the CEF transitions propagating through the lattice are called spin excitons [3][4][5][6][7] , analogous to electronic excitons that are bound states of an electron and a hole in a solid 9 . Spin excitons are fundamentally different from spin waves (magnons), which are strongly dispersive collective modes associated with spin precession on the lattice of magnetically ordered materials, and which disappear above the magnetic ordering temperature for isotropic Heisenberg magnets. In most cases, spin excitons originate from CEF levels of a localized single ion. Therefore, they are expected to be dispersionless in reciprocal space and well defined in both the magnetically ordered and paramagnetic states. However, when dispersive spin excitons are observed, the dispersion of these excitations can reveal unique information concerning magnetic exchange interactions between the localized ionic sites (spin-spin entanglement) and their relationship with the magnetically ordered state [3][4][5][6][7]10,11 .

Here we use thermodynamic and neutron scattering experiments to study stoichiometric honeycomb lattice antiferromagnetic (AF) ordered magnet Ni 2 Mo 3 O 8 , where Ni 2+ ions form a bipartite honeycomb lattice comprised of two triangular lattices, in the tetrahedral and octahedral crystalline environment, respectively (Fig. 1a-c) 12,13 . We find that CEF levels of Ni 2+ ions from both tetrahedral and octahedral environments have a nonmagnetic spin-singlet-ground state but with very different single-ion anisotropy energy scales for the two sites. Spin excitations of CEF levels (spin excitons) from the Ni 2+ triangular tetrahedral sites form a diffusive pattern around the Brillouin zone (BZ) boundary in the AF and paramagnetic states in momentum space. Therefore, Ni 2 Mo 3 O 8 realizes a novel situation in which the exchange energy falls between two very different single-ion energies, leading to a cooperative mechanism for magnetic order and strongly dispersing excitons associated with the larger single-ion anisotropy of the tetrahedral sites. Due to this hierarchy of energy scales, the excitons can persist even when the magnetic order is destroyed above the Néel temperature. In this regime, the excitons are strongly scattered from spin fluctuations and give rise to a distinct mechanism for dispersive diffusive scattering, hosting a unique coexistence of heavy particles (tetrahedral excitons) propagating in a frustrated background of light but dense (octahedral) spins due to spin entanglement and geometric frustrations.

Crystal structure, magnetic order, susceptibility, and specific heat of Ni 2 Mo 3 O 8

The M 2 Mo 3 O 8 (M = Fe, Mn, Ni, Co, Zn) compounds have drawn increasing attention due to their multiferroic properties [12][13][14][15][16] . The crystal structure of M 2 Mo 3 O 8 consists of magnetic bipartite honeycomb M-O layers, separated by sheets of Mo 4+ layers (Fig. 1a), where the Mo 4+ ions inside each layer are trimerized and form a singlet. The two M 2+ sites have different oxygen coordination, with one site being an MO 6 octahedron and the adjacent one being an MO 4 tetrahedron. In this family, Ni 2 Mo 3 O 8 was studied as a platform to explore the physics of geometrically frustrated lattice 12,13 . Neutron powder diffraction experiments reveal that both the MO 6 octahedron and the MO 4 tetrahedron each form perfect triangular lattices with no inter-site disorder, and the system orders antiferromagnetically with a Néel temperature of T N = 5.5 K 12,13 . The magnetic structure is stripe like within the Ni-O plane, and zig-zag like along the c-axis with different ordered moments for octahedral and tetrahedral Ni sites (Fig. 1c andd). Previous single-ion CEF analysis suggests that both octahedral and tetrahedral Ni sites have nonmagnetic singlet-ground states, and the first excited magnetic doublets are at 7.8 and 23 meV, respectively 12 .

Even though the Ni ions appear to form a simple (bipartite) honeycomb lattice in Ni 2 Mo 3 O 8 (Fig. 1b-d is assumed to arise from the relatively large magnetic exchange coupling between the octahedral and tetrahedral Ni sites (denoted as J 1 in Fig. 1d) in comparison with the energy of the CEF level of the Ni octahedral site 12 1 and Supplementary Information for details). The X-ray and neutron diffraction refinements also show that nonmagnetic dopant ions like Zn 2+ prefer to occupy the tetrahedral sites 12 . Consistent with previous neutron powder diffraction work 12,13 , our neutron single crystal diffraction refinements find that the spins of Ni 2+ ions form a stripy AF order (Fig. 1c andd) below T N = 5.5 K (Fig. 1e), but with the ordered moments of tetrahedral and octahedral Ni sites being 1.47 and 1.1 μ B , respectively, different from the previously reported values.

