  The quantum anomalous Hall (QAH) effect, first proposed by the Haldane model,
has become a paradigmatic example of application of band topology to condensed
matter physics. The recent experimental discoveries of high Chern number QAH
effect in pentalayer and tetralayer rhombohedral graphene highlights 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 the Hartree-Fock analysis to explore the phase diagram at
charge neutrality. We find that with Ising SOC on one outmost graphene layer,
the in-plane layer-antiferromagnetic order is the insulating ground state
without displacement field. Upon increasing the gate displacement field, we
find that the QAH state with Chern number being equal to the layer number
emerges between layer-antiferromagentic state and layer-polarized state, which
is consistent with experimental observations. We further study phase diagram
for different thickness and find pentalayer is optimal for the QAH effect.
Finally, we predict that QAH state is enlarged by engineering opposite Ising
SOC on the opposite outmost layers of rhombohedral graphene.
