Local polaron effects in mixed-valence systems: Exact model calculation in the limit of large degeneracy

The Anderson impurity model including a linear coupling to a local boson mode is solved exactly in the large-degeneracy limit for infinitely large $f-f$ Coulomb repulsion. Ground-state properties and the $f$-level Green's function are discussed. For a small boson frequency ${\omega }_0$, two different types of mixed-valence behavior occur, depending on the strength $v$ of the electron-phonon coupling. In the weak-coupling limit the "lattice" shows small fluctuations around its average position, while for a strong-coupling $v$ there exists a narrow regime of energies, ${\epsilon }_f$, of the $f$-level where mixed-valence behavior with large mean-square "lattice" deviations occurs. For finite phonon frequencies ${\omega }_0$, quantum fluctuations smooth out the first-order transition occurring in the limit ${\omega }_0\to 0$. In the weak-coupling limit, the mean-field approximation of the electron-phonon coupling, leading to a renormalization of the $f$-level position, provides a good description of the ground-state properties and the $f$-level spectrum. In the strong-coupling mixed-valence regime, some ground-state properties can be interpreted in terms of a renormalized $f$-electron-conduction-electron coupling $\tilde{\Delta }\to \tilde{\Delta }\exp \left(\frac{-v}{{\omega }_0}\right)\equiv \overline{\Delta }$. This renormalization does not occur for the width of $f$-level peak in the one-particle Green's function. In this regime, the $f$-level spectral function can be described as a superposition of two spectra. The relative weight of these two spectra varies rapidly when ${\epsilon }_f$ varies of the order $\overline{\Delta }$ in the transition regime, while the individual spectra change little. The large-boson-frequency limit ("plasmon case") is also discussed with special emphasis on the renormalization occurring in this limit.