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Motivated by the recent experimental developments in van der Waals heterostructures, we investigate the emergent magnetism in Mott insulator - semimetal moir\'e superlattices by deriving effective spin models and exploring their phase diagram by Monte Carlo simulations. Our analysis indicates that the stacking-dependent interlayer Kondo interaction can give rise to different types of magnetic order, forming domains within the moir\'e unit cell. In particular, we find that the AB (AA) stacking regions tend to order (anti)ferromagnetically for an extended range of parameters. The remaining parts of the moir\'e unit cell form ferromagnetic chains that are coupled antiferromagnetically. We show that the decay length of the Kondo interaction can control the extent of these phases. Our results highlight the importance of stacking-dependent interlayer exchange and the rich magnetic spin textures that can be obtained in van der Waals heterostructures.
Keskiner et al. (Wed,) studied this question.