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We study the quantum many-body instabilities of interacting electrons withSU(2)$\times$SU(2) symmetry in spin and orbital degrees of freedom on thetriangular lattice near van-Hove filling. Our work is motivated by effectivemodels for the flat bands in hexagonal moir\'e heterostructures like twistedbilayer boron nitride and trilayer graphene-boron nitride systems. We consideran extended Hubbard model including onsite Hubbard and Hund's couplings, aswell as nearest-neighbor exchange interactions and analyze the differentordering tendencies with the help of an unbiased functional renormalizationgroup approach. We find three classes of instabilities controlled by thefilling and bare interactions. For a nested Fermi surface at van-Hove filling,Hund-like couplings induce a weak instability towards spin or orbital densitywave phases. An SU(4) exchange interaction moves the system towards a Cherninsulator, which is robust with respect to perturbations from Hund-likeinteractions or deviations from perfect nesting. Further, in an extended rangeof fillings and interactions, we find topological $d\pm id$ and(spin-singlet)-(orbital-singlet) $f$-wave superconductivity.

Competing phases of interacting electrons on triangular lattices in moir\'e heterostructures