On the applicability of the equations-of-motion technique for quantum dots

TL;DR

This paper analyzes the equations-of-motion technique for quantum dots, providing exact solutions in certain limits and evaluating its accuracy in describing phenomena like the Kondo effect, with implications for Green function calculations.

Contribution

It offers a high-order decoupling approximation for EOM Green functions in quantum dots and compares results with exact solutions, highlighting the method's strengths and limitations.

Findings

Exact solutions at specific limits and the particle-hole symmetric point.

High accuracy of the approximation away from the Kondo regime.

Qualitative failure to capture the Kondo effect and violations of the Friedel sum rule.

Abstract

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.