Towards a Nonperturbative Many-Body Nuclear Theory: Benchmarking FRG-DFT with the One-Dimensional Fermi Gas

A deep microscopic understanding of quantum many-body theories would revolutionize many fields from nucleosynthesis and reactor design to neutron-star physics and ultracold atom experiments. Density Functional Theory (DFT) is the only approach applicable across nearly the entire nuclear chart, making its derivation from first principles a hot topic in recent years. Modern theoretical tools such as the Functional Renormalization Group (FRG) have enabled a first-principles formulation of DFT, known as the FRG-DFT. Though applied to 2D and 3D electrons with Coulomb interactions, a theoretical benchmark and connection to experiment remain missing. We address this gap by applying FRG-DFT to the exactly solvable Gaudin-Yang model: a strongly coupled 1D Fermi gas experimentally realizable with ultracold atoms. We derive the exact infinite hierarchy of coupled flow equations for the ground state energy, and explicit spin dependence enables extension to the spin-unsaturated electron gas. We truncate the hierarchy to compare our first-principles equation of state calculation to perturbation theory, competing methods, and the exact solution. Our results establish a clear theoretical benchmark for FRG-DFT and lay foundation for its extension to realistic higher-dimensional many-body systems.