Scaling Theory of the Asymmetric Anderson Model

A scaling theory is used to show that for temperatures $T\ll U$, the properties of the asymmetric Anderson model ($U\gg \left|E_d\right|, \Delta$) are universal functions of the scaling invariants $\Delta$ and ${E_d}^{*}=E_d+\left(\frac{\Delta }{\pi }\right)\ln \left(\frac{W_0}{\Delta }\right)$, where $W_0$ is the conduction electron bandwidth or $U$, whichever is smaller. Crossovers between various regimes of simple behavior as the temperature changes are described. $\left|{E_d}^{*}\right|\lesssim \Delta$ is identified as the criterion for a "mixed-valence" ground state, where the susceptibility $\approx {\Delta }^{-1\mathrm{}}$. For $-{E_d}^{*}\gg \Delta$, there is a local-moment regime with a Kondo temperature $T_{\mathrm{K}}\approx \Delta \exp \left(\frac{\pi {E_d}^{*}}{2\Delta }\right)$.