Local Electron Distribution in the Singlet Ground State Due to thes-dExchange Interaction

Abstract

On the basis of the theory of singlet ground state for a localized spin developed so far, the total charge and spin localized around the impurity are calculated in detail. It is shown that for ¢a-component associated with the localized up-spin state half of a down-spin electron and half of an up-spin hole are trapped by the impurity. This leads to a conclusion with the aid of the Friedel sum rule that a phase shift of the conduction electrons at the Fermi level is ±n/2. It is further shown in general that the localized charge in the ground state completely vanishes for the present s-d exchange Hamiltonian. 111 non-magnetic metals forms a singlet (non-degenerate) state coupled with the conduction electrons by the s-d exchange interaction. In this singlet state, the spins of the. conduction electrons are localized around the impurity spin, and form a singlet bound state with it. The energy of the singlet ground state is lower by the binding energy IE I than the normal-state energy JE which can be obtained by the usual perturbation calculation which starts from the doubly degenerate free state of a localized spin and the conduction electrons. Our theory dealing with the present system consisting of a localized spin and the conduction electrons is based on the perturbation method.*> It starts from a singlet state in which one electron (or hole) excited above (or below) the Fermi sea is coupled with the localized spin instead of starting with the doubly degenerate free state, and calculates the ground-state energy and the wave function (and also other quantities) perturbed by the s-d exchange interaction in a power series of the exchange coupling among which the most divergent terms are retained. As particularly shown in a previous paper by the present authors,7) the charge density at the impurity center which is :finite in the start­ ing approximation completely vanishes in the :final stage of pertu:rbation. This fact indicates that the localized charge around the impurity vanishes and only the spin-correlation density remains to be localized. The main purpose of this paper is to calculate the total localized charge