Correlations and electronic order in a two-orbital honeycomb lattice model for twisted bilayer graphene

The recent observation of superconductivity in proximity to an insulating phase in twisted bilayer graphene (TBG) at small `magic' twist angles has been linked to the existence of nearly-flat bands, which make TBG a fresh playground to investigate the interplay between correlations and superconductivity. The low-energy narrow bands were shown to be well-described by an effective tight-binding model on the honeycomb lattice (the dual of the triangular Moir\'e superlattice) with a local orbital degree of freedom. In this paper, we perform a strong-coupling analysis of the proposed $\left(p_{x},\,p_{y}\right)$ two-orbital extended Hubbard model on the honeycomb lattice. By decomposing the interacting terms in the particle-particle and particle-hole channels, we classify the different possible superconducting, magnetic, and charge instabilities of the system. In the pairing case, we pay particular attention to the two-component ($d$-wave) pairing channels, which admit vestigial phases with nematic or chiral orders, and study their phenomenology. Furthermore, we explore the strong-regime by obtaining a simplified spin-orbital exchange model which may describe a putative Mott-like insulating state at quarter-filling. Our mean-field solution reveals a rich intertwinement between ferro- and antiferro-magnetic orders with different types of nematic and magnetic orbital orders. Overall, our work provides a solid framework for further investigations of the phase diagram of the two-orbital extended Hubbard model in both strong- and weak-coupling regimes.