International Journal of Modern Physics B Vol. 17, Nos. 18, 19 & 20 (2003) 3381–3386 c  World Scientific Publishing Company WILSON’S NUMERICAL RENORMALIZATION GROUP CALCULATION OF THE ANDERSON HOLSTEIN MODEL: PHONON PROPAGATOR HAN-YONG CHOI and TAE-HO PARK * Department of Physics, BK21 Physics Research Division, CNNC, and Institute of Basic Science Research, Sung Kyun Kwan University, Suwon 440-746, Korea * hychoi@skku.ac.kr GUN SANG JEON The Pennsylvania State University, Department of Physics, University Park, PA 16802, USA Received 16 January 2003 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 & gco 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, can be either positive or negative depending on U and the hybridization Δ. Keywords : Numerical renormalization group; Anderson–Holstein model; phonon Green’s function. 1. Introduction An important problem in interacting many particle systems is understanding the interplay between the electron-electron and electron-phonon interactions. Although some perturbative approaches have been proposed to understand the interplay, 1 there has not been much progress in this field partially because there are no reliable calculation schemes applicable in strong coupling regime where standard perturbation methods break down. Some non-perturbative techniques have been developed for strongly correlated impurity model in the context of the Kondo problem. Among them, Wilson’s numerical renormalization group (NRG) is particularly powerful in that it is non-perturbative in nature so that it can cover the 3381