Retardation effects and the Coulomb pseudopotential in the theory of superconductivity

TL;DR

This paper investigates how retardation effects influence the Coulomb pseudopotential in electron-phonon superconductivity, showing that higher-order corrections can increase the pseudopotential and decrease the transition temperature, with results supported by dynamical mean-field theory.

Contribution

It develops a perturbation expansion for the Coulomb pseudopotential within the Hubbard-Holstein model, incorporating renormalization and retardation effects beyond conventional approaches.

Findings

Retardation effects persist at higher orders but are less effective due to bandwidth reduction.

Higher-order corrections can lead to larger pseudopotential values.

Perturbative results agree with non-perturbative dynamical mean-field calculations.

Abstract

In the theory of electron-phonon superconductivity both the magnitude of the electron-phonon coupling $\lambda$ as well as the Coulomb pseudopotential $\mu^*$ are important to determine the transition temperature $T_c$ and other properties. We calculate corrections to the conventional result for the Coulomb pseudopotential. Our calculation are based on the Hubbard-Holstein model, where electron-electron and electron-phonon interactions are local. We develop a perturbation expansion, which accounts for the important renormalization effects for the electrons, the phonons, and the electron-phonon vertex. We show that retardation effects are still operative for higher order corrections, but less efficient due to a reduction of the effective bandwidth. This can lead to larger values of the pseudopotential and reduced values of $T_c$. The conclusions from the perturbative calculations are corroborated up to intermediate couplings by comparison with non-perturbative dynamical mean-field results.