Many-Body Perturbation Theory for Moiré Systems: Symmetry-Breaking States in Twisted Bilayer Graphene

Moiré systems, such as magic-angle twisted bilayer graphene, have attracted significant attention due to their ability to host strongly correlated phenomena, including superconductivity and correlated insulating states. In this work, we systematically develop a many-body perturbation theory framework to address correlations beyond the usual mean-field Hartree-Fock approaches. As a specific example, we first analyze twisted bilayer graphene within the Hartree-Fock approximation, deriving analytical solutions for symmetry-breaking states at integer fillings and the finite-temperature metal-insulator transition. These results closely match previously known numerical results in the literature. Moving beyond Hartree-Fock, we incorporate self-consistent GW corrections, demonstrating that first-order diagrams can overestimate electronic compressibility -- in some cases as much as 50 percent. Our framework provides a comprehensive pathway for exploring strong electronic correlations in moiré systems, offering new insights into the interplay of symmetry breaking and electron correlations.