Selective measurements of intertwined multipolar orders: Non-Kramers doublets on a triangular lattice

Motivated by the rapid experimental progress on the spin-orbit-coupled Mott insulators, we propose and study a generic spin model that describes the interaction between the non-Kramers doublets on a triangular lattice and is relevant for triangular lattice rare-earth magnets. We predict that the system supports both pure quadrupolar orders and intertwined multipolar orders in the phase diagram. Besides the multipolar orders, we explore the magnetic excitations to reveal the dynamic properties of the systems. Due to the peculiar properties of the non-Kramers doublets and the selective coupling to the magnetic field, we further study the magnetization process of the system in the magnetic field. We point out the selective measurements of the static and dynamic properties of the intertwined multipolarness in the neutron scattering, NMR, and $\ensuremath{\mu}\mathrm{SR}$ probes and predict the experimental consequences. The relevance to the existing materials such as ${\mathrm{TmMgGaO}}₄$, Pr-based, and Tb-based magnets, and many ternary chalcogenides is discussed. Our results not only illustrate the rich physics and the promising direction in the interplay between strong spin-orbit-entangled multipole moments and the geometrical frustration, but also provide a general idea to use noncommutative observables to reveal the dynamics of the hidden orders.