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We propose an exactly solvable model for $j_{\text{eff}}=\frac32$ localmoments on the honeycomb lattice. Our construction is guided by a symmetryanalysis and by the requirement of an exact solution in terms of a Majoranafermion representation for multipole operators. The main interaction in themodel can be interpreted as a bond-dependent quadrupole-quadrupole interaction.When time reversal symmetry is explicitly broken, we obtain a gapped spinliquid with a single chiral Majorana edge mode. We also investigate anothersolvable model in which the time-reversal-invariant spin liquid is coupled toconduction electrons in a superconductor. In the presence of a Kondo-likecoupling that involves the octupole moment of the localized spins, theitinerant electrons hybridize with the emergent Majorana fermions in the spinliquid. This leads to spontaneous time reversal symmetry breaking and generatesodd-frequency pairing. Our results suggest that $j_{\text{eff}}=\frac32$systems with strong quadrupole-quadrupole interactions may provide a routetowards non-Abelian quantum spin liquids and unconventional superconductivity.

Quadrupolar spin liquid, octupolar Kondo coupling and odd-frequency superconductivity in an exactly solvable model