One-, Two-, and Three-channel Kondo effects for a model Ce$^{3+}$ Impurity in a Metal
We present studies of a simple Anderson model Hamiltonian for Ce$^{3+}$ ions in cubic symmetry with three configurations ($f^0$,$f^1$,$f^2$). Our Hamiltonian includes: (i) One-channel Anderson model; (ii) Two-channel Anderson model. Using the third order scaling analysis, we study stability of the non-Fermi liquid fixed point of the two-channel Kondo model for Ce$^{3+}$ ions in cubic symmetry against the one-channel Kondo interaction. Using the non-crossing approximation (NCA), we also report detailed studies of our simplified model of one-channel \& two-channel Anderson model which exhibits competition between the Fermi-liquid fixed point of the one-channel Kondo model and the non-Fermi fixed point of the two-channel Kondo model. We provide the phase diagram in the model parameter space and study the thermodynamics and the transport properties of our simplified model Hamiltonian. Thermodynamics and transport coefficients show the distinct behaviors for different numbers of channels. We confirm in detail that the NCA is a valid numerical method for the overcompensated multichannel $S_I=1/2$ Anderson models. Our model study might be relevant to the non-Fermi liquid alloy Ce$_{1-x}$La$_x$Cu$_{2.2}$Si$_2$.