Feynman-graph theory of the Kondo effect. I. Exact Summation of parquet graphs without divergences

Abstract

The Feynman-graph formalism is used to calculate the self-energy of conduction electrons exchange interacting with a single $S=\frac{1}{2}$ impurity atom, and hence the Kondo contribution to the resistivity $R$ in dilute magnetic alloys. To eliminate the effect of spurious states that result from second quantization, we derive a new limiting procedure which satisfies the linked-cluster expansion and maintains a fixed impurity-spin magnitude. A novel set of diagrammatic equations is found which exactly sums the set of non-self-consistent parquet graphs without any multiple counting. The resulting integral equations are solved in an approximation which omits certain diagrams, but still sums the remainder exactly. The self-energy so obtained yields the simple form $R=2{\ensuremath{\pi}}^{2}{R}_{u}{[4{\mathrm{ln}}^{2}(\frac{T}{{T}_{K}})+{\ensuremath{\pi}}^{2}]}^{\ensuremath{-}1}$ for the resistivity $R$ as a function of the temperature $T$, with ${R}_{u}$ the unitarity limit and ${T}_{K}$ the Kondo temperature. Although containing the results of various previous authors as approximations to it, this expression is free of the divergences found by them. While we succeed in eliminating the divergences by accurate summation, we find that the observed saturation of $R$ at zero temperature is not given by a non-self-consistent calculation. The self-consistent case is covered in the following paper.