Title: Quadratic optical response of CrSBr controlled by spin-selective interlayer coupling

Abstract: The optical properties of the layered magnet CrSBr are dominated by intralayer excitons: the antiferromagnetic order between the layers makes layer-to-layer charge hopping, and therefore interlayer excitons, spin-forbidden. An external magnetic field, however, continuously drives the magnetic order towards layer-to-layer ferromagnetic, which opens spin-allowed charge-transfer channels between the layers. Here we elaborate how their admixture changes the composition and nature of the excitons, leading to an extension over many layers, and causes a quadratic red-shift with respect to the external magnetic field. We address these effects by ab-initio $GW$-BSE calculations as a function of magnetic field and cast the data into a minimal four-band model to elucidate the interplay between the various interaction and coupling mechanisms. Our findings should be generally valid for antiferromagnetic layered magnets with and without external magnetic fields, and moreover for any couple of layers with different spin directions. Our insights help to systematically address excitons and predict their optical signatures in such systems.