General Many-Body Perturbation Framework for Moir\'e Systems

Moir\'e superlattices host a rich variety of correlated topological states, including interaction-driven integer 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 moir\'e 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 moir\'e systems.