The pseudofermion functional renormalization group (pffRG) is a computational method for determining zero-temperature phase diagrams of frustrated quantum magnets. In a recent methodological advance, the commonly employed Katanin truncation of the flow equations was extended to include multiloop corrections, thereby capturing additional contributions from the three-particle vertex [ arXiv:2011.01268 , arXiv:2011.01269 ]. This development has also stimulated significant progress in the numerical implementation of pffRG, allowing one to track the evolution of pseudofermion vertices under the renormalization group flow with unprecedented accuracy. However, cutting-edge solvers differ in their integration algorithms, heuristics to discretize Matsubara frequency grids, and more. To lend confidence in the numerical robustness of state-of-the-art multiloop pffRG codes, we present and compare results produced with two independently developed and algorithmically distinct solvers for Heisenberg models on three-dimensional lattice geometries. Using the cubic lattice Heisenberg (anti)ferromagnet with nearest and next-nearest neighbor interactions as a generic benchmark model, we find the two codes to quantitatively agree, often up to several orders of magnitude in digital precision, both on the level of spin-spin correlation functions and renormalized fermionic vertices for varying loop orders. These benchmark calculations further substantiate the usage of multiloop pffRG solvers to tackle unconventional forms of quantum magnetism.
Multiloop functional renormalization group for general models
我们提出了多回路流方程的功能重整化群（fRG）框架的四点顶点和自能，制定了一个一般的费米子多体问题。这推广了先前引入的顶点流[F. B. Kugler和J. von Delft，Phys. Rev. Lett. 120，057403（2018）]，并提供了必要的修正，以完成所有涉及截断fRG流图的衍生自能流。由于其迭代的单循环结构，多循环流非常适合于数值算法，从而能够改进许多fRG计算。我们证明了它的等价性的解决方案（一阶）镶木地板方程结合Schwinger-Dyson方程的自能。
KeldyshQFT: A C++ codebase for real-frequency multiloop functional renormalization group and parquet computations of the single-impurity Anderson model
BosonFlow: A C++ codebase for dynamic fRG and single-boson exchange in correlated fermion systems
Ruhr-University Bochum（波鸿鲁尔大学）；Max-Planck-Institut für Festkörperforschung（马克斯·普朗克固体研究所）；Vienna University of Technology（维也纳工业大学）；Universität Tübingen（蒂宾根大学）
