The quantum anomalous Hall (QAH) effect, first proposed in the Haldane model, is a paradigmatic example of the application of band topology in condensed matter physics. The recent experimental discoveries of the high Chern number QAH effect in pentalayer and tetralayer rhombohedral graphene highlight the intriguing interplay between strong interactions and spin-orbit coupling (SOC). Here, we propose a minimal interacting model for spin-orbit-coupled rhombohedral graphene and use Hartree-Fock analysis to explore the phase diagram at charge neutrality. We find that, with Ising SOC on one outmost graphene layer, in-plane layer-antiferromagnetic (LAF) order is the insulating ground state without a displacement field. Upon increasing the gate displacement field, we find that the QAH state with the Chern number being equal to the layer number emerges between the LAF and layer-polarized states, which is consistent with experimental observations. Remarkably, we study the phase diagram for different thicknesses and find the pentalayer is optimal for the QAH effect. Finally, we propose that the QAH state is enlarged by engineering opposite Ising SOCs on the opposite outmost layers of rhombohedral graphene. These results will facilitate the realization of QAH states in rhombohedral graphene with different thicknesses. This paper serves as a foundation for further exploration of correlated physics of the insulating state in rhombohedral graphene.

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• Electronic structure
• Quantum anomalous Hall effect
• Spin-orbit coupling
• Valley degrees of freedom
• Graphene
• van der Waals systems
• Hartree-Fock methods
• Tight-binding model