We investigate the properties of a minimal two-orbital $e_g$ model of transition metal compounds featuring bond-dependent spin-orbit terms arising from the atomic spin-orbit coupling at the ligand site. We demonstrate that this model hosts a rich array of phenomena. In the noninteracting, 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. 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 studies of doping, strain, or pressure on compounds like nickel dihalides, $\mathrm{Ni}{X}_2$ $(X=\mathrm{Cl}, \mathrm{Br}, \mathrm{I})$, and related triangular lattice materials.

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• Magnetism
• 2-dimensional systems
• Magnetic insulators
• Multiferroics
• Density functional calculations
• Topology