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We develop a new formulation of the functional renormalization group (RG) forinteracting fermions. Our approach unifies the purely fermionic formulationbased on the Grassmannian functional integral, which has been used in recentyears by many authors, with the traditional Wilsonian RG approach to quantumsystems pioneered by Hertz [Phys. Rev. B 14, 1165 (1976)], which attempts todescribe the infrared behavior of the system in terms of an effective bosonictheory associated with the soft modes of the underlying fermionic problem. Inour approach, we decouple the interaction by means of a suitableHubbard-Stratonovich transformation (following the Hertz-approach), but do noteliminate the fermions; instead, we derive an exact hierarchy of RG flowequations for the irreducible vertices of the resulting coupled field theoryinvolving both fermionic and bosonic fields. The freedom of choosing a momentumtransfer cutoff for the bosonic soft modes in addition to the usual band cutofffor the fermions opens the possibility of new RG schemes. In particular, weshow how the exact solution of the Tomonaga-Luttinger model emerges from thefunctional RG if one works with a momentum transfer cutoff. Then the Wardidentities associated with the local particle conservation at each Fermi pointare valid at every stage of the RG flow and provide a solution of an infinitehierarchy of flow equations for the irreducible vertices. The RG flow equationfor the irreducible single-particle self-energy can then be closed and can bereduced to a linear integro-differential equation, the solution of which yieldsthe result familiar from bosonization. We suggest new truncation schemes of theexact hierarchy of flow equations, which might be useful even outside the weakcoupling regime.Comment: 27 pages, 15 figures; published version, some typos correcte

Collective fields in the functional renormalization group for fermions, Ward identities, and the exact solution of the Tomonaga-Luttinger model