Figure 1f shows the temperature dependence of the d.c. susceptibility χ(T) with 1-8 T magnetic fields applied along the [1,1,0] direction perpendicular to the c-axis. The data in the 1 T field (orange line) shows a clear peak around T N ≈ 6 K consistent with neutron data in Fig. 1e, while the data in the 8 T field (blue line) shows no evidence of a magnetic transition. In-plane susceptibility is much larger than the caxis susceptibility (see Supplementary Fig. 3), indicating easyplane anisotropy. The Curie-Weiss fitting to the in-plane and c-axis susceptibility above 200 K gives θ CW⊥ = -454.12 ± 0.53 K and θ CW,// = -100.52 ± 0.11 K, respectively, suggesting a quasi-2D system where the in-plane magnetic exchange is larger than the c-axis exchange. The difference between our analysis and previous results on single crystals mainly comes from the direction of the in-plane magnetic fields 13 . Figure 1g shows the temperature dependence of the magnetic contribution to the specific heat as a function of applied magnetic fields perpendicular to the c-axis. At zero field (red line), we see a typical λ-shaped transition around T N ≈ 6 K. At 8 T field, there is no evidence of a magnetic transition above 2 K, consistent with Fig. 1f. Figure 1h shows the temperature-field phase diagram from our susceptibility and specific heat data.

To search for the CEF levels of the tetrahedral and octahedral Ni sites and demonstrate that the ground state of Ni 2 Mo 3 O 8 is indeed a spin singlet, we carried out INS experiments on single crystalline and powdered samples with incident energies (E i ) of 2.5 meV, 3.7 meV, 40 meV, 250 meV, and 1.5 eV. At E i = 2.5 meV, we see clear dispersive spin waves at 1.7 K (Fig. 2a) with two modes. This is consistent with the expectations from the linear spin-wave theory (LSWT) calculation, since there are four Ni 2+ sites in the magnetic unit cell, resulting in two doubly-degenerate modes. The small anisotropic spin gap (<0:3 meV) and overall spin-wave energy bandwidth of 1.5 meV are consistent with thermal dynamic data in Fig. 1f-h and T N = 5.5 K. Figure 2b andc shows a powder averaged INS S E,Q ð Þspectrum, where E and Q are energy and moment transfer, respectively, and a constant-|Q | cut with E i = 40 meV at temperatures below and above T N . At 1.7 K, there are at least three excitation peaks from the Ni 2+ spins at 13.8, 16.9, and 20.3 meV, and no visible excitations from 2 to 10 meV and above 40 meV. These excitations cannot be spin waves since there are already two modes below 1.5 meV. They must arise from the single-ion CEF levels which we analyze using the Stevens operator formalism, in which, due to the 12, the orbital ground state is the 3 A state which has a three-fold degeneracy that is further lifted by the spin-orbit coupling (SOC) producing a singlet and doublet (Fig. 2d). Since the Ni 2+ ions in NiZnMo 3 O 8 tend to occupy octahedral sites, we performed INS experiments on the powder sample of NiZnMo 3 O 8 to study the singleion crystal fields of octahedral sites 12 . We find that the intensity of the scattering in the range of 12-21 meV is considerably reduced (see Supplementary Fig. 4), consistent with the percentage of the Ni in the tetrahedral site of NiZnMo 3 O 8 determined from neutron powder diffraction. For comparison, the spin excitations in NiZnMo 3 O 8 are mostly centered below 2 meV, consistent with the notion that the energy of the CEF doublet levels from the octahedral site is below 2 meV (see Supplementary Fig. 5), which is clearly different from earlier low resolution electron spin resonance measurements and estimation from the point charge model 12 . Therefore, the three CEF levels observed in Ni 2 Mo 3 O 8 at 1.7 K near 17 meV are all from the first excited doublet of the tetrahedral Ni site. Since the magnetic unit cell doubles the structural unit cell in the ordered state and the molecular field from the ordered moments splits the doublets, one would expect up to four excitation modes from the Ni tetrahedral site below T N , while there is only one excitation above T N . On cooling below T N from 10 K, we see a clear splitting of the broad CEF peak at ~17 meV into two peaks around 13 meV and a broad peak around 20 meV, consistent with this picture (Fig. 2c andd). Combined with susceptibility data in Fig. 1f, we construct the Ni CEF levels as shown in Fig. 2d. While both Ni sites have singlet-ground states, the first excited state for the Ni tetrahedral and octahedral sites is at ~17 and ~1 meV, respectively. Since the energy bandwidth of spin waves in the AF ordered Ni 2 Mo 3 O 8 is less than 2 meV (Fig. 2a), we estimate that the magnetic exchange interactions between the Ni tetrahedral and octahedral sites J 1 , and second-neighbor tetrahedral (octahedral) and tetrahedral (octahedral) sites J T (J O ) to be less than 2 meV (Fig. 1d). Therefore, we identify Ni 2 Mo 3 O 8 as a spin-singletground-state system with magnetic order being induced by the exchange J 1 comparable to the CEF level of the Ni octahedral site (J 1 >D O ≈0:8 meV, Fig. 2e andf). For comparison, we note that Co 2 Mo 3 O 8 has a much higher T N ~40 K 17 and its upper band of spin waves is around 12 meV 18 .

