Parametrizations of local vertex corrections from weak to strong coupling: importance of the Hedin three-leg vertex

TL;DR

This paper demonstrates that local vertex corrections in correlated systems can be effectively parametrized using single-boson exchange, emphasizing the crucial role of the frequency-dependent Hedin vertex in capturing physics across weak to strong coupling regimes.

Contribution

It introduces a parametrization of local vertex corrections via single-boson exchange that accurately reproduces two-particle physics and highlights the importance of the Hedin three-leg vertex.

Findings

Fermion-boson coupling suppresses Néel temperature at weak coupling.

Large interactions enhance local spin-fluctuation exchange.

Neglecting the fermion-boson coupling leads to qualitative failures at strong coupling.

Abstract

In the study of correlated systems, approximations based on the dynamical mean-field theory (DMFT) provide a practical way to take local vertex corrections into account, which capture, respectively, particle-particle screening at weak coupling and the formation of the local moment at strong coupling. We show that in both limits the local vertex corrections can be efficiently parametrized in terms of single-boson exchange, such that the two-particle physics described by DMFT and its diagrammatic extensions is recovered to good approximation and at a reduced computational cost. Our investigation highlights the importance of the frequency-dependent fermion-boson coupling (Hedin vertex) for local vertex corrections. Namely, at weak coupling the fermion-spin-boson coupling suppresses the N\'eel temperature of the DMFT approximation compared to the static mean-field, whereas for large interaction it facilitates a huge enhancement of local spin-fluctuation exchange, giving rise to the effective-exchange energy scale $4t^2/U$. We find that parametrizations of the vertex which neglect the nontrivial part of the fermion-boson coupling fail qualitatively at strong coupling.