Sum rules and vertex corrections for electron-phonon interactions

TL;DR

This paper derives sum rules for electron-phonon interactions in the Holstein-Hubbard model, investigates their validity at finite Coulomb interactions, and emphasizes the importance of vertex corrections in different doping regimes.

Contribution

It introduces sum rules for self-energies in the Holstein-Hubbard model and analyzes the role of vertex corrections using exact diagonalization and dynamical mean-field theory.

Findings

Vertex corrections are crucial for the electron self-energy sum rule in undoped systems.

The phonon self-energy sum rule holds even without vertex corrections in doped systems.

Explicit vertex function results are provided for a two-site model.

Abstract

We derive sum rules for the phonon self-energy and the electron-phonon contribution to the electron self-energy of the Holstein-Hubbard model in the limit of large Coulomb interaction U. Their relevance for finite U is investigated using exact diagonalization and dynamical mean-field theory. Based on these sum rules, we study the importance of vertex corrections to the electron-phonon interactionin a diagrammatic approach. We show that they are crucial for a sum rule for the electron self-energy in the undoped system while a sum rule related to the phonon self-energy of doped systems is satisfied even if vertex corrections are neglected. We provide explicit results for the vertex function of a two-site model.