To determine the energy, wavevector, and temperature dependence of the Ni tetrahedral excitonic magnetism in Ni 2 Mo 3 O 8 , we coaligned high-quality single crystals in the [H, H, 0] × [-K, K, 0] scattering plane (Fig. 3a andb). Figure 3c-e is the E-Q dispersions of the spin excitons at 1.5, 10, and 120 K. We observed magnetic scattering in two separated energy regions, 12-16 and 18-22 meV, at 1.5 K. Above T N , the scattering below and above 17 meV merge and become dispersive. The dispersion persists up to 120 K, which is one of the signatures of excitons. Figure 3f-k shows reciprocal space maps of the spin excitations in the [H, K, 0] plane in the two energy regions at 1.5, 10, and 120 K. The maps at different temperatures show qualitatively the same features: For E = 20 ± 2 meV, the scattering show broad peaks Single crystal X-ray diffraction refinement on Ni2Mo3O8. Over 15,000 Bragg peaks are collected and refined with a space group P6 3 mc. The positions and occupation fractions are refined, yielding no magnetic/nonmagnetic disorder. The fitting results an R1 = 1.03%.

centered near the Brillouin zone center Γ points (Fig. 3f-h), while for E = 14 ± 2 meV, the scattering is like the complementary part of the high energy scattering that forms a diffusive pattern around the zone boundary (Fig. 3i-k). In both cases, the scattering below and above T N are similar, contrary to the expected broadening of spin waves in momentum space from a magnetically ordered state to a paramagnetic state across T N . Since spin waves in Ni 2 Mo 3 O 8 have a band top of ~1.5 meV at 1.7 K (Fig. 2a), the broad dispersive excitations in Fig. 3 f and i at 1.5 K well below T N cannot arise from spin waves of magnetic ordered Ni 2+ . Figure 3l and m shows the in-plane magnetic field dependence of SðE,QÞ at energies near the CEF levels of Ni tetrahedron at 2 K for zero and 5-T field-polarized ferromagnetic state, respectively. The wavevector dependence of spin excitations for zero (Fig. 3n) and 5-T (Fig. 3o) field at E = 20 ± 2 meV and 2 K are similar to zero field data at 1.5 K (Fig. 3f) and 10 K (Fig. 3g), respectively. The situation is similar at E = 14 ± 2 meV (Fig. 3i at 1.5 K and 3j at 10 K in zero field, Fig. 3p at zero field and 3q at 5-T both at 2 K). This reflects the fact that the paramagnetic and ferromagnetic states have the same periodicities and that the exchange interaction does not appreciably change with field or temperature, whereas the AF state has a different periodicity, as the magnetic unit cell is doubled that of the structural cell in the AF phase. However, the paramagnetic and ferromagnetic states differ in that the applied field splits the excited doublet in the ferromagnetic state while there is no splitting in the paramagnetic state. This splitting is not seen in the data because it is smaller (≈0.1 meV) than the instrument resolution (≈1 meV), whereas the splitting due to the magnetic ordering is expected to be an order of magnitude larger than that produced by the external field.

