This paper investigates the microscopic origin of magnetism in CrI₃ monolayer, both suspended and dielectrically embedded. Using down-folded generalized Hubbard models derived from ab initio calculations and the constrained random phase approximation (CRPA), the study reveals a multi-orbital super-exchange mechanism driven by ferro- and anti-ferromagnetic Cr-Cr d coupling channels, significantly influenced by ligand p orbitals. Environmental screening, like encapsulation with hexagonal boron nitride, alters Coulomb interactions and magnetic properties, leading to non-trivial effects on magnon dispersion and Curie temperature.

D. Soriano (Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands. Università di Pisa, Dipartimento di Ingegneria dell'Informazione, Via Caruso, 56122 Pisa, Italy), A. N. Rudenko (Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands. Department of Theoretical Physics and Applied Mathematics, Ural Federal University, 620002 Ekaterinburg, Russia), M. I. Katsnelson (Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands. Department of Theoretical Physics and Applied Mathematics, Ural Federal University, 620002 Ekaterinburg, Russia), M. Rösner (Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands)