Multipolar spin liquid in an exactly solvable model for $j_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{3}{2}$ moments

We study an exactly solvable model with bond-directional quadrupolar and octupolar interactions between spin-orbital entangled $j_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{3}{2}$ moments on the honeycomb lattice. We show that this model features a multipolar spin liquid phase with gapless fermionic excitations. In the presence of perturbations that break time-reversal and rotation symmetries, we find Abelian and non-Abelian topological phases in which the Chern number evaluates to $0$, $\pm 1$, and $\pm 2$. We also investigate quantum phase transitions out of the multipolar spin liquid using a parton mean-field approach and orbital wave theory. In the regime of strong integrability-breaking interactions, the multipolar spin liquid gives way to ferroquadrupolar-vortex and antiferro-octupolar ordered phases that harbor a hidden spin-$\frac{1}{2}$ Kitaev spin liquid. Our work unveils mechanisms for unusual multipolar orders and quantum spin liquids in Mott insulators with strong spin-orbit coupling.