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The ideal honeycomb lattice, featuring sublattice and SU(2) spin rotationsymmetries, is a fundamental model for investigating quantum matters withtopology and correlations. With the rise of the moir\'e-based design of modelsystems, realizing a tunable and symmetric honeycomb lattice system with anarrow bandwidth can open access to new phases and insights. We propose theABBA-stacked twisted double bilayer WSe$_2$ as a realistic and tunable platformfor reaching this goal. Adjusting the twist angle allows the bandwidth and theratio between hopping parameters of different ranges to be tuned. Moreover, thesystem's small bandwidth and spin rotation symmetry enable effective control ofthe electronic structure through an in-plane magnetic field. We construct anextended Hubbard model for the system to demonstrate this tunability andexplore possible ordered phases using the Hartree-Fock approximation. We findthat at a hole filling of $\nu = 2$ (two holes per moir\'e unit cell), anin-plane magnetic field of a few Tesla can ``dope" the system from a semimetalto a metal. Interactions then drive an instability towards a cantedantiferromagnetic insulator ground state. Additionally, we observe a competinginsulating phase with sublattice charge polarization. Finally, we discuss theexperimental signatures of these novel insulating phases.

Realizing a tunable honeycomb lattice in ABBA-stacked twisted double bilayer WSe$_2$