Wilson's numerical renormalization group calculation of the Anderson-Holstein model: Phonon propagator

We study the symmetric Anderson-Holstein (AH) model at zero temperature with Wilson's numerical renormalization group (NRG) technique to study the interplay between the electron-electron and electron-phonon interactions. An improved method for calculating the phonon propagator using the NRG technique is presented and applied to the AH model as the onsite Coulomb repulsion U and electron-phonon coupling constant g are varied. As g is increased, the phonon propagator is successively renormalized, and for g ≳ g_co crosses over to the regime where the mode splits into two components, one of which approaches back to the bare frequency and the other develops into a soft mode. The initial renormalization of the phonon mode, as g is increased from O can be either positive or negative depending on U and the hybridization Δ.