Understanding the microscopic origin of the magnetic interactions in CoNb$_2$O$_6$

Motivated by the on-going discussion on the nature of magnetism in the quantum Ising chain CoNb$_2$O$_6$, we present a first-principles-based analysis of its exchange interactions by applying an \textit{ab initio} approach with additional modelling that accounts for various drawbacks of a purely density functional theory ansatz. With this method we are able to extract and understand the origin of the magnetic couplings under inclusion of all symmetry-allowed terms, and to resolve the conflicting model descriptions in CoNb$_2$O$_6$. We find that the twisted Kitaev chain and the transverse-field ferromagnetic Ising chain views are mutually compatible, although additional off-diagonal exchanges are necessary to provide a complete picture. We show that the dominant exchange interaction is a ligand-centered exchange process - involving the $e_g$ electrons -, which is rendered anisotropic by the low-symmetry crystal fields environments in CoNb$_2$O$_6$, giving rise to the dominant Ising exchange, while the smaller bond-dependent anisotropies are found to originate from $d-d$ kinetic exchange processes involving the $t_{2g}$ electrons. We demonstrate the validity of our approach by comparing the predictions of the obtained low-energy model to measured THz and inelastic neutron scattering spectra.