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

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