Non-Fermi-liquid theory of a compactified Anderson single-impurity model

We consider a version of the symmetric Anderson impurity model (compactified) which has a non-Fermi-liquid weak-coupling regime. We find that in the Majorana fermion representation the perturbation theory can be conveniently developed in terms of Pfaffian determinants and we use this formalism to calculate the impurity free energy, self-energies, and vertex functions. We also derive expressions for the impurity and the local conduction-electron charge and spin-dynamical susceptibilities in terms of the impurity self-energies and vertex functions. In the second-order perturbation theory, a linear temperature dependence of the electrical resistivity is obtained, and the leading corrections to the impurity specific heat are found to behave as Tin T. The impurity static susceptibilities have terms in lnT to zero, first, and second order, and corrections of ln(2)T to second order as well. The conduction-electron static susceptibilities, and the singlet superconducting paired static susceptibility at the impurity site, are found to have second-order corrections lnT, which we interpret as an indication that a singlet conduction-electron pairing resonance forms at the Fermi level (the chemical potential). When the perturbation theory is extended to third order logarithmic divergences are found in the only vertex function Gamma(0,1,2,3)(0,0,0,0), which is nonvanishing in the zero-frequency limit. We use the multiplicative renormalization-group (RG) method to sum all the leading-order logarithmic contributions. These give rise to a weak-coupling low-temperature energy scale T-c=Delta exp[-(1/9)(pi Delta/U)(2)], which is the combination of the two independent coupling parameters. The RG scaling equation is also derived and shows that the dimensionless coupling constant (U) over bar=U/pi Delta is increased as the high-energy scale Delta is reduced, so our perturbational results can be justified in the regime T>T-c. Below T-c the perturbation theory breaks down.