Parquet decomposition calculations of the electronic self-energy

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 nonperturbative numerical calculations. We show the specific example of dynamical mean field theory and its cluster extensions [dynamical cluster approximation (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 [O. Gunnarsson et al., 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 a resonance valence bond state, which are typical hallmarks of a pseudogap state in DCA.