Moiré superlattices host a rich variety of correlated topological states, including interaction-driven integers and fractional Chern insulators. A common approach to study interacting ground states at integer fillings is the Hartree-Fock mean-field method. However, this method neglects dynamical correlations, which often leads to an overestimation of spontaneous symmetry breaking and fails to provide quantitative descriptions of single-particle excitations. This work introduces a general many-body perturbation framework for moiré systems, combining all-band Hartree-Fock calculations with $GW$ quasiparticle corrections and random-phase approximation (RPA) correlation energies. We apply this framework to hexagonal boron nitride-aligned rhombohedral pentalayer graphene and magic-angle twisted-bilayer graphene (MATBG). We show that incorporating RPA correlation energy and $GW$ self-energy corrections yields phase diagrams and single-particle spectra that quantitatively align with experimental measurements for both systems. Particularly, the ground state at charge neutrality of MATBG is predicted to be a nematic metal, which is stabilized over Kramers intervalley coherent insulator due to lower correlation energy. Our versatile framework provides a systematic beyond-mean-field approach applicable to generic moiré systems.

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• Graphene
• Twisted bilayer graphene
• GW method
• Random phase approximation