Electron transport in the four-lead two-impurity Kondo model: Nonequilibrium perturbation theory with almost degenerate levels

The eigenstates of an isolated nanostructure may get mixed by the coupling to external leads. This effect is the stronger, the smaller the level splitting on the dot and the larger the broadening induced by the coupling to the leads. We describe how to calculate the nondiagonal density matrix of the nanostructure efficiently in the cotunneling regime. As an example, we consider a system of two quantum dots in the Kondo regime, the two spins coupled by an antiferromagnetic exchange interaction and each dot tunnel coupled to two leads. Calculating the nonequilibrium density matrix and the corresponding current, we demonstrate the importance of the off-diagonal terms in the presence of an applied magnetic field and a finite bias voltage.

Figure 1

(Color online) Double quantum dot setup: two Kondo impurities, $L$ and $R$, both represented by a spin-$1∕2$ are coupled mutually by a spin exchange interaction $K$. The quantum dots are connected to two leads each, 1,2 and 3,4, respectively.

Figure 2

Diagram for the second order pseudoboson self-energy. Solid lines: the conduction electron Green’s functions; dashed lines: pseudoparticles of the double quantum dot system.

Figure 3

(Color online) Illustration of the symmetry in the off-diagonal self-energy ${\Sigma }_{s{t}_0}$. The two different paths from the singlet-state $\mid s⟩$ to the triplet-state $\mid t_0⟩$ over $\mid t_{-}⟩$ and $\mid t_{+}⟩$ come with opposite signs. Green (left) sign for interaction with left leads. Red (right) sign for interaction with right leads.

Figure 4

(Color online) Spectral functions $A_{ss},A_{s{t}_0}$ and $A_{t_0{t}_0}$ versus the frequency $\omega ∕B$ for an exchange spin interaction $K$ of the order of the level broadening, $K=0.05\sim {\Gamma }_{s{t}_0}$. Further parameters are $B=1.0$, $g_L=0.1$, $g_R=0.2$, and $T=0.001$. The off-diagonal spectral function $A_{s{t}_0}$ has total spectral weight zero and thus contains negative values.

Figure 5

(Color online) Spectral functions ${\tilde{A}}_{11}\left(\omega \right)$ and ${\tilde{A}}_{22}\left(\omega \right)$ versus the frequency $\omega ∕B$ in the rotated space for the same parameter set as in Fig. 4.

Figure 6

(Color online) The expectation value of $⟨s^{†}{t}_0+t_0^{†}s⟩$, i.e., the contribution of the off-diagonal Green’s function is of the order of the diagonal contributions as illustrated here for the cases of $K∕{\Gamma }_{s{t}_0}\simeq 1,0.2,0$. Further parameters of the plot are $B=1.0$, $g_L=0.1$, $g_R=0.2$, and $T=0.001$.

Figure 7

(Color online) The magnetization $M_L$ of the left quantum dot is strongly influenced by a voltage applied over the left dot. For $K\simeq {\Gamma }_{s{t}_0}$, the magnetization $M_R$ over the right dot shows only minor deviations from the thermodynamic value. Neglecting the off-diagonal contributions would give $M_L=M_R=M_{\mathit{tot}}∕2$. Further, parameters of the plot are $B=1.0$, $g_L=0.1$, $g_R=0.2$, and $T=0.001$.

Figure 8

(Color online) Differential conductance in units $\left(\frac{\pi }{8}\right)\left(\frac{e^2}{h}\right)g_{12}{g}_{21}$ of the case of small exchange interaction $K∕{\Gamma }_{s{t}_0}\simeq 1,0.2,0$. For $K\simeq {\Gamma }_{s{t}_0}$, the results including or neglecting the off-diagonal contributions are shown. The further parameters are chosen identically to the previous figures.