Effective electron-electron and electron-phonon interactions in the Hubbard-Holstein model

TL;DR

This paper uses the flow-equation method to analyze how electron-electron and electron-phonon interactions influence each other in the Hubbard-Holstein model, revealing momentum-dependent effective couplings and the conditions under which phonon-mediated attraction occurs.

Contribution

It provides analytical expressions for effective couplings in the Hubbard-Holstein model, highlighting the interplay between correlations and phonons across different regimes.

Findings

Electron-electron attraction peaks at low transferred momenta.

Phonons significantly modify the effective electron-electron interaction.

Characteristic energies influence the effective couplings.

Abstract

We investigate the interplay between the electron-electron and the electron-phonon interaction in the Hubbard-Holstein model. We implement the flow-equation method to investigate within this model the effect of correlation on the electron-phonon effective coupling and, conversely, the effect of phonons in the effective electron-electron interaction. Using this technique we obtain analytical momentum-dependent expressions for the effective couplings and we study their behavior for different physical regimes. In agreement with other works on this subject, we find that the electron-electron attraction mediated by phonons in the presence of Hubbard repulsion is peaked at low transferred momenta. The role of the characteristic energies involved is also analyzed.