SU(4)symmetry for strongly correlated electrons: Kondo and mixed-valence effects in terms of Gell-Mann matrices

Abstract

The concept of dynamical symmetries is used for formulation of the renormalization group approach to the Kondo effect in the Anderson model with repulsive and attractive interaction $U$ in the Kondo and mixed valence regimes. It is shown that the generic local dynamical symmetry of the Anderson Hamiltonian is determined by the $SU(4)$ Lie group. The Anderson Hamiltonian is rewritten in terms of the Gell-Mann matrices of fourth rank, which form the set of group generators and the basis for construction of the irreducible vector operators describing the excitation spectra in the charge and spin sectors. The multistage Kondo screening is interpreted as a consecutive reduction of local $SU(n)$ dynamical symmetries from $SU(4)$ to $SU(2)$. It is shown that the similarity between the conventional Kondo cotunneling effect for spin 1/2 in the positive $U$ model and the Kondo resonance for pair tunneling in the negative $U$ model is a direct manifestation of implicit $SU(4)$ symmetry of the Anderson/Kondo model. The relations between the local $SU(4)$ dynamical symmetry and the global $SO(4)$ symmetry in the Hubbard model are discussed in brief.