Correlated physics, charge and magnetic orders in moiré kagomé systems

Moiré systems have emerged as an ideal platform for exploring interaction effects and correlated states. However, most of the experimental systems are based on either triangular or honeycomb lattices. In this study, based on the self-consistent Hartree–Fock calculation, we investigate the phase diagram of the kagomé lattice in a recently discovered system with two degenerate Γ valley orbitals and strong spin–orbit coupling. By focusing on the filling factors of 1/2, 1/3 and 2/3, we identify various symmetry-breaking states by adjusting the screening length and dielectric constant. At the half filling, we discover that the spin–orbit coupling induces Dzyaloshinskii–Moriya interaction and stabilizes a classical magnetic state with 120° ordering. Additionally, we observe a transition to a ferromagnetic state with out-of-plane ordering. In the case of 1/3 filling, the system is ferromagnetically ordered due to the lattice frustration. Furthermore, for 2/3 filling, the system exhibits a pinned droplet state and a 120° magnetic ordered state at weak and immediate coupling strengths, respectively. For the strong coupling case, when dealing with non-integer filling, the system is always charge ordered with sublattice polarization. Our study serves as a starting point for exploring the effects of correlation in moiré kagomé systems.