Renormalization of the electron-phonon interaction: a reformulation of the BCS-gap equation

TL;DR

This paper applies a renormalization method to derive a BCS-like gap equation for electron-phonon systems, revealing that phonon-induced interactions are finite and energy-dependent, refining understanding of superconductivity mechanisms.

Contribution

It introduces a renormalization approach that yields a singularity-free, energy-dependent effective electron-electron interaction and a reformulated BCS gap equation.

Findings

Effective interaction is free of singularities.

Phonon contributions are limited to energy differences below phonon energy.

Derived a BCS-like gap equation from the renormalization framework.

Abstract

A recently developed renormalization approach is used to study the electron-phonon coupling in many-electron systems. By starting from an Hamiltonian which includes a small gauge symmetry breaking field, we directly derive a BCS-like equation for the energy gap from the renormalization approach. The effective electron-electron interaction for Cooper pairs does not contain any singularities. Furthermore, it is found that phonon-induced particle-hole excitations only contribute to the attractive electron-electron interaction if their energy difference is smaller than the phonon energy.