Tetrahedral triple-Q magnetic ordering and large spontaneous Hall conductivity in the metallic triangular antiferromagnet Co1/3TaS2

The triangular lattice antiferromagnet (TLAF) has been the standard paradigm of frustrated magnetism for several decades. The most common magnetic ordering in insulating TLAFs is the 120° structure. However, a new triple-Q chiral ordering can emerge in metallic TLAFs, representing the short wavelength limit of magnetic skyrmion crystals. We report the metallic TLAF Co1/3TaS2 as the first example of tetrahedral triple-Q magnetic ordering with the associated topological Hall effect (non-zero σxy(H = 0)). We also present a theoretical framework that describes the emergence of this magnetic ground state, which is further supported by the electronic structure measured by angle-resolved photoemission spectroscopy. Additionally, our measurements of the inelastic neutron scattering cross section are consistent with the calculated dynamical structure factor of the tetrahedral triple-Q state.


Introduction
Discovering magnetic orderings with novel properties and functionalities is an important goal of condensed matter physics.While ferromagnets are still the best examples of functional magnetic materials due to their vast spectrum of technological applications, antiferromagnets are creating a paradigm shift for developing new spintronic components 1,2 .A more recent case is the discovery of Skyrmion crystals induced by a magnetic field in different classes of materials.These chiral states, which result from a superposition of three spirals with ordering wave vectors that differ by a ± 120° rotation about a high-symmetry axis, can produce a substantial synthetic magnetic field that couples only to the orbital degrees of freedom of conduction electrons 3,4 .When conduction electrons propagate through a skyrmion spin texture, they exhibit spontaneous Hall effect.The origin of this effect becomes transparent in the adiabatic limit.Due to the exchange interaction, the underlying local moment texture aligns the spin of a conduction electron that moves in a loop, inducing a Berry phase in its wavefunction equal to half of the solid angle spanned by the local moments enclosed by the loop.This phase is indistinguishable from the Aharonov-Bohm phase induced by a real magnetic flux, and each skyrmion generates a flux quantum in the corresponding magnetic unit because the spins span the full solid angle of the sphere (4).
The triangular lattice Heisenberg model is a textbook example that can host diverse quantum states with small variations of short-range exchange interactions.The generic ground state for nearest-neighbour (NN) antiferromagnetic interactions is the three-sublattice 120° structure shown in Fig. 1a.This spiral structure is characterized by an ordering wave vector located at ± K-points of the hexagonal Brillouin zone (Fig. 1d).Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1b, whose ordering wave vector is one of the three M-points of the Brillouin zone (see Fig. 1e).Remarkably, for S = ½, these two phases seem to be separated by a quantum spin liquid state [5][6][7][8][9][10][11] whose nature is not yet fully understood.
However, small effective four-spin interactions can induce a fundamentally different chiral antiferromagnetic order in triangular lattice antiferromagnets.This state is the triple-Q version of the stripe order, where the three different M-ordering wave vectors (see Fig. 1f) coexist in the same phase giving rise to a noncoplanar four-sublattice magnetic ordering (see Fig. 1c).The spins of each sublattice point along the all-in or all-out principal directions of a regular tetrahedron.Theoretical studies suggest that this state can appear naturally in metallic TLAFs, where effective four-spin interactions arise from the exchange interaction between conduction electrons and localized spin degrees of freedom 12,13 .This state was predicted to appear in the Mn monolayers on Cu(111) surfaces by density functional theory, 14 and it was observed in the hcp Mn monolayers on Re(0001) using spin-polarized scanning tunnelling microscopy 15,16 .However, it has not been reported yet in bulk systems.
The tetrahedral ordering is noteworthy for its topological nature, as it can be viewed as the short-wavelength limit of a magnetic skyrmion crystal 17 .The three spins of each triangular plaquette span one-quarter of the solid angle of a sphere, implying that each skyrmion (one flux quantum) is confined to four triangular plaquettes.As illustrated in Fig. 1c, the two-dimensional (2D) magnetic unit cell of the tetrahedral ordering consists of eight triangular plaquettes, meaning there are two skyrmions per magnetic unit cell.In other words, the tetrahedral triple-Q ordering creates a very strong effective magnetic field of one flux quantum divided by the area of 4 triangular plaquettes in the adiabatic limit.Notably, this ordering does not have any net spin magnetization.However, the emergent magnetic field couples to the orbital degrees of freedom of the conduction electrons, giving rise to a uniform orbital magnetization and a large topological Hall effect characterized by scalar spin chirality (  = 〈  ⋅   ×   〉) 3,4 .Thus, this spin configuration is the simplest textbook example where non-trivial band topology is induced in the absence of relativistic spin-orbit coupling (the noncoplanar configuration generates an effective gauge field that couples to the orbital degrees of freedom of the conduction electrons).Moreover, the tetrahedral ordering can provide a potential route to realize the antiferromagnetic Chern insulator by properly adjusting the Fermi level of the system, as suggested in Refs. 12,13,18.
This work reports a four-sublattice tetrahedral triple-Q ordering as the only scenario known to the authors that is consistent with our data for the metallic triangular antiferromagnet Co1/3TaS2.Our key observations are the coexistence of long-range antiferromagnetic ordering with wave vector qm = (1/2, 0, 0), a weak ferromagnetic moment (Mz( = 0)), and a non-zero AHE (σ xy ( = 0) ) below TN2, which rules out the possibility of single-Q and double-Q ordering [19][20][21][22] .Based on the crystalline and electronic structure of Co1/3TaS2, we also provide a theoretical conjecture about the origin of the observed ordering, consistent with our angleresolved photoemission spectroscopy (ARPES) data.Moreover, the calculated low-energy magnon spectra of the tetrahedral ordering agree with the spectra measured by inelastic neutron scattering.Finally, we discuss the robustness of the tetrahedral ordering against an applied magnetic field in Co1/3TaS2.

