General Shiba mapping for on-site four-point correlation functions

By applying the Shiba mapping on the two particle level, we derive the relation between the local four-point correlation functions of bipartite lattice models with on-site electronic repulsion and those of the corresponding models with attractive interaction in the most general setting. In particular, we extend the results of [Phys. Rev. B, 101, 155148 (2020)], which were limited to the rather specific situation of the static limit in strictly particle-hole symmetric models, (i) by explicitly including both magnetic field and different values of the chemical potentials, and (ii) by considering the full dependence of the generalized susceptibilities on the transfer (bosonic) Matsubara frequency. The derived formalism is then applied, as a relevant benchmark, to the Hubbard atom, by investigating the general properties of the divergences of its irreducible vertex functions as a function of chemical potential and applied magnetic field. The resulting phase-diagrams provide an insightful compass for future studies of the breakdown of the self-consistent perturbation expansion beyond high-symmetric regimes.