Single-ion properties of the transverse-field Ising model material CoNb$_2$O$_6$

CoNb$_2$O$_6$ is one of the few materials that is known to approximate the one-dimensional transverse-field Ising model (1D-TFIM) near its quantum critical point. It has been inferred that Co$^{2+}$ acts as a pseudo-spin 1/2 with anisotropic exchange interactions that are largely Ising-like, enabling the realization of the TFIM. However, the behavior of CoNb$_2$O$_6$ is known to diverge from the ideal TFIM under transverse magnetic fields that are far from the quantum critical point, requiring the consideration of additional anisotropic, bond-dependent (Kitaev-like) terms in the microscopic pseudo-spin 1/2 Hamiltonian. These terms are expected to be controlled in part by single-ion physics, namely the wavefunction for the pseudo-spin 1/2 angular momentum doublet. Here, we present the results of both inelastic neutron scattering measurements and electron paramagnetic resonance spectroscopy on CoNb$_2$O$_6$, which elucidate the single-ion physics of Co$^{2+}$ in CoNb$_2$O$_6$ for the first time. We find that the system is well-described by an intermediate spin-orbit coupled Hamiltonian, and the ground state is a well-isolated Kramers doublet with an anisotropic $g$-tensor. We provide the approximate wavefunctions for this doublet, which we expect will be useful in theoretical investigations of the anisotropic exchange interactions.