  Twisted graphene bilayers develop highly localised states around AA-stacked
regions for small twist angles. We show that interaction effects may induce
either an antiferromagnetic (AF) and a ferromagnetic (F) polarization of said
regions, depending on the electrical bias between layers. Remarkably,
F-polarised AA regions under bias develop spiral magnetic ordering, with a
relative $120^\circ$ misalignment between neighbouring regions due to a
frustrated antiferromagnetic exchange. This remarkable spiral magnetism emerges
naturally without the need of spin-orbit coupling, and competes with the more
conventional lattice-antiferromagnetic instability, which interestingly
develops at smaller bias under weaker interactions than in monolayer graphene,
due to Fermi velocity suppression. This rich and electrically controllable
magnetism could turn twisted bilayer graphene into an ideal system to study
frustrated magnetism in two dimensions, with interesting potential also for a
range of applications.
