Effective strong-coupling Hamiltonians for bipolaron centers and magnetic impurities with on-site electron-phonon coupling

Abstract

We have studied a model of bipolaron (negative-U) centers and magnetic impurities which includes a repulsive on-site Coulomb interaction as well as a coupling of the electronic on-site energy to the displacement amplitude of a local phonon mode. By means of a generalized Schrieffer-Wolff transformation, we derive an effective strong-coupling (or weak-hybridization) Hamiltonian for the low-lying states of the system. In the case of a bipolaron center (where the phonon-mediated on-site attraction dominates over the Coulomb repulsion), the effective Hamiltonian describes conduction-electron scattering and electron-pair tunneling processes between the localized impurity orbital and the conduction band of the metallic host. In the limit of very strong electron-phonon coupling, the pair tunneling matrix elements become exponentially small, whereas conduction electron scattering is suppressed only algebraically. The impurity orbital occupation number is then almost ``frozen in'' and the bipolaron center becomes inefficient in enhancing the superconducting pairing correlations of the host. In the case of a magnetic impurity (with a predominant repulsive Coulomb interaction), the effective Hamiltonian reduces to the form of an ordinary Kondo spin-exchange model. The spin-exchange coupling constants are not affected by the presence of the electron-phonon interaction.