtain $c_T^{*}=4.86$ and $c_V^{*}=0.89$ in agreement with Ref. [42].