Starting with twisted bilayer graphene, graphene-based moiré materials have recently been established as a new platform for studying strong electron correlations. In this paper, we study twisted graphene monolayers on trilayer graphene and demonstrate that this system can host flat bands when the twist angle is close to the magic angle of $1.{16}^{\circ }$. When monolayer graphene is twisted on ABA trilayer graphene, the flat bands are not isolated, but are intersected by a Dirac cone with a large Fermi velocity. In contrast, graphene twisted on ABC trilayer graphene (denoted AtABC) exhibits a gap between flat and remote bands. Since ABC trilayer graphene and twisted bilayer graphene are known to host broken-symmetry phases, we further investigate the ostensibly similar magic-angle AtABC system. We study the effect of electron-electron interactions in AtABC using both Hartree theory and an atomic Hubbard theory to calculate the magnetic phase diagram as a function of doping, twist angle, and perpendicular electric field. Our analysis reveals a rich variety of magnetic orderings, including ferromagnetism and ferrimagnetism, and demonstrates that a perpendicular electric field makes AtABC more susceptible to magnetic ordering.

中文翻译：

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魔角单三层石墨烯中的平带、电子相互作用和磁序

从扭曲双层石墨烯开始，基于石墨烯的莫尔材料最近被建立为研究强电子相关性的新平台。在本文中，我们研究了三层石墨烯上的扭曲石墨烯单层，并证明当扭曲角接近$1.{16}^{\circ }$. 当单层石墨烯在 ABA 三层石墨烯上扭曲时，平带不是孤立的，而是与费米速度大的狄拉克锥相交。相比之下，在 ABC 三层石墨烯（表示为 AtABC）上扭曲的石墨烯在平坦和远程带之间表现出间隙。由于已知 ABC 三层石墨烯和扭曲双层石墨烯具有破坏对称相，我们进一步研究了表面上相似的魔角 AtABC 系统。我们使用 Hartree 理论和原子 Hubbard 理论研究 AtABC 中电子-电子相互作用的影响，以计算作为掺杂、扭曲角和垂直电场函数的磁相图。我们的分析揭示了丰富多样的磁序，包括铁磁性和亚铁磁性，