Accessing the ordered phase of correlated Fermi systems: Vertex bosonization and mean-field theory within the functional renormalization group

We present a consistent fusion of functional renormalization group and mean-field theory which explicitly introduces a bosonic field via a Hubbard-Stratonovich transformation at the critical scale, at which the order sets in. We show that a minimal truncation of the flow equations, that neglects order-parameter fluctuations, is integrable and fulfills fundamental constraints as the Goldstone theorem and the Ward identity connected with the broken global symmetry. To introduce the bosonic field, we present a technique to factorize the most singular part of the vertex, even when the full dependence on all its arguments is retained. We test our method on the two-dimensional attractive Hubbard model at half-filling and calculate the superfluid gap as well as the Yukawa couplings and residual two-fermion interactions in the ordered phase as functions of fermionic Matsubara frequencies. Furthermore, we analyze the gap and the condensate fraction for weak and moderate couplings and compare our results with previous functional renormalization group studies, and with quantum Monte Carlo data. Our formalism constitutes a convenient starting point for the inclusion of order-parameter fluctuations by keeping a full, nonsimplified, dependence on fermionic momenta and/or frequencies.