The Kondo problem. III. The susceptibility and relaxation rate

We complete the scaling equations derived in the previous two papers by restoring the spin-rotational invariance. We obtain two coupled nonlinear first-order differential equations for the invariant coupling and the resonance width. The solution for antiferromagnetic coupling is shown to be positive, independent of the model density of states and the order of the considered skeleton diagrams. At low temperatures the solution follows a Fermi-liquid behavior. For $T\to 0$ and $\omega \to 0$ the properties are seen to be universal over 3 orders of magnitude of the Kondo temperature. The static susceptibility and the impurity relaxation rate are calculated. At high temperatures they reproduce the perturbation theory in leading and next-leading logarithmic hierarchy. At low temperatures both ${\chi }_0$ and $T_1$ are finite, indicating the singlet ground state and a decrease with temperature as $T^2$. There is a monotonic and smooth crossover from the singlet state to the asymptotically free spin, giving evidence for the gradual breaking up of the spin compensation. The susceptibility is in fair agreement with Wilson's numerical diagonalization. The results are compared with experimental data for $\mathrm{Cu}\mathrm{Fe}$ alloys.