Real frequency approach to frustrated quantum magnetism using the functional renormalization group

Magnetic materials with predominant antiferromagnetic couplings exhibit a wide range of phenomena arising from the interplay of quantum fluctuations, competing interactions and the underlying lattice geometry. While static properties of quantum magnets have been extensively studied and are often well understood, many of the most distinctive signatures of unconventional magnetic phases – such as fractionalization, confinement, or emergent gauge structures – are encoded in their dynamical response. Predicting these responses in strongly frustrated systems, particularly in two and three dimensions, remains a major theoretical challenge, as analytic solutions are rare and numerical methods are often limited by either dimensionality or the degree of frustration. At the same time, experimental data – e.g. from inelastic neutron scattering – is increasingly available, creating a gap between experiment and theory. To bridge this gap, we develop a real-frequency extension of the PFFRG using the Keldysh formalism, enabling dynamical observables to be computed directly without relying on a numerically unstable analytical continuation required in the Matsubara PFFRG. We derive the full set of Keldysh-indexed flow equations and perform a comprehensive symmetry analysis that is used to substantially reduce the computational effort. Benchmark calculations of the Heisenberg model on the zero-dimensional dimer, the one-dimensional spin chain, and the two-dimensional square lattice as well as the honeycomb Kitaev model show that the method is best suited for systems near the paramagnetic regime, such as quantum spin liquids. We further identify methodological limitations and outline possible extensions, including the incorporation of pseudo-Majorana particles in place of complex pseudo-Fermions, for which we also derive the corresponding flow equations and symmetry relations. With these shortcomings in mind, we apply the Keldysh PFFRG to analyze quantum spin liquid systems on the trellis lattice and the Shastry-Sutherland model and compare to a projected symmetry group approach. For the trellis lattice, the Keldysh PFFRG provides insights into which mean-field Ansätze capture the different phases together with their low-energy excitation spectra. Using optimized density-functional-theory models, we reproduce the spinon-fractionalization and deconfinement signatures observed in inelastic neutron scattering for the trellis compound \( {CaCu2O3} \), and we predict the low-energy dynamical spin structure factor for \({SrCu2O3}\) and \({CaV2O5}\). In the Shastry-Sutherland model, the Keldysh PFFRG reveals the same Dirac-cone splitting predicted by the projected symmetry group approach, thereby confirming the connection between the U(1) square-lattice spin-liquid Ansatz proposed by Wen et al. [PRB 65, 165113 (2002)] and the parametrically small quantum spin liquid phase of the Shastry-Sutherland model.