There are several unusual features in the CEF levels of Ni 2 Mo 3 O 8 . First, the band top of low-energy spin waves is below 1.5 meV and the magnetic order is destroyed by an in-plane field of 8 T, indicating J 1 ≪D T ≈1:2 meV. In singlet-ground-state systems, one would expect a paramagnetic state at zero temperature, but the system orders below 6 K. Second, spin excitons and spin waves should only hybridize when they have similar energy scales, while excitons should be featureless in Q if CEF levels have much higher energy than spin waves. In Ni 2 Mo 3 O 8 , the excitonic bands at high energy (12-20 meV) are weakly dispersive in energy but show clear Q-dependence. Comparison of the magnetic exchange couplings and CEF energy levels for Ni ions in tetrahedral and octahedral sites, respectively. One expects a nonmagnetic ground state for tetrahedral site while a magnetic ordered state is expected in the octahedral sites.

To understand these results, we consider a S = 1 XXZ Hamiltonian for Ni 2 Mo 3 O 8 :

where the first-nearest neighbor coupling is

the second-and third-neighbor coupling are

and the SIA term is

Here S i is the spin operator at site i, γ the anisotropic exchange, êa is the unit vector linking the spins at site i and i + a, D T and D O are SIA at tetrahedral and octahedral sites, respectively. For J = 0, the Hamiltonian in Eq. ( 1) has a unique gapped ground state (S z i = 0 on all states). Perturbation for J≪D preserves the gap and the system remains in a unique, trivial ground state. For J≫D, the single-ion terms are unimportant, and we expect an ordered ground state. Consequently, a quantum phase transition from paramagnetic to ordered state is The data is integrated along the L direction since the scattering above 12 meV has no modulation along the L direction. Dashed lines indicate the energy integration range in f-h, Momentum dependence of the magnetic scattering at 1.5, 10, and 120 K, where high intensity is near the zone center. The energy integration range is 18-22 meV. i-k Momentum dependence of the scattering at 1.5, 10, and 120 K, where high intensity is near the zone boundary. The energy integration range is 12-16 meV. The scattering becomes sharper in the paramagnetic state at 10 K. l-q In-plane magnetic field dependence of S E,Q ð Þat 0 and 5-T at 2 K. The missing data in h-q is due to the narrower detector coverage when Ni 2 Mo 3 O 8 is in a magnet.

expected as a function of increasing J. We capture the transition by a Curie-Weiss mean-field approach (see Supplementary Information). At the simplest level, if we only consider nonzero J 1 ,γ and D T ,D O , the critical value of J 1 is found to be

This equation shows that order can be induced when the exchange is intermediate between the two single-ion energies. In Ni 2 Mo 3 O 8 , D T ≈ 16 meV and D O ≈ 1 meV, according to our CEF analysis. Therefore, the scale of magnetic exchange required to induce moment is largely reduced due to the bipartite nature of the lattice. From Eq. ( 5), we have e J≈ 2.06 meV, which is consistent with the low-energy scale of the magnetic order. The anisotropic exchange γ, which is originated from the combined effects of crystal field and SOC, can be used to energetically favor the experimentally observed four-sublattice state. The Dzyaloshinskii-Moriya interaction d is responsible for the small out-of-plane spin canting 19 , and a finite value above the threshold (d≈ e J) is necessary 19 to stabilize the stripy (as opposed to the Néel) ordered state that is observed experimentally.

The Q-dependent scattering for Ni tetrahedral CEF levels in Fig. 3f, g and i, j suggest short-range ferromagnetic and AF correlations of Ni tetrahedrons, respectively. We first consider three tetrahedral Ni atoms on the vertices of an equilateral triangle and the spins forms a 120°configuration (Fig. 4a). The Fourier transform of the spins on this cluster is:

where S 0 is the direction of one of the spins, R 1=3 is the 120°rotational matrix, F Ni is the magnetic form factor of Ni 2+ , and M is the magnetization matrix. The observed scattering SðE,QÞ intensity is proportional to |κ × ðS c × κÞ| 2 , where κ Q=Q. Figure 4b shows the calculated structural factor SðQÞ for the 120°AF spin configuration, where the J T in Ni tetrahedron triangles dominates. Similarly, the structural factor for a ferromagnetic spin configuration can be calculated by canceling the offdiagonal term of M, which is shown in Fig. 4d. The calculated factors fit well with the observed spin excitations in the INS experiments, indicating that spin excitons have spin-spin correlations.

