Half and quarter metals in rhombohedral trilayer graphene.

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d-orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems1–5. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene6 drive spontaneous ferromagnetic polarization of the electron system into one or more spin- and valley flavors. Using capacitance and transport measurements we observe a cascade of density- and electronic displacement field-tuned transitions between phases in which quantum oscillations have either four-fold, two-fold, or one-fold degeneracy, associated with a spin and valley degenerate normal metal, spin-polarized ‘half-metal’, and spin and valley polarized ‘quarter metal’, respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model7 that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moire superlattice using a rotationally aligned hexagonal boron nitride substrate5,8. Remarkably, we find the isospin order is only weakly perturbed, with the moire potential catalyzing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter metal states occur at half- or quarter superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moire materials4,5,9,10 to be fundamentally itinerant in nature.