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Recent experiments have observed possible spin- and valley-polarizedinsulators and spin-triplet superconductivity in twisted double bilayergraphene, a moire structure consisting of a pair of Bernal-stacked bilayergraphene. Besides the continuously tunable band widths controlled by an applieddisplacement field and twist angle, these moire bands also possess van Hovesingularities near the Fermi surface and a field-dependent nesting which is farfrom perfect. Here we carry out a perturbative renormalization group analysisto unbiasedly study the competition among all possible instabilities in twisteddouble bilayer graphene and related systems with a similar van Hove fermiologyin the presence of weak but finite repulsive interactions. Our key finding isthat there are several competing magnetic, valley, charge, and superconductinginstabilities arising from interactions in twisted double bilayer graphene,which can be tuned by controlling the displacement field and the twist angle.In particular, we show that spin- or valley-polarized uniform instabilitiesgenerically dominate under moderate interactions smaller than the band width,whereas $p$-wave spin-triplet topological superconductivity and exoticspin-singlet modulated paired state become important as the interactionsdecrease. Realization of our findings in general moire systems with a similarvan Hove fermiology should open up new opportunities for manipulatingtopological superconductivity and spin- or valley-polarized states in highlytunable platforms.

physical review. b./physical review. b

Topological superconductivity, ferromagnetism, and valley-polarized  phases in moire systems: Renormalization group analysis for twisted double  bilayer graphene