On the Method of Anomalous Green Functions in the Kondo Problem at Zero Temperature

Abstract

The method of anomalous Green functions introduced by Takano and Ogawa into the Kondo problem of sod interaction is reformulated more carefully. It is found that the self­ consistency equation should contain the energy shift caused by the sod interaction as a para­ meter and therefore can have a non-trivial solution even in the limit of vanishing anomalous Green functions. The equation in this limit has precisely the same form as Yoshimori's secular equation, provided that the most divergent corrections are included into the vertex part. Abrikosov formulation,5) in which the localized spin is described in terms of de­ struction and creation operators of We introduce the Green function which describes the propagation of the pair of an extra electron and a quasifermion injected into the Fermi sea of the free electron gas. The bound state is represented by an isolated pole of the Green function on the complex energy plane. The standard perturbation expansion has been applied. When the most divergent vertex corrections are included, our secular equation to deter­ mine the isolated pole takes exactly the same form as the Yoshimori equation. 4 ) Therefore, Yoshimori's conclusion that only the singlet bound state is possible and his expression for the binding energy apply also to our secular equation. Now, by use. of the same Abrikosov formulation, Takano and Ogawa 6 ) pro-' posed a non-perturbational approach. In analogy to the Gorkov 7 ) formulation of BCS theory, they· assumed that anomalous amplitudes such as <claa,e), <a,et Clva) do not vanish, where c, c t are destruction and creation operators of conduction electrons and a, at are those of quasifermions. Takano and Ogawa. made the Hartree-Fock-Gorkov approximation and obtained non-trivial solutions both for singlet and triplet states which have essentially the same binding energies as those obtained by the lowest order Yosida theory.S) Abrikosov 8 ) extended the Takano-Ogawa theory by including anomalous amplitudes such as <at c t ), <ca) and also.' by including the most divergent correc-