  The parquet decomposition of the self-energy into classes of diagrams, those
associated with specific scattering processes, can be exploited for different
scopes. In this work, the parquet decomposition is used to unravel the
underlying physics of non-perturbative numerical calculations. We show the
specific example of dynamical mean field theory (DMFT) and its cluster
extensions (DCA) applied to the Hubbard model at half-filling and with hole
doping: These techniques allow for a simultaneous determination of two-particle
vertex functions and self-energies, and hence, for an essentially "exact"
parquet decomposition at the single-site or at the cluster level. Our
calculations show that the self-energies in the underdoped regime are dominated
by spin scattering processes, consistent with the conclusions obtained by means
of the fluctuation diagnostics approach [Phys. Rev. Lett. 114, 236402 (2015)].
However, differently from the latter approach, the parquet procedure displays
important changes with increasing interaction: Even for relatively moderate
couplings, well before the Mott transition, singularities appear in different
terms, with the notable exception of the predominant spin-channel. We explain
precisely how these singularities, which partly limit the utility of the
parquet decomposition, and - more generally - of parquet-based algorithms, are
never found in the fluctuation diagnostics procedure. Finally, by a more
refined analysis, we link the occurrence of the parquet singularities in our
calculations to a progressive suppression of charge fluctuations and the
formation of an RVB state, which are typical hallmarks of a pseudogap state in
DCA.
