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An emergent $\mathrm{SU}(4)$ symmetry discovered in the microscopic model for$d^1$ honeycomb materials [M.~G.~Yamada, M.~Oshikawa, and G.~Jackeli, Phys.Rev. Lett. \textbf{121}, 097201 (2018).] has enabled us to tailor exotic$\mathrm{SU}(4)$ models in real materials. In the honeycomb structure, theemergent $\mathrm{SU}(4)$ Heisenberg model would potentially have a quantumspin-orbital liquid ground state due to the \textit{multicomponentfrustration}, and we can expect similar spin-orbital liquids also inthree-dimensinal versions of the honeycomb lattice. In such quantumspin-orbital liquids, both the spin and orbital degrees of freedom becomefractionalized and entangled together due to the strong frustrated interactionsbetween them. Similarly to spinons in pure quantum spin liquids, quantumspin-orbital liquids can host not only spinon excitations, but also fermionic\textit{orbitalon} excitations at low temperature.

$\mathrm{SU}(4)$-Symmetric Quantum Spin-Orbital Liquids on Various Lattices