Ground State Energy of Conduction Electrons Interacting with a Localized Spin

Abstract

875 In previous publications it has been shown that there exists a singlet bound state for a system consisting of conduction electrons and a localized spin coupled by an antiferromagnetic exchange interaction, and its binding energy has been calculated by a modified perturbation method. In this paper the structure of this modified perturbation method is analysed and it is shown in detail that the energy of the state containing the bound state is lower by the binding energy of the latter than that obtained by the usual perturbation method. The case of a localized spin whose magnitude is greater than one-half is also briefly discussed. § 1. Introduction In two previous publications by the present author 1) and by Okiji 2) (hence-forth these two papers are referred to as I and II) it has been shown that there exists a singlet bound state for a system consisting of conduction electrons and a localized spin coupled by an antiferromagnetic exchange interaction, and its binding energy has been calculated by a modified perturbation method. According to the results obtained in these two papers I and II, the binding energy of this singlet bound state has an exponential form-Dexp(-aN/p]JI), where D denotes half the band width, p the density of states of the band, which is assumed to be energ~-independent, J the coupling constant of the s-d exchange interaction which is taken to be negative for an antiferromagnetic interaction, and 1V the total number of atoms in the crystal; a represents a numerical factor near to unity. This form of binding energy is singular at J = 0 and it is quite natural that the usual perturbation expansion 3) for the free energy in a power series in J misses this exponential term. This binding energy should, of course, be interpreted as an energy difference between the state containing a bound state and the state without it. In the perturbation calculations carried out in I and II, only those terms were retained which possess a logarithmic singularity as the binding energy tends to zero and other regular terms were neglected because these terms did not play any decisive role as far as the binding energy is concerned. However, the question might arise whether or not the energy-lowering due to the ordinary scattering by the exchange interaction would be affected by the existence of the bound state. The purpose of this paper is to clarify the meaning of the terms neglected