To quantitatively understand the S E,Q ð Þspectrum of the CEF levels, we modeled the INS spectra using an effective Hamiltonian in Eqs. (1-5) to describe a ground state singlet and excited doublet on each octahedral and tetrahedral site with an effective S = 1 spins, SIAs, symmetric and anti-symmetric exchange interactions (see Supplementary Information). The large SIA on the tetrahedral site gives rise to the apparent high energy CEF modes, which disperse due to the exchange couplings. The spectral function of the excitations in the ordered state is calculated using a flavor wave expansion based on the SU(3) representation of the triplet of levels on each site. This method captures the partial suppression of ordered moment by the tendency to single-ion singlet formation, while still describing the ordering and associated spin waves.

Focusing on the high energy excitations, the flavor wave calculation predicts the formation of four bands, which are spin-split and folded due to the AF enlargement of the unit cell. The predominant momentum dependence arises from the AF exchange coupling between the closest pairs of tetrahedral sites (J T ), which leads to high intensity at the lower edge of the band near the zone boundary, and at the upper edge of the band near the zone center (right panels in Fig. 4b-d). The exchange parameters from the calculation are summarized in the Supplementary Information.

Our single crystal X-ray and neutron diffraction refinements find that Ni tetrahedrons in Ni 2 Mo 3 O 8 form an ideal 2D triangular lattice without the magnetic and nonmagnetic disorder (Table 1). As the energy scale of the CEF spin excitons from Ni tetrahedrons is much larger than the magnetic exchange interactions determined from spin waves of Ni 2 Mo 3 O 8 (Fig. 2a), the presence of static AF order only slightly modifies the continuum-like Q-dependent scattering by making it less well-defined in the ordered state possibly due to mixing of the dispersive spin waves with CEF levels. In induced-moment systems with a singlet-ground state [3][4][5][6][7] , spin excitons can become highly dispersive and couple strongly to the ground state with a large magnetic exchange coupling J. When the ground state is not a spin singlet but a pseudospin doublet, spin excitons at high energies in some d 3 transition-metal oxides can also be dispersive and have unusual properties due to strong SOC. For example, in the classic Mott insulator CoO, where the strength of SOC is comparable to the magnetic exchange coupling 20 , the Q-dependence of spin excitons at high energies decay faster with Q than the Co 2+ magnetic form factor, suggesting a breakdown of the localized spin excitons towards spatially extended magnetism 21 . More recently, in A-type AF ordered CoTiO 3 with T N ≈ 38 K 22 , the dispersive spin excitons around ~27 meV due to SOC become softer and acquire a larger bandwidth on warming from the AF (5 K) to the paramagnetic state (60-120 K) 23,24 . For comparison, the ~17 meV CEF doublet in the Ni tetrahedral site in Ni 2 Mo 3 O 8 induced by SOC (Fig. 2d) is similar in Qspace around the BZ boundary (see Supplementary Fig. 8 for the detailed cuts and comparisons) and narrower in energy bandwidth on warming from the AF (1.5 K) to the paramagnetic (10 and 120 K) state (Fig. 3c-k). While weakly dispersive spin waves above a large spin gap seen in the spin-chain compound Sr 3 NiIrO 6 25 and the 1D magnet BaMo(PO 4 ) 2 26 also survive to temperatures well above their perspective T N s, they originate from magnons (not spin excitons) and do not have the line-shapes in Q-space as we observe in Ni 2 Mo 3 O 8 .

Therefore, our results highlight the novel physics associated with two magnetic species, each on a frustrated triangular lattice, with very different single-ion anisotropies, and expose Ni 2 Mo 3 O 8 as a promising venue to explore the propagation of spin excitons in a dense highly fluctuating magnetic background. Most importantly, they indicate that CEF levels in an ideal triangular lattice magnet can produce dispersive spin excitons irrespective of static magnetic order, and the origin of this phenomenon is most likely due to spin entanglement and geometric frustrations without invoking the quantum spin liquid paradigm [27][28][29][30] .

