Renormalization-group approach to the Anderson model of dilute magnetic alloys. I. Static properties for the symmetric case

The temperature-dependent impurity susceptibility for the symmetric Anderson model is calculated for all physically relevant values of its parameters $U$ (the Coulomb correlation energy) and $\Gamma$ (the impurity-level width). It is shown that, when $U>\mathrm{}\pi \Gamma$, for temperatures $T<\mathrm{}\frac{U}{\left(10\mathrm{}{k}_B\right)}$ the symmetric Anderson model exhibits a local moment and that its susceptibility maps neatly onto that of the spin-½ Kondo model with an effective coupling given by $\rho J_{\mathrm{eff}}=-\frac{8\Gamma }{\pi U}$. Furthermore, this mapping is shown for remarkably large values of $\left|\rho J_{\mathrm{eff}}\right|$. At very low temperatures (much smaller than the Kondo temperature) the local moment is frozen out, just as for the Kondo model, leading to a strong-coupling regime of constant susceptibility at zero termperature. The results also depict the formation of a local moment from the free orbital as $T$ drops below $U$, a feature not present in the Kondo model. Finally, when $U\ll \pi \Gamma$ there is a direct transition from free-orbital regime for $T\gg \Gamma$ to the strong-coupling regime for $T\ll \Gamma$. The calculations were performed using the numerical renormalization group originally developed by Wilson for the Kondo problem. In addition to the actual numerical calculations, analytic results are presented. In particular, the effective Hamiltonians, i.e., fixed-point Hamiltonian plus relevant and marginal operators, are constructed for the free-orbital, local-moment, and strong-coupling regimes. Analytic formulas for the impurity susceptibility and free energy in all three regimes are developed. The impurity specfic heat in the strong-coupling regime is calculated.