Functional renormalization with interaction flows: A single-boson exchange perspective and application to electron-phonon systems

The functional renormalization group (fRG) is acknowledged as a powerful tool in quantum many-body physics and beyond. On the technical side, conventional implementations of the fRG rely on regulators for bare propagators only. Starting from Schwinger–Dyson and Bethe–Salpeter equations, we develop here an fRG formulation where both bare propagators and bare interactions can be dressed with regulators. The approach thus obtained is an extension of the multiloop fRG recently introduced for many-fermion systems. Using the single-boson exchange decomposition, we show that the underlying flow equations are simply interpreted as adding a regulator to the bosonic propagator and that such an extension scarcely changes the original structure of the flow equations. Overall, we provide a framework for implementing approaches that cannot be realized with conventional fRG methods, such as temperature flows for models with retarded interactions. For concrete applications, we analyze the loop convergence of our scheme against conventional cutoff schemes for the Anderson impurity model. Finally, we devise a new temperature-flow scheme that implements a cutoff in both the propagator and the bare interaction, and demonstrate its validity on a model of an Anderson impurity coupled to a phonon.

💡 Research Summary

The paper introduces a novel functional renormalization group (fRG) scheme that extends the conventional approach by applying regulators not only to the bare propagator but also to the bare interaction. Starting from the Schwinger‑Dyson (SD) and Bethe‑Salpeter (BS) equations, the authors construct flow equations in which both the non‑interacting Green’s function (G_0) and the bare four‑point interaction (U) are dressed with scale‑dependent regulator functions (\Lambda_G) and (\Lambda_U). This “interaction flow” provides a flexible framework that can accommodate retarded (frequency‑dependent) interactions and non‑local interactions, which are difficult to treat with standard fRG where only (G_0) is regulated.

A central technical ingredient is the single‑boson exchange (SBE) decomposition of the full two‑particle vertex (V). In the SBE picture, (V) is expressed in terms of bosonic propagators (w_r) and fermion‑boson (Yukawa) vertices (\lambda_r) for each channel (r) (particle‑hole, crossed particle‑hole, particle‑particle). The decomposition is based on the concept of (U)-reducibility: diagrams that can be split by cutting a bare interaction line are grouped into a term (\nabla_r) that factorizes as (\lambda_r , w_r , \lambda_r). The remaining part, (M_r), contains the (U)-irreducible contributions. By inserting this factorization into the parquet decomposition, the authors obtain a set of self‑consistent SBE equations (Eqs. 12a‑f) that closely resemble the Bethe‑Salpeter equations but are now expressed in terms of bosonic objects.

When the regulator is added to the bosonic propagator (w_r) (or equivalently to the interaction channel), the flow equations retain essentially the same structure as the multiloop fRG derived previously for bare‑propagator regulation only. The multiloop hierarchy is preserved, and each loop order can be interpreted as adding higher‑order parquet diagrams. Importantly, the interaction flow does not introduce new diagrammatic classes; it merely modifies the internal lines of existing diagrams, which greatly simplifies implementation.

To benchmark the method, the authors study two impurity models. First, the standard Anderson impurity model (AIM) is used to compare loop convergence between the new interaction‑flow scheme and conventional cutoff schemes. The results show comparable or improved convergence, confirming that the additional regulator does not spoil the multiloop summation. Second, the Anderson–Holstein impurity model (AHIM), which couples the impurity electrons to a local phonon mode, serves as a test case for retarded interactions. By employing a temperature flow where the temperature acts as the RG scale, and by adding a regulator to the bare electron‑phonon interaction, the authors capture the temperature‑dependent renormalization of the effective electron‑phonon coupling. The computed spectral functions and susceptibilities agree with expectations from exact or numerically exact methods, demonstrating that the scheme can handle frequency‑dependent interactions that are inaccessible to conventional fRG.

Beyond these applications, the paper discusses how the interaction‑flow framework can be combined with dynamical mean‑field theory (DMFT) to form a DMF(^2)RG approach for lattice systems with extended or retarded interactions, and how the B‑reducibility concept allows the treatment of non‑local interactions within the same formalism. The authors also outline future directions, including extensions to multi‑band systems, real‑time dynamics via 2PI‑fRG, and integration with machine‑learning techniques for optimal regulator design.

In summary, the work delivers (i) a rigorous derivation of multiloop fRG flow equations that incorporate regulators on both propagators and interactions, (ii) a clear interpretation of these flows in terms of bosonic propagator regulation within the SBE decomposition, (iii) numerical evidence that the method converges efficiently and captures retarded interaction physics, and (iv) a versatile platform for future developments in strongly correlated electron systems where conventional fRG faces limitations. This contribution significantly broadens the applicability of functional renormalization group techniques to a wider class of quantum many‑body problems.