Singular dynamics of underscreened magnetic impurity models

We give a comprehensive analysis of the singular dynamics and of the low-energy fixed point of one-channel impurity s-d models with ferromagnetic and underscreened antiferromagnetic couplings. We use the numerical renormalization group (NRG) to perform calculations at T=0. The spectral densities of the one-electron Green's functions and t-matrices are found to have very sharp cusps at the Fermi level (omega=0), but do not diverge. The approach of the Fermi level is governed by terms proportional to 1/ln(2)(omega/T-0) as omega -> 0. The scaled NRG energy levels show a slow convergence as 1/(N+C) to their fixed point values, where N is the iteration number and C is a constant dependent on the coupling J from which the low energy scale T-0 can be deduced, with a renormalized coupling (J) over tilde(omega)=1/ln(omega/T-0). We calculate also the dynamical spin susceptibility, and the elastic and inelastic scattering cross sections as a function of omega. The inelastic scattering goes to zero as omega -> 0, as expected for a Fermi liquid, but anomalously slowly compared to the fully screened case. We obtain the asymptotic forms for the phase shifts for elastic scattering of the quasiparticles in the high-spin and low-spin channels. The asymptotic forms found for the one-electron Green's functions and t-matrices do not simply correspond to a perturbation expansion in powers of (J) over tilde(omega).