  We generalize our recently developed super-field functional renormalization
group (RG) method involving both Fermi and Bose fields [F. Schuetz, L.
Bartosch, and P. Kopietz, Phys. Rev. B 72, 035105 (2005)] to include the
possibility that some bosonic components of the field have a finite vacuum
expectation value. We derive an exact hierarchy of flow equations for the
one-line irreducible vertices and the vacuum expectation value of the field. We
apply our method to an interacting Fermi system where the interaction can be
decoupled in the zero-sound channel and is then mediated by a collective
bosonic field. The vacuum expectation value of the zero-frequency and
zero-momentum component of the bosonic field is then closely related to the
fermionic density. This can be exploited to calculate the compressibility of
the interacting system. By using a cutoff in the bosonic propagator, the RG can
be set up such that the self-consistent Hartree approximation is imposed as
initial condition for the RG flow and the corrections to this approximation are
generated as the remaining degrees of freedom are successively eliminated by
the RG procedure.
