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Recent experiments on multilayer graphene materials have discovered aplethora of correlated phases, including ferromagnetism and superconductivity,in the absence of a moir\'{e} potential. These findings pose an intriguingquestion of whether an underlying moir\'{e} potential plays a key role indetermining the phases realizable in tunable two-dimensional quantum materials,or whether it merely acts as a weak periodic potential that perturbs anunderlying correlated many body state. In this work, employing a Hartree-Fockmean field analysis, we examine this question theoretically by quantitativelystudying the effects of an hexagonal Boron Nitride (h-BN) substrate onABC-stacked trilayer graphene (ABC-TLG). For the topologically trivial regime,we find that the moir\'{e} potential leads to a strong suppression of theferromagnetism of the underlying metal. Further, band insulators appear solelyat full filling of the moir\'{e} unit cell, with a moir\'{e} potential strongerthan is conventionally assumed. Thus the observed correlated insulating phasesin ABC-TLG aligned with h-BN cannot be understood through band folding of theferromagnetic metal found without the moir\'{e} potential. For thetopologically non-trivial regime, we discover the appearance of prominentincompressible states when fractional hole fillings (of the moir\'{e} unitcell) coincide with the occurrence of fractional-metallic states in themoir\'{e}-less setting, as well as a slight weakening of the ferromagneticnature of the phases; however this once again requires a moir\'{e} potentialstronger than is conventionally assumed. Our findings highlight the importanceof interactions in renormalizing the electronic bandstructure, and emphasizesthe key role played by the moir\'{e} potential in determining the strongcorrelation physics.

Strong correlations in ABC-stacked trilayer graphene: Moir\'{e} is important