In this work we present a comprehensive analysis of collective electronic fluctuations and their effect on single-particle properties of the Hubbard model. Our approach is based on a standard dual fermion/boson scheme with the interaction truncated at the two-particle level. Within this framework we compare various approximations that differ in the set of diagrams (ladder vs exact diagrammatic Monte Carlo), and/or in the form of the four-point interaction vertex (exact vs partially bosonized). This allows to evaluate the effect of all components of the four-point vertex function on the electronic self-energy. In particular, we observe that contributions that are not accounted for by the partially bosonized approximation for the vertex have only a minor effect on electronic degrees of freedom in a broad range of model parameters. In addition, we find that in the regime, where the ladder dual fermion approximation provides an accurate solution of the problem, the leading contribution to the self-energy is given by the longitudional bosonic modes. This can be explained by the fact that contributions of transverse particle-hole and particle-particle modes partially cancel each other. Our results justify the applicability of the recently introduced dual triply irreducible local expansion (D-TRILEX) method that represents one of the simplest consistent diagrammatic extensions of the dynamical mean-field theory. We find that the self-consistent D-TRILEX approach is reasonably accurate also in challenging regimes of the Hubbard model, even where the dynamical mean-field theory does not provide the optimal local reference point (impurity problem) for the diagrammatic expansion.
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