Two-orbital e_g model with bond-dependent spin-orbit coupling: A playground for emergent band topology, Kitaev magnetism, and magnetoelectricity

Inspired by the electronic structure of compounds like nickel dihalides Ni$X_2$ ($X$=Cl, Br, I), we propose a low-energy two-orbital $e_g$ model featuring bond-dependent spin-orbit terms, driven by atomic spin-orbit coupling on the ligand $X$. We demonstrate that this model hosts a rich array of phenomena. In the non-interacting band limit, spin-orbit-derived spin-dependent and spin-flip hopping terms produce topological bands with spin-Chern numbers $C_s=\pm 2, \pm 4$, and higher order topological states with fractional corner charges, respectively. In the half-filled Mott insulator limit, we recover a spin-$1$ Hamiltonian with bond-dependent Kitaev exchange interactions. We explore the magnetoelectric effect in this two-orbital model using symmetry-based perspective and microscopic calculations, going beyond the generalized Katsura-Nagaosa-Balatsky theory for the single-orbital case. Our work may be relevant to study of doping, strain, or pressure on Ni$X_2$ and related materials.