Emergent charge order instabilities in a moiré superlattice at fractional band filling

Recent experiments have established the flat bands of ABC stacked trilayer graphene aligned with boron nitride as a platform for ferromagnetism and strong correlations. Depending on the direction of the gate-voltage, the topology of the flat bands manifested in their Chern numbers can be either trivial or non-trivial tuning the system between two distinct physical regimes. We provide evidence of possible charge order instabilities at fractional band fillings corresponding to non-integer number of electrons per moiré unit cell. Remarkably, the charge order can be understood as a nesting instability of an emergent Fermi surface from the purely interacting problem. The competition between the emergent Fermi liquids and the charge ordered states is found to be sensitive to the filling fraction and the direction of the gate-voltage. We further propose an effective action that describes the low energy physics in order to interpret the phases analytically. Our results provide an explicit example of strong-weak coupling dualities in a condensed matter system and sheds the light on the non trivial role of particle-hole asymmetry in moiré superlattices.