Ferromagnetism and Spin-Valley liquid states in Moir\'{e} Correlated Insulators

Motivated by the recent observation of evidences of ferromagnetism in correlated insulating states in systems with Moir\'{e} superlattices, we study a two-orbital quantum antiferromagnetic model on the triangular lattice, where the two orbitals physically correspond to the two valleys of the original graphene sheet. For simplicity this model has a SU(2)$^s$$\otimes$SU(2)$^v$ symmetry, where the two SU(2) symmetries correspond to the rotation within the spin and valley space respectively. Through analytical argument, Schwinger boson analysis and also DMRG simulation, we find that even though all the couplings in the Hamiltonian are antiferromagnetic, there is still a region in the phase diagram with fully polarized ferromagnetic order. We argue that a Zeeman field can drive a metal-insulator transition in our picture, as was observed experimentally. We also construct spin liquids and topological ordered phases at various limits of this model. Then after doping this model with extra charge carriers, the system most likely becomes spin-triplet/valley-singlet $d+id$ topological superconductor as was predicted previously.