New approach to the mixed-valence problem

A new formulation of the mixed-valence problem is presented in which the singlet valence state of a rare-earth ion is represented by a zero-energy boson and the spinning state by a spin-$j$ fermion. This representation avoids the need to use Hubbard operators with awkward algebras and avails itself of standard techniques for dealing with interacting quantum systems. In particular, a Feynman-diagram expansion for the thermodynamic variables and spectral functions can be developed. The advantages of the approach are illustrated for the mixed-valence impurity problem. Vertex corrections are found to be $O(\frac{1}{{N}^{2}})$, where $N$ is the degeneracy of the rare-earth ion, allowing a self-consistent calculation of the $f$-electron spectral function to order $O(\frac{1}{{N}^{2}})$ that is valid in both the mixed-valence and Kondo regimes. The extension to the lattice is outlined and some preliminary results reported.