We show that the traditional classification of quantum inpurity models based on thermodynamics is insufficient to probe the nature of their low-energy dynamics. We propose an analysis based on scattering theory, dividing Fermi liquids into regular Fermi liquids and singular Fermi liquids. In both cases electrons at the Fermi energy scatter elastically off the impurity, but in the case of regular Fermi liquids the scattering has analytical properties in the vicinity of the Fermi energy, while for singular Fermi liquids it does not, resulting in a breakdown of Nozi\`eres' Fermi-liquid picture and singular thermodynamic behavior. Using the Bethe ansatz and numerical renormalization group, we show that the ordinary Kondo model is a regular Fermi liquid whereas the underscreened Kondo model is a a singular Fermi liquid. Conventional regular Fermi liquid behavior is reestablished in an external magnetic field
H, but with a density of states which diverges as
1∕H. Our results may be relevant for the recently observed field-tuned quantum criticality in heavy electron materials.
The mechanisms leading to instability of the non-Fermi-liquid state of a Luttinger liquid with two-level impurities are proposed. Since exchange scattering in 1D systems is two-channel scattering in a certain range of parameters, several types of non-Fermi-liquid excitations with different quantum numbers exist in the vicinity of the Fermi level. These excitations include, first, charge density fluctuations in the Luttinger liquid and, second, many-particle excitations due to two-channel exchange interaction, which are associated with band-type as well as impurity fermion states. It is shown that mutual scattering of many-particle excitations of various types leads to the emergence of an additional Fermi-liquid singularity in the vicinity of the Fermi level. The conditions under which the Fermi-liquid state with a new energy scale (which is much smaller than the Kondo temperature) is the ground state of the system are formulated.
By a mix of form factors and analyticity techniques, we determine some fundamental scattering amplitudes in non-Fermi-liquid systems. These include the reflection and transmission amplitudes for Laughlin quasiparticles at a point contact between two
g=21 Luttinger liquids, and the $\stackrel{\ensuremath{\rightarrow}}{e}e,$ $\mathrm{lq}\stackrel{\ensuremath{\rightarrow}}{p}\mathrm{lqp}$ and $\stackrel{\ensuremath{\rightarrow}}{e}{e}_{\mathrm{edge}}$ at a point contact between a Fermi liquid and a
g=31 Luttinger liquid (all liquids chiral or not). These results are obtained in closed form, and give rise to rather simple expressions for the probabilities of the most basic processes of non-Fermi-liquid physics at these special values of the couplings.