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The enhancement of the spin-space symmetry from the usual $\mathrm{SU}(2)$ to$\mathrm{SU}(N)$ with $N>2$ is promising for finding nontrivial quantum spinliquids, but the realization of $\mathrm{SU}(N)$ spin systems in real materialsis still challenging. Although there is a proposal in cold atomic systems, inmagnetic materials with a spin-orbital degree of freedom it is difficult toachieve the $\mathrm{SU}(N)$ symmetry by fine tuning. Here we propose a newmechanism by which the $\mathrm{SU}(4)$ symmetry emerges in the strongspin-orbit coupling limit. In $d^1$ transition metal compounds withedge-sharing anion octahedra, the spin-orbit coupling gives rise to stronglybond-dependent and apparently $\mathrm{SU}(4)$-breaking hopping between the$J_\textrm{eff}=3/2$ quartets. However, in the honeycomb structure, a gaugetransformation maps the system to an $\mathrm{SU}(4)$-symmetric Hubbard model,which means that the system has a hidden symmetry in spite of its largespin-orbit coupling. In the strong repulsion limit at quarter filling, asexpected in $\alpha$-ZrCl$_3,$ the low-energy effective model is the$\mathrm{SU}(4)$ Heisenberg model on the honeycomb lattice, which cannot have atrivial gapped ground state and is expected to host a gapless spin-orbitalliquid. In such quantum spin-orbital liquids, both the spin and orbital degreesof freedom become fractionalized and correlated together at low temperature dueto the strong frustrated interactions between them. Similarly to spinons inpure quantum spin liquids, quantum spin-orbital liquids can host not onlyspinon excitations, but also fermionic "orbitalon" excitations at lowtemperature, which we have named here in distinction from orbitons in thesymmetry-broken Jahn-Teller phases.

PhD Thesis: Searching for an emergent SU(4) symmetry in real materials