BosonFlow: A C++ codebase for dynamic fRG and single-boson exchange in correlated fermion systems

BosonFlow: A Codebase for Dynamic fRG and Single-Boson Exchange in Correlated Fermion Systems

Introduction and Motivation

The theoretical study of strongly correlated fermionic systems—such as high-$T_c$ cuprates and unconventional superconductors—relies critically on the unbiased resolution of competing instabilities. Functional renormalization group (fRG) and parquet approaches provide a comprehensive many-body framework for these challenges, enabling non-perturbative summation over fluctuation channels. However, the practice has long been limited by the exponential scaling in the dynamic, momentum- and frequency-dependent two-particle vertex, forcing a trade-off—sacrificing frequency dependence for spatial resolution, or vice versa. This has major implications, for instance, leading to unphysical phase transitions in two dimensions when dynamics are neglected.

The presented work introduces BosonFlow, a C++ codebase implementing fRG and parquet schemes in the single-boson exchange (SBE) formalism. Through an efficient truncated-unity approach to momentum dependence and controlled frequency discretization, BosonFlow is constructed to resolve full dynamic vertex functions and self-energies for both equilibrium tight-binding lattice models and quantum impurity problems. The code incorporates state-of-the-art algorithms and multiple methodological advances, supporting a wide class of models and allowing for systematic control of approximations and numerical parameters (2604.04232).

Algorithmic Framework: fRG, Parquet, and SBE

BosonFlow realizes both conventional (one-loop) and multiloop fRG schemes as well as iterative parquet solvers, all within the SBE decomposition. In the standard fRG, the scale-derivative of the vertex and fermionic self-energy are hierarchically coupled, with the propagator regulated by a parameter $\Lambda$. The code supports multiple flow schemes (e.g., frequency, temperature, interaction cutoffs), as well as interpolated initial conditions, as used in DMF$^2$RG and its generalizations.

The SBE approach, closely related to the seminal works of Krien, Bonetti, and co-workers [Krien2019, Bonetti2022], recasts the four-point vertex as a sum of bosonic propagators and fermion-boson couplings (the "Hedin vertices") in the particle-hole, particle-particle, and crossed channels, plus a residual multiboson term and double-counting corrections. In spin-rotation-invariant systems, physical channels (magnetic, density, pairing) are defined. SBE achieves a significant reduction in numerical complexity because the bosonic objects retain simpler frequency and momentum structure than the original vertex, while the residual (multiboson) rest function can often be neglected or approximated.

Multiloop extensions [Kugler2018a, Fraboulet2025] are fully implemented, allowing the code to sum all parquet diagrams systematically and remove regulator dependencies endemic to one-loop approximations. This directly enables quantitative results for pseudogap phenomena, Mermin-Wagner physics, and criticality in two-dimensional models.

Notably, BosonFlow incorporates further recent developments such as $G$+$U$ flows, temperature-driven interaction scaling, and the $\mathcal{B}+\mathcal{F}$ splitting of retarded/nonlocal interactions, allowing for ab initio analysis of coupled electron-phonon, long-range Coulomb, and more exotic models.

Code Architecture and Usage

The architecture of BosonFlow is highly modular:

• Models: The code supports paradigmatic problems—Hubbard, Anderson impurity, Holstein-Hubbard, extended Hubbard, among others—via standardized interfaces, admitting arbitrary lattice geometries and interaction profiles.
• Flow and Solver: User-configurable compile-time flags select between ODE-driven RG flow or self-consistent parquet/SBE fixed-point iteration. Flow schemes for frequency, temperature, or interaction are abstracted into dedicated classes.
• Grid and Resolution: Technical parameters control Matsubara frequency boxes, bosonic/fermionic momentum discretization, and the form-factor expansion used in the truncated-unity decomposition.
• Observables and I/O: The output scheme is based on HDF5, facilitating storage and post-processing of high-dimensional objects (self-energy, vertex functions, susceptibilities). Dedicated tools for momentum- and frequency-resolved cuts, symmetry-point interpolation, and detailed fluctuation diagnostics are provided.

