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

Abstract

We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertzapproach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In particular, we show how the exact solution of the Tomonaga-Luttinger model (i.e., one-dimensional fermions with linear energy dispersion and interactions involving only small momentum transfers) emerges from the functional RG if one works with a momentum transfer cutoff. Then the Ward identities associated with the local particle conservation at each Fermi point are valid at every stage of the RG flow and provide a solution of an infinite hierarchy of flow equations for the irreducible vertices. The RG flow equation for the irreducible single-particle self-energy can then be closed and can be reduced to a linear integro-differential equation, the solution of which yields the result familiar from bosonization. We suggest new truncation schemes of the exact hierarchy of flow equations, which might be useful even outside the weak coupling regime.

Aspects of the functional renormalisation group

“…Flows with the coupling to composite operators have been considered in e.g. [21,41,77,[79][80][81][82]. Flows for the 2PI effective action have been studied in [77,79,82].…”

“…This includes the bosonisation of fermionic degrees of freedom [72,73,78,80], e.g. in low-energy QCD, where the relevant degrees of freedom are mesons and baryons instead of quarks.…”

“…In theories with a complicated phase structure this might necessitate introducing a large number of vertices to the effective action in terms of the fundamental fields. A way to avoid such a drawback is to reparameterise the theory in terms of the relevant degrees of freedom [41,[72][73][74][75][76][78][79][80][81][82].…”

Dynamic structure factor of Luttinger liquids with quadratic energy dispersion and long-range interactions

“…In appendix D we justify this procedure using a functional renormalization group approach 36,37 . With this substitution, Eq.…”

“…We use the momentum-transfer cutoff scheme proposed in Ref. [36], where only the free bosonic part S 0 [φ] is regularized by suppressing bosonic fluctuations with momenta q smaller than a certain cutoff Λ. One possibility is to introduce the cutoff as a multiplicative Θ-function 42 by replacing in Eq.…”