Flow Equations for Electron-Phonon Interactions

TL;DR

This paper introduces a continuous unitary transformation method to eliminate electron-phonon interactions, resulting in less singular behavior and consistent attractive interactions for Cooper pairs, improving upon traditional Fr"ohlich transformations.

Contribution

It develops a new flow equation approach that generalizes and improves the Fr"ohlich transformation for electron-phonon systems, especially in problematic regions.

Findings

Less singular interaction equations compared to Fr"ohlich's.

Always predicts attractive interactions between electrons in Cooper pairs.

Effective elimination of electron-phonon interaction even where Fr"ohlich's method fails.

Abstract

A recently proposed method of continuous unitary transformations is used to eliminate the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fr\"ohlich-transformations. The two approaches are compared. Our result will turn out to be less singular than Fr\"ohlich's. Furthermore the interaction between electrons belonging to a Cooper-pair will always be attractive in our approach. Even in the case where Fr\"ohlich's transformation is not defined (Fr\"ohlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter.