Magnetic impurities in gapless Fermi systems: perturbation theory

We consider a symmetric Anderson impurity model with a soft-gap hybridization vanishing at the Fermi level: Delta(I) proportional to \ w \(r) with r > 0. Three facets of the problem are examined. First the non-interacting limit, which despite its simplicity contains much physics relevant to the U > 0 case: it exhibits both strong coupling (SC) states (for r < 1) and local moment states (for r > 1); with characteristic signatures in both spectral properties and thermodynamic functions. Second, we establish general conditions upon the interaction self-energy for the occurence of a SC state for U > 0. This leads to a pinning theorem, whereby the modified spectral function A(w) = \ w \(r) D(w) is pinned at the Fermi level w = 0 for any U where a SC state obtains. it generalizes to arbitrary r the pinning condition upon D(w = 0) familiar in the normal r = 0 Anderson model. Finally, we consider explicitly spectral functions at the simplest level: second order perturbation theory ill U, which we conclude is applicable for r < 1/2 and r > 1 but not for 1/2 < r < 1. Characteristic spectral features observed in numerical renormalization group calculations are thereby recovered, for both SC and LM phases; and for the SC state the modified spectral functions are found to contain a generalized Abrikosov-Suhl resonance exhibiting a characteristic low-energy Kondo scale with increasing interaction strength.