Effects of Orbital Degeneracy on a Magnetic Impurity in a Nonmagnetic Metal

Abstract

We study the Green's-function equations for a version of the two-orbital Anderson model of a magnetic impurity, in an approximation scheme that displays the logarithmic anomalies and enables the Kondo temperature $T_K$ to be calculated as a function of the parameters. The nature of the solutions is explicitly studied for large Hund's-rule exchange constant $J$, in which case we obtain the following results: If the internal exchange is treated in the Hartree-Fock approximation, it has the same effect as an enormous applied magnetic field, and the logarithmic resonances are removed far from the Fermi level, i.e., $T_K\to \infty$. However, if exchange is properly treated in a rotationally invariant manner, the nondiagonal terms cancel most of the effect of the Hartree-Fock terms, and the Kondo effect is restored. For large $J$, our model has an analog in the far simpler $s-d$ exchange model, as we show by a transformation of the Schrieffer-Wolff type. For small $J$, however, the complicated and extremely structured solution of equations involving some 24 coupled Green's functions is required, and no substantial simplification appears possible in general. An exception is the limit where the transfer matrix element $V_{\mathrm{k}d}\equiv 0$, for which we display explicit and exact solutions.