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The $S=3/2$ Kitaev honeycomb model (KHM) is unique among the spin-$S$ Kitaevmodels due to a massive ground state quasi-degeneracy that hampered previousnumerical and analytical studies. In a recent work~\cite{jin2022unveiling}, weshowed how an SO(6) Majorana parton mean-field theory of the $S=3/2$ isotropicKHM explains the anomalous features of this Kitaev spin liquid (KSL) in termsof an emergent low-energy Majorana flat band. Away from the isotropic limit,the $S=3/2$ KSL generally displays a quadrupolar order with gapped or gaplessMajorana excitations, features that were quantitatively confirmed by DMRGsimulations. In this paper, we explore the connection between the $S = 3/2$ KHMwith Kugel-Khomskii models and discover new exactly soluble examples for thelatter. We perform a symmetry analysis for the variational parton mean-field\emph{Ans{\"a}tze} in the spin and orbital basis for different quantum liquidphases of the $S=3/2$ KHM. Finally, we investigate a proposed time-reversalsymmetry breaking spin liquid induced by a {[}111{]} single ion anisotropy andelucidate its topological properties as well as experimental signatures, e.g.an unquantized thermal Hall response.

Quantum liquids of the S=3/2 Kitaev honeycomb and related Kugel-Khomskii models