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

TL;DR

The paper develops an extended functional renormalization group framework that incorporates regulators in both the bare propagator and bare interaction, enabling retarded interactions and nonlocal starting points. It unites the Bethe–Salpeter/Schwinger–Dyson-based multiloop fRG with the single-boson exchange decomposition and the B-reducibility concept, recasting the two-particle vertex in terms of bosonic propagators w_r and fermion-boson vertices λ_r. A key advance is the generalization to non-local interactions and the derivation of regulator-driven flow equations for w_r, λ_r, λ̄_r, M_r, and the self-energy, including a Katanin-type substitution via a polarization P_r. The framework is validated numerically on AIM and AHIM, showing loop convergence and parquet PA accuracy, and enabling temperature flows for retarded interactions, with implications for starting from correlated reference frames (eg, DMF2RG) and building Ward identity–respecting RG schemes.

Abstract

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.

Figures (11)

• Figure 1: Examples of diagrams contributing to the parquet decomposition \ref{['eq:parquetDecomposition']}. The vertices $\phi_r$ contain all diagrams that are 2PR in channel $r$, whereas the diagrams that are 2PI (i.e., not 2PR in any channel) are included in $I^{\text{2PI}}$. By definition, the diagrams contributing to $\phi_r$ can be split into two disconnected parts by cutting two propagator lines that form the $\Pi_r$ bubbles defined by Eqs. \ref{['eq:Bubbles']}. The bubbles $\Pi_{ph}$, $\Pi_{\overline{ph}}$ and $\Pi_{pp}$ are respectively made of two transverse antiparallel, two antiparallel and two parallel lines.
• Figure 2: Diagrammatic classification underlying the SBE decomposition of the two-particle vertex $V$ in the $ph$ channel. (a) Examples of diagrams of the $U$-reducible vertex $\nabla_{ph}$ and the SBE rest function $M_{ph}$ introduced in the SBE decomposition based on $U$-reducibility, i.e., Eq. \ref{['eq:SBEDecomposition']}. The vertex $\nabla_{ph}$ is the sum of all diagrams of $V$ which are 2PR and $U$-reducible in the $ph$ channel whereas the SBE rest function $M_{ph}$ contains all 2PR diagrams in the $ph$ channel that are not $U$-reducible. (b) Splitting of the bare interaction $U$ into a bosonic part $\mathcal{B}_{ph}$ and a fermionic part $\mathcal{F}_{ph}$, which corresponds to Eqs. \ref{['eq:UrBrFr']} and \ref{['eq:UrBrFrfull']} for $r=ph$. (c) Examples of diagrams contributing to the $\mathcal{B}_{ph}$-reducible vertex $\nabla^{(\mathcal{B})}_{ph}$ and the corresponding SBE rest function $M^{(\mathcal{B})}_{ph}$ introduced in the SBE decomposition based on $\mathcal{B}$-reducibility, i.e., Eq. \ref{['eq:SBEDecompositionBreducibility']}. The vertices $\nabla^{(\mathcal{B})}_{ph}$ and $M^{(\mathcal{B})}_{ph}$ include all $\mathcal{B}_{ph}$-reducible and 2PR $\mathcal{B}_{ph}$-irreducible diagrams in the $ph$ channel, respectively.
• Figure 3: Bosonic frequency dependence of susceptibilities and fermion-boson vertices for the AIM in the magnetic and density channels at $\beta = 5$, $U = 1.4$, $\Delta_0 = \pi/5$ and at half filling. The results are shown at $1\ell$ and $\infty \ell$ (fully converged) multiloop order for the two flow schemes specified in the text. Note that the $\mathrm{D}$ and $\mathrm{SC}$ channels are degenerate, i.e., $\chi_\mathrm{D}=\chi_\mathrm{SC}$ and $\lambda_\mathrm{D}=\lambda_\mathrm{SC}$. The insets in the left-hand panels show the same susceptibilities as the corresponding main panels, but magnified around $\Omega=0$. It is observed that, like the $U$-flow (2004), the $U$-flow (2025) converges to the PA solution, which, by construction, does not depend on the choice of a cutoff scheme. The panels on the right-hand side show the average number of loop corrections per self-energy iteration $\left\langle N_{\ell} \right\rangle_{\Sigma}$ and the corresponding number of self-energy iterations $N_{\Sigma}$ at each step of the flow.
• Figure 4: Imaginary part of the self-energy from the $1\ell$ fRG and the converged multiloop fRG ($\infty\ell$) compared to the self-consistent result in the PA. The results are obtained for the AIM with the same parameters as in Fig. \ref{['fig:aim_loopconvergence']} for two flow schemes.
• Figure 5: Susceptibilities and fermion-boson vertices for the AHIM at inverse temperature $\beta = 5$, hybridization parameter $\Delta_0 = \pi/5$, Hubbard interaction $U = 1.4$, phonon frequency $\omega_0 = 1$, electron-phonon coupling $g = 0.5$, at half filling from the fully converged multiloop equations ($\infty\ell$). The $T$-flow (2001) that does not implement the necessary flow of the interaction fails qualitatively whereas the $T$-flow (2025) yields results that lie on top of the self-consistent solution of the SBE equations in the PA.