Moiré systems such as magic-angle twisted bilayer graphene have attracted significant attention due to their ability to host correlated phenomena including superconductivity and strongly correlated insulating states. By defining the single-particle Green's function in the band basis, 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. We derive analytical solutions for symmetry-breaking states at integer fillings and the finite-temperature metal-insulator transition that closely match previously known numerical results in the literature. Moving beyond Hartree-Fock, we incorporate self-consistent $GW$ corrections demonstrating that first-order diagrams significantly overestimate the filling-dependent fluctuations in the electronic compressibility. This framework provides a comprehensive pathway for exploring strong electronic correlations in moiré systems beyond the mean-field theory, giving new insights into the interplay of symmetry breaking and electron correlations.

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• Electronic structure
• Feynman diagrams
• Strongly correlated systems
• Twisted bilayer graphene
• Twisted heterostructures
• Green's function methods
• Random phase approximation