Linear Theory of the Vertex Function in Spin Dependent s-d Scattering

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Abstract
Following the field theoretic techniques of Abrikosov,1 we have derived a linear equation for the vertex function associated with the scattering of conduction electrons from magnetic impurities. We consider the model of a general nonlocal interaction consisting of the standard spin‐independent impurity potential plus an s‐d exchange interaction. In addition to the development of the linear theory, we have made an exact calculation of those terms in the electron proper self‐energy which are linear in the density of the impurities to fourth order in the coupling constant for the restricted model of only an s‐d exchange interaction. We show that to logarithmic accuracy (in the sense of Abrikosov1), not only Abrikosov's vertex function, but also that of Suhl2 and the one which satisfies a linear integral equation all give the same result for the proper self‐energy. However, the less divergent terms in any order of perturbation theory are not correctly reproduced by any of the vertex functions, which among themselves lead to different contributions to these terms in the self‐energy. Thus, the use of the various vertex functions to compute the self‐energy must be considered as inequivalent interpolation procedures for obtaining a substitute for the exact perturbation‐theory results. We have solved the linear equation for separable s‐wave interactions using both the ordinary and s‐d terms in the potential. The linear equations do not manifestly exhibit the absence of singularities in the upper half‐energy plane, and for weak, long‐range separable interactions such singularities occur despite the interference between the spin‐flip and non‐spin‐flip scattering. Although the linear theory is manifestly invariant under rotations, the resulting vertex functions do not satisfy single‐channel, energy‐shell unitarity2 in channels labeled by the coupled angular momentum of the incident electron plus impurity. The origin of this result is the linear theory's incorporation of correlations in the wavefunction of Fermi gas for which the incident electron couples to both the impurity and a particle‐hole excitation to give other states of the same total angular momentum as that of the incident electron plus ``bare'' impurity. The states associated with such correlations assume the role of additional channels for the decay of an initial state with fixed angular momentum.