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Intertwined dipolar and multipolar order in the triangular-lattice magnet TmMgGaO4

A phase transition is often accompanied by the appearance of an order parameter and symmetry breaking. Certain magnetic materials exhibit exotic hidden-order phases, in which the order parameters are not directly accessible to conventional magnetic measurements. Thus, experimental identification and theoretical understanding of a hidden order are difficult. Here we combine neutron scattering and thermodynamic probes to study the newly discovered rare-earth triangular-lattice magnet TmMgGaO4. Clear magnetic Bragg peaks at K points are observed in the elastic neutron diffraction measurements. More interesting, however, is the observation of sharp and highly dispersive spin excitations that cannot be explained by a magnetic dipolar order, but instead is the direct consequence of the underlying multipolar order that is “hidden” in the neutron diffraction experiments. We demonstrate that the observed unusual spin correlations and thermodynamics can be accurately described by a transverse field Ising model on the triangular lattice with an intertwined dipolar and ferro-multipolar order.

Supplementary Note 1: Sample synthesis and characterization.
High quality single-crystalline TmMgGaO 4 samples were synthesized using the floating-zone technique. The single crystal is transparent with shiny cleaved surfaces ( Supplementary Fig. 1a). Sharp and clear diffraction spots can be seen in the X-ray Laue pattern ( Supplementary Fig. 1b). In the single-crystal X-ray diffraction measurement, a series of reflections along the L direction can be indexed and the full-width at the half maximum (FWHM) of the rocking scan across Q=(0, 0, 18) is around 0.025 degrees ( Supplementary Fig. 1c). These results indicate the high crystallization quality of the single crystals.
Supplementary Note 2: Calculations of the spin excitations in TmMgGaO 4 .
In order to understand the observed spin excitation spectra in TmMgGaO 4 at low temperature, we calculate the spin wave dispersion by linear spin wave theory (LSW) using the spinw program 1 . Here we present the calculated results of five representative scenarios.
First, we examine the possibilities of conventional dipolar ordering. The corresponding spin Hamiltonian is assumed as: where S ± = S x ± iS y and J 1 and J 2 are nearest and next-nearest neighboured magnetic interactions, respectively, as illustrated in Supplementary Fig. 2a. The three-fold magnetic twins are included for all calculations.
We consider the isotropic Heisenberg model (J zz 1 = 2J ± 1 = 0.9 meV, J zz 2 = 0 meV) and anisotropic XY model (J zz 1 = J zz 2 = 0 meV, J ± 1 = 0.4 meV) that support the 120 degree Néel order and lead to strong magnetic Bragg peaks at K points ( Supplementary   Fig. 3a). We calculate both the total scattering function (S xx + S yy + S zz ) and the longitudinal component (S zz ) for both of the models (Supplementary Fig. 4a-d). Although the calculated spin waves catch the feature of Goldstone mode stemmed from K points, the branches around Γ points go as a minimum in the calculation instead of a maximum which is observed in the neutron experiments. Moreover, the branches around the M points are located at higher energies than the experimental data.
Another scenario is the Ising model with a stripe order that has been proposed in a recent research 2 ( Supplementary Fig. 3b).
The parameters are chosen to be J zz 1 = 0.8 meV and J zz 2 = 0.076 meV. The proposed stripe antiferromagnetic structure will lead to magnetic Bragg peaks at M points. We measured the elastic scattering at M points with the similar statistic with that at Γ point in the main text. As shown in Supplementary Fig. 5, no indication of peak-like feature can be distinguished. For the spin excitations, since all spins are aligned along the c direction, the S zz sector vanishes for local spins and the transverse components are essentially dispersionless, clearly inconsistent with our data ( Supplementary Fig. 4e).
In addition, we calculate the energy dependence of the spectral intensity at Γ and M points with various values of J zz 1 /J ± 1 . It is shown that the simulated spin wave energy at the Γ point is either comparable or lower than that at the M point ( Supplementary   Fig. 6). This is inconsistent with our data where the spin wave energy at the Γ point exhibits a maximum with an energy 3 significantly higher than that at the M point (Fig. 3a). Thus, the anisotropic XXZ model with a dipolar order by no means can describe the spin wave dispersion in TmMgGaO 4 .
We also calculate the scattering function based on the quasi-doublet scenario that is proposed in the main text ( Supplementary   Fig. 2b). We show that the longitudinal excitations (S zz channel) of the intertwined multipolar order are in excellent agreement with our data (Fig. 2g-l, Fig. 3b).
Finally, we consider the intertwined multipolar order raised from a non-Kramers doublet system 3 . It is described by the where γ i j are phase factors that depend on the directions of the bonds. In this scenario, the effective spin components S x and S y show hidden quadrupolar behavior while the out-of-plane component S z shows dipolar behavior, similar to the quasi-doublet scenario that is proposed for TmMgGaO 4 in the main text. However, the observed large Landé g-factor (12.11) is inconsistent with a non-Kramers doublet state. Moreover, the heat capacity of the highly diluted Tm 0.04 Lu 0.96 MgGaO 4 shows a finite zerotemperature limit of C m /T , which is consistent with the quasi-doublet state, but inconsistent with the non-Kramers doublet state 2 .
Supplementary Note 3: Magnetic form factor.
For conventional dipolar moments, the magnetic form factor decreases with increasing Q while the multipoles may display an anisotropic and non-monotonic Q-dependence of the magnetic form factor, because multipoles have complex magnetization density 4 . In Supplementary Fig. 7, we present the constant energy slices of the simulated spin wave excitations considering a dipolar Tm 3+ form factor. The simulated spin excitation signal in the second Brillouin zone is considerably weaker than that of the first zone, which is clearly inconsistent with the measured data in Fig. 2a-f. The constant energy cuts through the first and second Brillouin zones further prove that the spin excitation intensity does not follow the dipolar Tm 3+ form factor ( Supplementary Fig. 8).
Correspondence and requests for materials should be addressed to J.Z. (zhaoj@fudan.edu.cn) or G.C.