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 systems^1,2,3,4,5. 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 flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold 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 moiré superlattice using a rotationally aligned hexagonal boron nitride substrate^5,8. Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing 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 moiré materials^4,5,9,10 to be fundamentally itinerant in nature.

铁磁性在过渡金属化合物中最常见，其中电子占据高度局部化的d轨道。然而，铁磁顺序也可能出现在低密度二维电子系统^1,2,3,4,5中。^{在这里，我们展示了菱形三层石墨烯6}中的栅极调谐范霍夫奇点将电子系统的自发铁磁极化驱动为一种或多种自旋和谷味。使用电容和传输测量，我们观察到调整到相之间的密度和电子位移场的级联跃迁，其中量子振荡具有四倍、二倍或一倍简并，与自旋和谷简并正常金属相关，自旋极化'半金属”，以及自旋和谷极化的“四分之一金属”。^{对于电子掺杂，现象学斯通纳模型7}很好地捕捉到了数据的显着特征其中包括谷-各向异性相互作用。对于孔填充，我们观察到更丰富的相图，其特征是费米表面拓扑中的破缺对称和跃迁之间的微妙相互作用。最后，我们使用旋转对齐的六方氮化硼基板^5,8引入莫尔超晶格。值得注意的是，我们发现同位旋顺序仅受到微弱的扰动，当流动的半或四分之一金属态出现在半或四分之一超晶格带填充时，莫尔电位催化拓扑非平凡带隙态的形成。我们的结果表明，菱面体石墨烯是对多体理论进行良好控制测试的理想平台，并揭示了莫尔材料^{4、5、9、10}中的磁性本质上是流动的。