Polycrystalline samples of Ni 2 Mo 3 O 8 , NiZnMo 3 O 8 , and Zn 2 Mo 3 O 8 were synthesized using a solid-state method. Stoichiometric powders of NiO, ZnO, Mo, and MoO 3 were mixed and pressed into pellets, and then sintered at 1050 °C for 24 h. Single crystalline Ni 2 Mo 3 O 8 was synthesized using the chemical vapor transport method. Powder X-ray diffraction measurements performed on powder samples and ground single crystals reveal that the samples have a pure phase, with a space group P6 3 mc and the lattice parameter a = b = 5.767 Å and c = 9.916 Å. The structural information of the Ni 2 Mo 3 O 8 single crystal was investigated using a Rigaku XtaLAB PRO diffractometer equipped with a HyPi x-6000HE detector at ORNL. A molybdenum anode was used to generate X-rays with wavelength λ = 0.7107 Å. The samples were cooled by cold nitrogen flow provided by an Oxford N-Helix cryosystem. Single crystal X-ray refinements on Ni 2 Mo 3 O 8 reveal that the Ni and Mo are in fully occupied positions with no magnetic and nonmagnetic site disorder (Table 1).

Specific heat measurements were conducted using a thermalrelaxation method in a physical property measurement system (Quantum Design).

Neutron powder diffraction experiments were performed at room temperature using the high resolution powder diffractometer BT-1, at the NIST center for neutron research. 5.0 grams of NiZnMo 3 O 8 powder was used. Powder neutron refinements reveal that Zn prefers to occupy tetrahedral sites (88.1%) and the rest of the tetrahedral sites are occupied by Ni (11.9%). Detailed results of the refinement are shown in Supplementary Table 1. Single crystal neutron diffraction experiments were carried out using the elastic diffuse scattering spectrometer,CORELLI 31 , at the Spallation Neutron Source, ORNL. One small piece of single crystalline Ni 2 Mo 3 O 8 was aligned in the [H, 0, L] plane. 53 structural Bragg peaks and 32 magnetic Bragg peaks at 2 K were used for the refinement.

INS experiments were carried out on polycrystalline Ni 2 Mo 3 O 8 (6.0 g), NiZnMo 3 O 8 (4.24 g), and Zn 2 Mo 3 O 8 (4.33 g), on the chopper spectrometer, MARI, and the cold neutron multi-chopper spectrometer, LET, at ISIS neutron and muon source. We collected data with 40 meV, 250 meV, and 1.5 eV incident energy (E i ) at 4 K on MARI, and with 1.8, 3.7, and 12.1 meV E i at 2 and 12 K on LET.

We co-aligned more than 200 pieces of single crystals of Ni 2 Mo 3 O 8 (1.5 g) to carry out inelastic neutron experiments on the cold neutron chopper spectrometer, CNCS, the fine-resolution fermi chopper spectrometer 32 , SEQUOIA 33 and ARCS 34 thermal chopper spectrometers, at the Spallation Neutron Source, Oak Ridge National Laboratory. The sample assembly was aligned in [H, K, 0] scattering plane on CNCS and SEQUOIA. We performed 180°rotational scans at 1.7, 3.5, 4.5, 5.5, and 6.5 K with 2.5 meV E i on CNCS and at 1.5, 10, and 120 K with 40 meV E i on SEQUOIA. On ARCS, we aligned the sample in [-K, K, L] scattering plane and measured with 26 and 40 meV E i at 2 K.

Each mean-field state of the quantum magnet corresponds to a product state where the wave function on each site lives in the spin-1 Hilbert space. One can design a trial wave function to describe the magnetic order, with the variational parameters determined by minimization of the mean-field energy. The excitations on top of the ground state can then be suitably described by the "flavor waves" making use of the SU(3) flavor rotation in the spin-1 Hilbert space (see Supplementary Information for details). The dispersive excitations are well-defined in both ordered and disordered phases and can exhibit the experimentally observed behaviors in structure factors. In Fig. 4, we have used the parameters D t = 16 meV, D o = 1:0 meV, d = 0:3, γ = 0:5, J 1 = 2:0 meV, J t = 0:5 meV, J o = 0:15 meV, J 3 = 0:0 meV:

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