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We study the orbitally frustrated singlet-triplet models that emerge in thecontext of spin-orbit coupled Mott insulators with $t_{2g}^4$ electronicconfiguration. In these compounds, low-energy magnetic degrees of freedom canbe cast in terms of three-flavor "triplon" operators describing the transitionsbetween spin-orbit entangled $J=0$ ionic ground state and excited $J=1$ levels.In contrast to a conventional, flavor-isotropic O(3) singlet-triplet models,spin-orbit entangled triplon interactions are flavor-and-bond selective andthus highly frustrated. In a honeycomb lattice, we find close analogies withthe Kitaev spin model -- an infinite number of conserved quantities, nomagnetic condensation, and spin correlations being strictly short-ranged.However, due to the bosonic nature of triplons, there are no emergent gaplessexcitations within the spin gap, and ground state is a strongly correlatedparamagnet of dense triplon pairs with no long-range entanglement. Using exactdiagonalization, we study the bosonic Kitaev model and its various extensions,which break exact symmetries of the model and allow magnetic condensation oftriplons. Possible implications for magnetism of ruthenium oxides arediscussed.

Highly frustrated magnetism in relativistic $\boldsymbol{d}^\mathbf{4}$ Mott insulators: Bosonic analog of Kitaev honeycomb model