  Recent experiments have observed possible spin- and valley-polarized
insulators and spin-triplet superconductivity in twisted double bilayer
graphene, a moire structure consisting of a pair of Bernal-stacked bilayer
graphene. Besides the continuously tunable band widths controlled by an applied
displacement field and twist angle, these moire bands also possess van Hove
singularities near the Fermi surface and a field-dependent nesting which is far
from perfect. Here we carry out a perturbative renormalization group analysis
to unbiasedly study the competition among all possible instabilities in twisted
double bilayer graphene and related systems with a similar van Hove fermiology
in the presence of weak but finite repulsive interactions. Our key finding is
that there are several competing magnetic, valley, charge, and superconducting
instabilities 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 instabilities
generically dominate under moderate interactions smaller than the band width,
whereas $p$-wave spin-triplet topological superconductivity and exotic
spin-singlet modulated paired state become important as the interactions
decrease. Realization of our findings in general moire systems with a similar
van Hove fermiology should open up new opportunities for manipulating
topological superconductivity and spin- or valley-polarized states in highly
tunable platforms.
