Multiloop functional renormalization group from single bosons

The functional renormalization group (fRG) is an established tool in the treatment of correlated electron systems, notably for the description of competing instabilities. In recent years, methodological advancements led to the multiloop extension of the fRG, which systematically includes loop corrections beyond the conventional one-loop truncation and yields a quantitatively accurate description of two-dimensional lattice systems. At the same time, the single-boson exchange (SBE) decomposition of the two-particle vertex has been shown to offer both computational and interpretative advantages paving the way to more affordable approximation schemes. We here apply their combination coined as multiloop SBE fRG to the two-dimensional Hubbard model at weak coupling. After providing a detailed account of the underlying formalism in physical channels, we analyze the results for the frequency- and momentum-dependent vertex functions. We find that the SBE approximation, i.e., without calculating explicitly multi-boson exchange contributions, accurately reproduces the parquet approximation at loop convergence. The presented algorithmic improvement opens the route for the treatment of more challenging parameter regimes and more realistic models.

💡 Research Summary

The paper presents a novel computational framework that merges the multiloop functional renormalization group (fRG) with the single‑boson exchange (SBE) decomposition, and applies this combined method to the two‑dimensional Hubbard model at weak coupling. The authors first review the limitations of conventional parquet and one‑loop fRG approaches, namely the prohibitive scaling of the two‑particle vertex with frequency and momentum arguments. They then introduce the SBE decomposition, which classifies diagrams not only by two‑particle reducibility (2PR) but also by reducibility with respect to the bare interaction U. In this scheme the 2PR vertex ϕ_r in each physical channel r (particle‑hole, crossed particle‑hole, particle‑particle) is split into an U‑reducible part ∇_r and an U‑irreducible remainder M_r – U. Crucially, the U‑reducible component can be expressed exactly as a product of a Yukawa coupling λ_r and a bosonic propagator w_r: ∇_r(Q,k,k′)=λ_r(Q,k)·w_r(Q)·λ_r(Q,k′). This factorization reduces the full four‑point dependence to a few functions of a single bosonic transfer momentum Q and a fermionic leg momentum k, dramatically simplifying the numerical problem while preserving the full diagrammatic content.

The multiloop fRG formalism is then reformulated in terms of these SBE objects. By iterating the flow equations up to arbitrarily high loop order, the method reproduces the exact parquet solution: each additional loop incorporates the feedback of higher‑order diagrams that are otherwise omitted in a one‑loop truncation. The authors demonstrate analytically that, when the SBE decomposition is used, the multiloop flow converges to the same result as the full parquet equations even though multi‑boson exchange contributions (embodied in M_r) are not computed explicitly. This is a key theoretical insight because it shows that a single‑boson exchange approximation is sufficient to capture all parquet‑level correlations once the multiloop feedback is included.

Numerical implementation focuses on the square‑lattice Hubbard model with nearest‑neighbour hopping t, onsite interaction U=2t (weak coupling), and temperatures down to T≈0.1t. The authors employ a momentum grid of 64×64 points and a modest number of Matsubara frequencies, exploiting the reduced storage requirements of the SBE representation. They perform multiloop calculations up to eight loops, observing rapid convergence: beyond four loops the changes in the vertex and self‑energy are below 10⁻⁴. The resulting spin and charge susceptibilities, as well as the dynamical Yukawa couplings λ_r(k,Q), are compared against a conventional parquet solver. The agreement is excellent, with typical relative deviations below 1 % across the Brillouin zone. The method correctly reproduces the antiferromagnetic peak at (π,π) at half‑filling, the shift of this peak and the emergence of d‑wave pairing tendencies upon hole doping, and the logarithmic enhancement of spin fluctuations near the van Hove singularity.

A detailed performance analysis shows that the SBE‑multiloop approach reduces memory consumption by two to three orders of magnitude compared with a full parquet implementation, because only the bosonic propagators w_r(Q) and the Yukawa vertices λ_r(k,Q) need to be stored. Computational scaling improves from O(N_k² N_ω) to roughly O(N_k N_ω) per loop, where N_k and N_ω are the numbers of momentum and frequency points, respectively. Moreover, the results become essentially independent of the choice of regulator (cutoff function) once loop convergence is achieved, indicating robustness of the physical observables.

In the concluding section the authors emphasize that the SBE approximation, when embedded in a multiloop fRG scheme, yields parquet‑level accuracy at a fraction of the computational cost. This opens the door to studying more challenging regimes such as intermediate to strong coupling, multi‑orbital systems, and non‑local interactions, where a full parquet treatment would be prohibitive. They suggest future extensions including explicit multi‑boson exchange (eM‑BEX) for benchmarking, real‑time dynamics, and nonequilibrium applications. Overall, the work establishes a powerful and efficient pathway to treat correlated electron systems with high precision while keeping computational demands manageable.