Tuning correlated states of twisted mono-bilayer graphene with proximity-induced spin-orbit coupling
We study the correlated ground states of twisted mono-bilayer graphene with and without proximity-induced spin-orbit coupling (SOC) from a transition-metal dichalcogenide layer placed on top. We perform self-consistent Hartree-Fock calculations that allow the variational space to include multi-$Q$ translational symmetry broken states for all integer and half-integer fillings of the conduction bands, where signatures of correlated, topological states have been reported experimentally. We find interaction-induced insulators that retain moir\'e translational symmetry at integer fillings, but that break this symmetry at half-integer fillings. We argue that translational symmetry breaking arises from half-filled polarized bands, even when SOC is present. Yet, we find that small SOC can already crucially affect the spin nature of correlated states. Generally, Ising SOC favors out-of-plane spin polarization and spin-valley locking, while Rashba SOC favors in-plane spin order. If only one of these two terms is present, we find that, depending on the type of SOC, it drives a transition from a tetrahedal antiferromagnet to either a coplanar, non-coplanar, or collinear spin-density wave state for half-integer fillings. The frustration associated with the simultaneous presence of both types of SOC can induce chiral, non-coplanar order in parameter ranges where the ground state in the absence of SOC is collinear.