We use the framework setup recently to compute nonperturbatively inelastic scattering from quantum impurities [G. Zar\'and et al., Phys. Rev. Lett. 93, 107204 (2004)] to study the energy dependence of the single-particle
S matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin
S=1∕2 two-channel Kondo model, the Anderson model, and the usual
S=1∕2 single-channel Kondo model. We discuss the difference between non-Fermi-liquid and Fermi-liquid models and study how a crossover between the non-Fermi-liquid and Fermi-liquid regimes appears in the case of channel anisotropy for the
S=1∕2 two-channel Kondo model. We show that for the most elementary non-Fermi-liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the
T matrix to local correlation functions are also presented.
We investigate a modified Anderson model at the large-N limit, where the Coulomb interaction is replaced by the Sachdev–Ye–Kitaev random interaction. The resistivity of conduction electron ρ c has a minimum value around temperature T*, which is similar to the Kondo system, but the impurity electron’s density of state A d(ω) demonstrates no sharp-peak like the Kondo resonance around the Fermi surface. This provides a counterintuitive example where resistivity minimum exists without Kondo resonance. The impurity electron’s entropy S d and specific heat capacity C v show a crossover from Fermi liquid to a non-Fermi liquid behavior dependent on temperature. The system is a Fermi liquid at T < T*, and becomes a non-Fermi liquid at T > T*, and then becomes a Fermi gas at sufficiently high temperatures T ≫ T*. The non-Fermi liquid at the intermediate-T regime does not occur in the standard Anderson model. We also make a renormalization group analysis, which confirms the crossover from Fermi liquid to the non-Fermi behavior. It is emphasized that the resistivity minimum emerges in our model when the system behaves as a non-Fermi liquid rather than Fermi liquid, which provides an alternative example showing resistivity minimum in condensed matter physics.