Intrinsic (non)-Gilbert damping in magnetic insulators calculated from a minimal model and \textit{ab initio} spin Hamiltonians

We present an analytically solvable minimal model for the relaxation of low-frequency magnons in magnetic insulators arising from magnon-phonon and magnon-magnon interactions. The model establishes a direct connection between microscopic relaxation processes and Gilbert damping, and reveals how magnon decay evolves from bulk systems to the monolayer limit. We find that magnon-phonon coupling produces Gilbert damping of comparable magnitude in three- and two-dimensional magnets, with qualitative differences between flexural phonons in free-standing monolayers and three-dimensional phonons in substrate-supported layers. By contrast, non-Gilbert damping due to four-magnon scattering is strongly enhanced in two dimensions, where it becomes independent of spin-orbit coupling. To benchmark the model against real materials, we introduce a numerical approach for computing magnon damping from ab initio-derived spin Hamiltonians. We demonstrate that the central conclusions of the model remain valid for magnons in bulk YIG and in a monolayer of the van der Waals magnetic insulator CrSBr.