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In the study of correlated systems, approximations based on the dynamicalmean-field theory (DMFT) provide a practical way to take local vertexcorrections into account, which capture, respectively, particle-particlescreening at weak coupling and the formation of the local moment at strongcoupling. We show that in both limits the local vertex corrections can beefficiently parametrized in terms of single-boson exchange, such that thetwo-particle physics described by DMFT and its diagrammatic extensions isrecovered to good approximation and at a reduced computational cost. Ourinvestigation highlights the importance of the frequency-dependentfermion-boson coupling (Hedin vertex) for local vertex corrections. Namely, atweak coupling the fermion-spin-boson coupling suppresses the N\'eel temperatureof the DMFT approximation compared to the static mean-field, whereas for largeinteraction it facilitates a huge enhancement of local spin-fluctuationexchange, giving rise to the effective-exchange energy scale $4t^2/U$. We findthat parametrizations of the vertex which neglect the nontrivial part of thefermion-boson coupling fail qualitatively at strong coupling.

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