Similarity renormalization of the electron--phonon coupling

TL;DR

This paper applies similarity renormalization to the electron-phonon Hamiltonian to derive an effective interaction that promotes superconductivity, avoiding degeneracy issues and calculating critical temperatures.

Contribution

It introduces a similarity renormalization approach to compute effective electron-electron interactions mediated by phonons, providing a new method to analyze superconductivity.

Findings

Effective electron-electron interaction is attractive across parameter space.

Renormalization affects electronic energies and critical temperature.

Method avoids singularities due to degeneracies.

Abstract

We study the problem of the phonon-induced electron-electron interaction in a solid. Starting with a Hamiltonian that contains an electron-phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitely. The effective interactions are calculated to second order in the electron-phonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. For a simple Einstein model we calculate the renormalization of the electronic energies and the critical temperature of superconductivity.