Quantitative calculation of the spatial extension of the Kondo cloud

Abstract

A recently developed compact solution for the singlet state of the Friedel-Anderson and the Kondo impurity is applied to investigate the old question of a Kondo cloud in the Kondo ground state. Wilson's states with an exponentially decreasing frame of energy cells toward the Fermi level are used. The Wilson states are expressed as free electron waves with a linear dispersion and integrated over the width of their energy cells. For the magnetic state of the Friedel-Anderson impurity, one finds essentially no spin polarization in the vicinity of the $d$ impurity. However, for the magnetic component of the singlet state, a spin polarization cloud is observed which screens the spin (magnetic moment) of the $d$ electron. The range ${\ensuremath{\xi}}_{K}$ of this polarization cloud is investigated in detail for the Kondo impurity. The range is inversely proportional to the Kondo energy ${\ensuremath{\Delta}}_{K}$. The extent of the electron density in real space is a detector for a resonance in energy. The spatial extension $\ensuremath{\xi}$ and the resonance width $\ensuremath{\Delta}$ are reciprocal and given by the relation $\ensuremath{\xi}\ensuremath{\Delta}\ensuremath{\approx}\ensuremath{\hbar}{v}_{F}$.