Quantum anomalous, spin, and valley Hall effects in pentalayer rhombohedral graphene moiré superlattices

Recent experiments on pentalayer rhombohedral graphene moir\'e superlattices have observed the quantum anomalous Hall effect at a moir\'e filling factor of $\ensuremath{\nu}=1$ 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 $\ensuremath{\nu}=2$, 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 $\ensuremath{\nu}=3$ and 4, the ground state can be a combination of topologically trivial and nontrivial states, unlike the cases of $\ensuremath{\nu}=1$ and 2. This contrasts with the conventional quantum Hall effect in graphene where the ground state at filling factor $\ensuremath{\nu}$ is given as the particle-hole counterpart at $4\ensuremath{-}\ensuremath{\nu}$.