Hidden multipolar orders of dipole-octupole doublets on a triangular lattice

Motivated by the recent development in strong spin-orbit-coupled materials, we consider the dipole-octupole doublets on the triangular lattice. We propose the most general interaction between these unusual local moments. Due to the spin-orbit entanglement and the special form of its wave function, the dipole-octupole doublet has a rather peculiar property under the lattice symmetry operation. As a result, the interaction is highly anisotropic in the pseudospin space, but remarkably, is uniform spatially. We analyze the ground state properties of this generic model and emphasize the hidden multipolar orders that emerge from the dipolar and octupolar interactions. We clarify the quantum mutual modulations between the dipolar and octupolar orders. We predict the experimental consequences of the multipolar orders and propose the rare-earth triangular materials as candidate systems for these unusual properties.

Figure 1

(a) The six surfaces of the cuboid in the parameter space. The parameters ${\mathcal{J}}_{\mu }$ are found in the Hamiltonian of Eq. (4). $I_{\mu }$ ($O_{\mu }$) refers to the inner (outer) surface with ${\mathcal{J}}_{\mu }=-1$ (${\mathcal{J}}_{\mu }=1$). We have marked the $I_x$ and $O_x$ surfaces. (b) The magnetization of the ferro-dipolar (FD) state on the $I_z$ surface with $({\mathcal{J}}_x,{\mathcal{J}}_y,{\mathcal{J}}_z)=(-0.5,-0.2,-1)$ and $\theta =\pi /3$. The FD transition is at $T_d=1.5|{\mathcal{J}}_z|$. (c) Inverse magnetic susceptibility ${\chi }^{zz}$ of the ferro-octupolar (FO) state on the $I_x$ surface with $({\mathcal{J}}_x,{\mathcal{J}}_y,{\mathcal{J}}_z)=(-1,-0.2,-0.5)$ and $\theta =\pi /3$. The FO transition is at $T_o=1.5|{\mathcal{J}}_x|$. (d) Octupolar-wave excitation with the same parameters in (c).

Figure 2

The phase diagram on the $O_x$ surface (${\mathcal{J}}_x=1$). Solid (dashed) lines indicate first (continuous) order phase transitions.

Figure 3

(a) The ordering pattern in the AFO-$\mathrm{FD}{}_y$ and AFO-$\mathrm{AFD}{}_y$ phases. The local moments are in the $xy$ plane of the pseudospin space. The pseudospin configurations of phases across the diagonal line of the phase diagram in Fig. 2 are related by interchanging $y$ and $z$ components of $\mathbf{m}$. The inset is the coordinate system for the pseudospins. (b) The original Brillouin zone and the folded Brillouin zone due to the three-sublattice ordering. (c), (d) Excitation spectrum by linear spin-wave theory with dominant antiferromagnetic ${\mathcal{J}}_x$, for (c) $({\mathcal{J}}_x,{\mathcal{J}}_y,{\mathcal{J}}_z)=(1,0.4,0)$ in AFO-$\mathrm{AFD}{}_y$, and (d) $({\mathcal{J}}_x,{\mathcal{J}}_y,{\mathcal{J}}_z)=(1,-0.4,0)$ in AFO-$\mathrm{FD}{}_y$.