Vertex bosonization in the Hubbard model: a new perspective

The two-particle vertex is an essential ingredient in the study of strongly correlated electrons as it provides useful information on the collective excitations of the system. A suitable representation of the vertex as a sum of bosonic propagators not only reduces its complexity, but also helps to identify those collective fluctuations which play a decisive role in the phase transitions to ordered states. We analyze a functional renormalization group (fRG) flow in which the effective interaction diverges before reaching the final scale, signaling the onset of spontaneous symmetry breaking. While keeping the full frequency dependencies, we continue the flow by explicitly introducing a bosonic field and deriving a reduced set of flow equations for the calculation of the order parameter in a mean-field-like fashion. This method satisfies fundamental constraints, such as the Ward identities, and it can be easily combined with the most recent fRG truncations. We also extend the present formulation to the newly developed DMF2RG, which combines the dynamical mean-field theory (DMFT) and the fRG. Our method represents a convenient starting point for the inclusion of bosonic fluctuations and for the access of the symmetry broken phases also with strong coupling methods.