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

Abstract

The temperature-dependent impurity susceptibility for the asymmetric Anderson model is calculated over a broad, physically relevant range of its parameters ${\epsilon }_d$ (the energy of impurity orbital), $U$ (the Coulomb correlation energy), and $\Gamma$ (the impurity-level width). Within the context of renormalization-group theory four fixed points and their associated regimes are identified: (i) the free-orbital regime which is unstable and flows into (ii) or (iv); (ii) the valence-fluctuation regime which is characteristic of the asymmetric Anderson model. Properties are dominated by a temperature-dependent impurity-orbital energy $E_d\mathrm{}\left(T\right)\mathrm{}\cong {\epsilon }_d+\left(\frac{\Gamma }{\pi }\right)\ln \left(\frac{U}{T}\right)$. If $E_d\mathrm{}\left(T\right)\mathrm{}$ is negative and large compared to $\Gamma$ as $T$ decreases, the system is unstable with respect to (iii), otherwise it flows to (iv); (iii) the local-moment regime is similar to that in the symmetric Anderson model except that it has potential scattering. That is, this regime maps onto the Kondo model with potential scattering, the latter having little effect on the susceptibility; (iv) the frozen-impurity regime, into which all the regimes above flow, is stable, having only irrelevant operators. Furthermore, in the valence-fluctuation regime nonuniversal properties are observed for $-E_d\mathrm{}\left(T\right)<\Gamma$. These conclusions are supported with extensive analytic and numerical calculations, the latter based on the numerical renormalization-group approach. Analytic formulas for the impurity susceptibility and free energy in all four regimes are presented, together with the impurity-specific heat in the frozen-impurity regime.