Title: Nature of excitons and their ligand-mediated delocalization in nickel dihalide charge-transfer insulators

Abstract: The fundamental optical excitations of correlated transition-metal compounds are typically identified with multielectronic transitions localized at the transition-metal site, such as $dd$ transitions. In this vein, intense interest has surrounded the appearance of sharp, below band-gap optical transitions, i.e. excitons, within the magnetic phase of correlated Ni$^{2+}$ van der Waals magnets. The interplay of magnetic and charge-transfer insulating ground states in Ni$^{2+}$ systems raises intriguing questions on the roles of long-range magnetic order and of metal-ligand charge transfer in the exciton nature, which inspired microscopic descriptions beyond typical $dd$ excitations. Here we study the impact of charge-transfer and magnetic order on the excitation spectrum of the nickel dihalides (NiX$_2$, X $=$ Cl, Br, and I) using Ni-$L_3$ resonant inelastic x-ray scattering (RIXS). In all compounds, we detect sharp excitations, analogous to the recently reported excitons, and assign them to spin-singlet multiplets of octahedrally-coordinated Ni$^{2+}$ stabilized by intra-atomic Hund's exchange. Additionally, we demonstrate that these excitons are dispersive using momentum resolved RIXS. Our data evidence a ligand-mediated multiplet dispersion, which is tuned by the charge-transfer gap and independent of the presence of long-range magnetic order. This reveals the mechanisms governing non-local interactions of on-site $dd$ excitations with the surrounding crystal/magnetic structure, in analogy to ground state superexchange. These measurements thus establish the roles of magnetic order, self-doped ligand holes, and intersite coupling mechanisms for the properties of $dd$ excitations in charge-transfer insulators.