Applicability of the equations-of-motion technique for quantum dots

The equations-of-motion (EOM) hierarchy satisfied by the Green functions of a quantum dot embedded in an external mesoscopic network is considered within a high-order decoupling approximation scheme. Exact analytic solutions of the resulting coupled integral equations are presented in several limits. In particular, it is found that at the particle-hole symmetric point the EOM Green function is temperature independent due to a discontinuous change in the imaginary part of the interacting self-energy. However, this imaginary part obeys the Fermi liquid unitarity requirement away from this special point, at zero temperature. Results for the occupation numbers, the density of states, and the local spin susceptibility are compared with exact Fermi liquid relations and the Bethe ansatz solution. The approximation is found to be very accurate far from the Kondo regime. In contrast, the description of the Kondo effect is valid on a qualitative level only. In particular, we find that the Friedel sum-rule is considerably violated, up to 30%, and the spin susceptibility is underestimated. We show that the widely used simplified version of the EOM method, which does not account fully for the correlations on the network, fails to produce the Kondo correlations even qualitatively.

Figure 1

The zero-temperature occupation number $n_0$ as function of the renormalized energy $E_d$ calculated self-consistently by the EOM method (solid line), and from the Friedel sum-rule, Eq. (17) (dashed line). Open circles show the exact Bethe ansatz results (Ref. 36).

Figure 2

The local spin susceptibility $\chi$ as function of the renormalized energy level $E_d$, for $T=0$. The EOM result (solid line) is close to the Bethe ansatz one (Ref. 36) (circles) in the empty orbital and mixed valence regimes, but deviates significantly in the strongly correlated region (large negative values of $E_d∕\Gamma$). Inset: The scaling of the susceptibility with $T∕T_K^{*}$ at three fixed energy level positions (marked by arrows in the main graph). The solid lines indicate the high-temperature asymptotic behavior, $\chi \left(T\right)={\left(g{\mu }_B\right)}^2∕\left(6T\right)$.

Figure 3

Data from Fig. 2 compared to the spin susceptibility given by the MWL approximation in the temperature range from $T=1\Gamma$ down to $T={10}^{-4}\Gamma$. The inset shows the MWL susceptibilities only, on a linear scale, allowing for the negative values.

Figure 4

(Color online) The scaling of the local density of states at the Fermi level as function of the reduced temperature $T∕T_K^{*}$, is demonstrated by plotting $\rho (\omega =0,T)$ for three different values of the renormalized single-electron energy on the dot, $E_d$ [Eq. (43)]. The dashed line at $\rho =2∕\left(\pi \Gamma \right)$ corresponds to the limit dictated by the Friedel sum-rule at $E_d∕\Gamma ⪡-1$. The inset shows the melting of the Kondo peak as the temperature is increased at fixed $E_d∕\Gamma =-5$. Temperature values represented in the inset are marked by arrows in the main graph.

Figure 5

The equilibrium occupation number $n_0={\sum }_{\sigma }⟨n_{\sigma }⟩$ for the same parameters as in Fig. 1, and $D=100\Gamma$ (since second-order perturbation theory does not scale with $E_d$, the bandwidth has to be specified), calculated from the self-consistently truncated EOM (solid lines), and by perturbation theory to first order in $\Gamma$, Eqs. (55, 57) (dashed line). The perturbational result diverges at ${ϵ}_0=0$.