Bosonic effective action for interacting fermions

We compare different versions of a bosonic description for systems of interacting fermions, with particular emphasis on the free-energy functional. The bosonic effective action makes the issue of symmetries particularly transparent. A systematic expansion for the bosonic effective action starts with a solution to the lowest-order Schwinger-Dyson or gap equation. We propose a two-particle irreducible formulation of an exact functional renormalization group equation for computations beyond leading order. On this basis we suggest a renormalized gap equation. This approach is compared with functional renormalization in a partially bosonized setting