Numerical renormalization-group study of the symmetric Anderson-Holstein model: Phonon and electron spectral functions

We study the symmetric Anderson-Holstein (AH) model at zero temperature with Wilson’s numerical renormalization-group (NRG) technique to investigate the interplay between the electron-electron and electron-phonon interactions. An improved method for calculating the phonon propagator using the NRG technique is presented. It is more accurate and reliable than the previous methods in that it is written in the form of the Dyson equation which calculates the phonon renormalization explicitly, and satisfies the boson sum rule better. The method is applied to calculate the renormalized phonon propagators along with the electron propagators as the on-site Coulomb repulsion U and electron-phonon coupling constant g are varied. As g is increased from 0, the phonon mode gets softened (hardened) for U≳(≲)Us(Δ) depending on the parameters of the AH model, U, and the hybridization Δ. And, for g≳g_co, one crosses over to the regime where the phonon mode splits into two components, one of which developing into a soft mode and the other approaching back to the bare frequency as g is further increased. Correlated with the emergence of the soft mode, the central peak of the electron spectral function is severely suppressed. These NRG calculations are compared with the standard Green’s-function results for the weak-coupling regime to understand the phonon renormalization and soft mode.