Ground-State Energy Shift due to thes−dInteraction

Abstract

A calculation of the ground-state energy of a localized spin embedded in a normal metal is described. The unperturbed state is taken as the state of the unperturbed Fermi sphere multiplied by a spin state with ${S}_{z}=S$. Configurations with 1, 2, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} electron-hole pairs excited are taken into account by a perturbation theory. Then a self-consistency equation to determine the ground-state energy is obtained. This equation has a solution which reduces to the Rayleigh-Schr\"odinger perturbation theory. In the case of antiferromagnetic exchange interaction, however, there is another solution which is lower than the Rayleigh-Schr\"odinger theory by a BCS-type expression. Thus the ground-state energy is expressed for small and negative $J$ as $E=\ensuremath{-}4(\mathrm{ln}2)S(S+1){J}^{2}{\ensuremath{\rho}}^{2}D\ensuremath{-}kD{e}^{\frac{1}{2J\ensuremath{\rho}}}$ for a band structure described in the text. Here $k$ is a constant of order unity. A consideration of the spin configuration of the perturbed state is given.