Quantum anomalous, spin, and valley Hall effects in pentalayer rhombohedral graphene moir\'e superlattices

Recent experiments on pentalayer rhombohedral graphene moir\'e superlattices have observed the quantum anomalous Hall effect at moir\'e filling factor of $\nu = 1$ and various fractional values. These phenomena are attributed to a flat Chern band induced by electron-electron interactions. In this study, we demonstrate that at $\nu = 2$, many-body effects can lead to the emergence of quantum spin Hall and quantum valley Hall states, in addition to the quantum anomalous Hall state, even in the absence of spin-orbit coupling or valley-dependent potentials. These three topological states can be selectively induced by the application and manipulation of a magnetic field. Furthermore, we show that at $\nu = 3$ and $4$, the ground state can be a combination of topologically trivial and nontrivial states, unlike the cases of $\nu=1$ and 2. This contrasts with the conventional quantum Hall effect in graphene where the ground state at filling factor $\nu$ is given as the particle-hole counterpart at $4-\nu$.