The completeness problem in the impurity Anderson model

With the recent development of the nanoscopic technology, the impurity Anderson model (AIM) was experimentally realized in quantum dot devices, and there is renewed interest in the study of the Kondo physics of the AIM. Several Green's functions approximations by the equation of motion method (EOM), that incorporates the Kondo effect through a digamma function, have been presented in the literature as an adequate tool to describe, at least qualitatively, the Kondo effect. However, these approximations present several drawbacks: they are no longer valid as the temperature decreases below the Kondo temperature, because the logarithmic divergence of the digamma function makes the spectral density at the chemical potential to vanish, and the Friedel sum rule and the completeness in the occupation numbers are not fulfilled. In this work we present a critical discussion comparing the results of digamma approximations GF with the atomic approach, recently developed by some of us, that satisfy the completeness and the Friedel sum rule. We present results for the density of states, the Friedel sum rule and the completeness