The codebase emphasizes separation of concerns and extensibility: new models, cutoff schemes, or state variables can be incorporated with minimal cross-interference. Concurrency via OpenMP, exploitation of lattice symmetries, and precomputed projections ensure tractable computational scaling.

Numerical Applications and Results

BosonFlow has been benchmarked on several archetypal models and problems:

• 2D Hubbard Model: Applications demonstrate that in the weak-coupling regime, as the system approaches the spurious antiferromagnetic transition of one-loop fRG, only the magnetic bosonic propagator diverges while the multiboson rest function remains finite. This supports neglect of the rest function for broad parameter regions, simplifying the dynamic vertex structure and enabling efficient high-resolution computations [Fraboulet2022].
• Multiloop fRG: Implementations in the SBE formalism achieve convergence to the parquet solution, confirmed by correspondence with the parquet approximation in the loop limit [Fraboulet2025]. The code thus enables rigorous verification of diagrammatic completeness and consistency with conservation laws and sum rules.
• Strong-Coupling, Extended/Retarded Interactions: The SBE approach is generalized to treat bare interactions with nontrivial frequency/momentum dependence ($\mathcal{B}+\mathcal{F}$ splitting). The framework performs temperature scans and fluctuation diagnostics in electron-phonon coupled and non-local interaction models [AlEryani2024, AlEryani2025].
• Self-Energy and Pseudogap Physics: Channel-decomposed Schwinger-Dyson equations in the SBE basis provide efficient calculation and analysis of self-energy effects, critical in studies of pseudogap opening and non-Fermi-liquid features.

Key numerical claims:

• The SBE truncation, with neglected multiboson rest function, yields quantitatively accurate two-particle vertices and susceptibilities in the weak-to-intermediate coupling regime for both one-loop and multiloop fRG [Fraboulet2022, Fraboulet2025].
• The code demonstrates precise fulfillment of the Mermin-Wagner theorem in two dimensions at the multiloop parquet-consistent limit [AlEryani2026].
• Unphysical overestimation of ordering temperatures observed in static-vertex fRG is absent in dynamic SBE/multiloop calculations.

Broader Implications and Future Prospects

BosonFlow directly addresses the major computational bottleneck in diagrammatic many-body methods: the need for full dynamical (frequency-dependent) vertex resolution at high spatial momentum resolution. By integrating SBE ideas with truncated unity and flexible flow/init schemes, it becomes possible to utilize diagrammatic, RG-based techniques beyond qualitative instability analysis—delivering quantitative results even in the physically relevant intermediate and strong coupling regimes for high-dimensional models.

Practically, this opens the path to controlled simulation of phase diagrams, susceptibilities, spectral functions, and response properties in a wide array of models central to condensed matter and quantum materials research.

Theoretically, BosonFlow enables benchmarking of new RG approximations (e.g., different rest function closures, channel decompositions), and direct study of quantum criticality, symmetry-constrained systems, and systems with coupled bosonic and fermionic modes (e.g., electron-phonon, nematic, or multiband systems).

Future technical developments suggested include integration of compressed frequency bases (intermediate representation, discrete Lehmann), orbital degrees of freedom, implementation of sharp cutoffs, and expansion to non-equilibrium real-frequency solvers.

Conclusion

BosonFlow constitutes a versatile and robust open-source implementation of dynamic fRG and parquet approaches within the SBE formalism, offering both a reference for methodological advances and a practical platform for quantitative analysis of strongly correlated fermionic systems. Its flexible architecture and comprehensive model coverage enable both methodological research and application to paradigmatic correlated lattice and impurity problems. Continued development along avenues such as efficient frequency parameterizations and novel flow schemes is both straightforward and anticipated to yield further advances (2604.04232).