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 $\mu \mathrm{SR}$ 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.

Figure 1

The combination of spin-orbit coupling and the $D_{3d}$ crystal electric field generates a non-Kramers doublet ground state for the ${\mathrm{Pr}}^{3+}$ ion. Here SOC refers to spin-orbit coupling, and CEF refers to crystal electric field. Other ions such as ${\mathrm{Tm}}^{3+}$ and ${\mathrm{Tb}}^{3+}$ could potentially support non-Kramers doublets.

Figure 2

(a) The triangular lattice with three distinct neighboring bonds and interactions. The phase parameter ${\gamma }_{ij}$ depends on the bond orientation, which reflects the spin-orbit-entangled nature of the local moments. (b) The definition of the Brillouin zone for the triangular lattice.

Figure 3

The mean-field ground state phase diagram of the model in Eq. (1) with $J_{zz}>0$. We find the ${\mathrm{Stripe}}_{\mathit{\text{y}}}$ state for large $J_{\pm \pm }$ regardless of the sign of $J_{\pm }$, the ${\mathrm{F}}_{\text{xy}}$ state for large negative $J_{\pm }$, and the Néel state for a large and positive $J_{\pm }$. Thick curves refer to first-order transitions, and thinner curves refer to second-order transitions. The dashed phase boundaries are determined by comparing the energy of ${\mathrm{AF}}_{\mathit{\text{z}}}{\mathrm{Stripe}}_{\mathit{\text{y}}}$ states with those of ${\mathrm{AF}}_{\mathit{\text{z}}}{\mathrm{F}}_{\mathit{\text{xy}}}$ and ${\mathrm{AF}}_{\mathit{\text{z}}}{\mathrm{AF}}_{\mathit{\text{xy}}}$. The transitions across the dashed lines are complicated and may involve other competing states that are not well captured by our mean-field approach. The spin configurations of all ordered states are illustrated in Fig. 4.

Figure 4

Real-space spin configurations of the mean-field ground states found in Fig. 3. (a) The ferromagnetic quadrupolar order with spins aligned in $xy$ plane, which we name the ${\mathrm{F}}_{\mathit{\text{xy}}}$ order. There is a global $U\left(1\right)$ degeneracy in the $xy$ plane. (b) The antiferromagnetic quadrupolar stripe order with spins aligned in $y$ direction, which we dub ${\mathrm{Stripe}}_{\mathit{\text{y}}}$. (c), (d) The ${\mathrm{AF}}_{\mathit{\text{z}}}{\mathrm{F}}_{\mathit{\text{xy}}}$ and ${\mathrm{AF}}_{\mathit{\text{z}}}{\mathrm{AF}}_{\mathit{\text{xy}}}$ orders in Fig. 3 for small $J_{\pm \pm }$ and small $J_{\pm }$. Both orders have a three-sublattice structure, consistent with results from Refs. [37, 38, 39]. As in (b), the component in $xy$ plane has a global $U\left(1\right)$ degeneracy for reasons explained in the text. (e) The ${120}^{\circ }$-Néel order stabilized by a large positive $J_{\pm }$. In all figures, we draw the coordinate system of the spin space to help visualization. The coordinate system of the real space always takes the same convention in Fig. 2.

Figure 5

Energy per spin taking into account quantum zero-point energy vs the azimuth angle $\theta$ of spins for (a) the ${\mathrm{F}}_{\mathit{\text{xy}}}$ state and (b) the ${120}^{\circ }$ Né${\mathrm{el}}_{\mathit{\text{xy}}}$ state. Here we take the parameter $J_{\pm }=0.4J_{zz}, J_{\pm \pm }=0.4J_{zz}$ for the ${\mathrm{F}}_{\mathit{\text{xy}}}$ state and parameter $J_{\pm }=0.9J_{zz}, J_{\pm \pm }=0.2J_{zz}$ for the Né${\mathrm{el}}_{\mathit{\text{xy}}}$ state. The zero-point energy is calculated within the linear spin-wave method.

Figure 6

Dynamic spin structure factors for the phases discussed in Sec. 3, obtained from the linear spin-wave theory. The representative parameters for different subfigures are given. The plots here are intensity plots. We also plot the full spin-wave dispersions in Appendix pp3.

Figure 7

(a) The magnetic suscpetibility ${\chi }^{zz}$ at finite temperatures for different field strengths. There is a transition at $B_c\simeq 7.2J_{zz}$. (b) The antiferroquadrupolar order parameter at finite transversal fields and finite temperatures. (c) The dipolar moment at finite fields and finite temperatures. (d) The ordering temperature of the antiferroquadrupolar ${\mathrm{Stripe}}_{\mathit{\text{y}}}$ order as a function of transversal fields. In all figures we choose $J_{\pm }=0.1J_{zz}$, and $J_{\pm \pm }=J_{zz}$, so that the ground state at zero field is in the ${\mathrm{Stripe}}_{\mathit{\text{y}}}$ phase.

Figure 8

The non-Kramers ground state doublet from splitting the spin-1 triplets.

Figure 9

The complete spin-wave dispersions for different phases. The parameters are chosen to be the same as in Fig. 6. The plots of the full dispersions here are to be compared with the intensity plots in Fig. 6.