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

TL;DR

This paper uses an improved numerical renormalization group method to analyze the interplay of electron-electron and electron-phonon interactions in the symmetric Anderson-Holstein model, revealing phonon mode renormalization and soft mode emergence.

Contribution

An enhanced NRG approach for calculating phonon propagators explicitly, improving accuracy and satisfying the boson sum rule, applied to study phonon and electron spectral functions.

Findings

Phonon mode renormalization depends on U and g.

Soft mode appears when g exceeds a critical value.

Electron spectral function shows a suppressed central peak with soft mode emergence.

Abstract

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, which turns out to be more accurate and reliable than the previous works in that it 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 onsite Coulomb repulsion $U$ and electron-phonon coupling constant $g$ are varied. As $g$ is increased, the phonon mode is successively renormalized, and for $g \gtrsim g_{co}$ 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 0, depends on $U$ and the hybridization $\Delta$; it gets softened (hardened) for $U \gtrsim (\lesssim) U_s (\Delta)$. Correlated with the emergence of the soft mode is the central peak of the electron spectral function severely suppressed. These NRG calculations will be compared with the standard Green's function results for the weak coupling regime to understand the phonon renormalization and soft mode.