Removing the Divergence at the Kondo Temperature by Means of Self-Consistent Perturbation Theory

We consider a dilute alloy consisting of conduction $s$ electrons exchange interacting with magnetic impurity $d$ electrons. Perturbation calculations of the $s-d$ scattering amplitude $\Gamma$ by Abrikosov, Duke, and Silverstein show a divergence at the Kondo temperature $T_K$; this implies a breakdown of perturbation theory and transition to a bound state. However, Hamann's calculations, utilizing decoupled Green's-function equations of motion, reveal no divergence at $T_K$. It has been proposed that the disagreement is due to the fact that the perturbation calculations are restricted to a sum over only the leading logarithmic terms ("parquet" diagrams) in each order. To investigate this idea, we have followed a suggestion by Doniach and extended the perturbation sum to include nonparquet diagrams by self-consistently clothing all s-electron propagators. We first clothe just the simple ladder parquets, where the sum can be carried out exactly, then extend the argument to include all parquets. In both cases, it is found that the clothing pushes the divergence in $\Gamma$ down to $T=0\mathrm{}$, showing that perturbation theory is valid for all temperatures greater than zero. At $T=0\mathrm{}$, there is a bound state. The resistance calculated from $\Gamma$ turns out to have the Hamann form, thus producing agreement between the perturbation-theoretic and decoupled equations-of-motion results.