Renormalization-group study of Anderson and Kondo impurities in gapless Fermi systems

Abstract

Thermodynamic properties are presented for four magnetic impurity models describing the scattering of fermions from a localized orbital at an energy-dependent rate $\ensuremath{\Gamma}(\ensuremath{\epsilon})$ which vanishes precisely at the Fermi level, $\ensuremath{\epsilon}=0$. Specifically, it is assumed that for small $|\ensuremath{\epsilon}|$, $\ensuremath{\Gamma}(\ensuremath{\epsilon})\ensuremath{\propto}|\ensuremath{\epsilon}{|}^{r}$ with $r>0$. The cases $r=1$ and $r=2$ describe dilute magnetic impurities in unconventional $(d$- and $p$-wave) superconductors, ``flux phases'' of the two-dimensional electron gas, and certain zero-gap semiconductors. For the nondegenerate Anderson model, the main effects of the depression of the low-energy scattering rate are the suppression of mixed valence in favor of local-moment behavior and a marked reduction in the exchange coupling on entry to the local-moment regime, with a consequent narrowing of the range of parameters within which the impurity spin becomes Kondo screened. The precise relationship between the Anderson model and the exactly screened Kondo model with power-law exchange is examined. The intermediate-coupling fixed point identified in the latter model by Withoff and Fradkin (WF) is shown to have clear signatures both in the thermodynamic properties and in the local magnetic response of the impurity. The underscreened, impurity-spin-1 Kondo model and the overscreened, two-channel Kondo model both exhibit a conditionally stable intermediate-coupling fixed point in addition to unstable fixed points of the WF type. In all four models, the presence or absence of particle-hole symmetry plays a crucial role in determining the physics both at strong coupling and in the vicinity of the WF transition. These results are obtained using an extension of Wilson's numerical renormalization-group technique to treat energy-dependent scattering. The strong- and weak-coupling fixed points of each model are identified and their stability is analyzed. Algebraic expressions are derived for the fixed-point thermodynamic properties and for low-temperature corrections about the stable fixed points. Numerical data are presented confirming the algebraic results, identifying and characterizing intermediate-coupling (non-Fermi-liquid) fixed points, and exploring temperature-driven crossovers between different physical regimes.