Results and Discussion
Co1/3TaS2 is a Co-intercalated metal comprising triangular layers of magnetic Co 2+ ions (Fig. 1g).Previous studies on Co1/3TaS2 in the 1980s reported the bulk properties of a metallic antiferromagnet with S = 3/2 (a high-spin d 7 configuration of Co 2+ ), [23][24][25] including a neutron diffraction study that reported an ordering wave vector qm = (1/3, 1/3, 0) characteristic of a 120° ordering 25 .More recently, an experimental study on single-crystal Co1/3TaS2 observed a significant anomalous Hall effect (AHE) comparable to those in ferromagnets below 26.5 K, which is the second transition temperature (TN2) of the two antiferromagnetic phase transitions at TN1 = 38 K and TN2 = 26.5 K (see Fig. 1h) 26 .Based on the 120° ordering reported in Ref. 25 and a symmetry argument, the authors of Ref. 26 suggested that the observed AHE, σ xy ( = 0) ≠ 0, can be attributed to a ferroic order of cluster toroidal dipole moments.However, our latest neutron scattering data reported in this work reveals an entirely different picture: Co1/3TaS2 has a magnetic structure with ordering wave vectors of the M-points (qm = (1/2, 0, 0) and symmetry-related vectors) instead of qm = (1/3, 1/3, 0).The most likely scenario for such distinct outcomes would be a difference in Co composition; while we confirmed that our experimental results are consistently observed in CoxTaS2 with 0.299(4) < x < 0.325(4), the reported composition value for Ref. 25 's sample is x=0.29 (see Ref. 24 ).Additional Co disorder, beyond variations in composition, could also be a contributing factor.However, the limited data available for the sample in Ref. 25 prevents us from forming a definitive assessment in this regard (see Supplementary Notes).In any case, the theoretical analysis in Ref. 26 based on Ref. 25 is not valid for qm = (1/2, 0, 0) since a different ordering wave vector leads to a qualitatively different scenario.This forced us to conduct a more extensive investigation and come up with a scenario consistent with our comprehensive experimental datasets.
Figs. 2a and 2b show the neutron diffraction patterns of powder and single-crystal Co1/3TaS2 for T < TN.Magnetic reflections appear at the M-points of the Brillouin zone for both T < TN2 and TN2 < T < TN1, implying that the ordering wave vector is qm = (1/2, 0, 0) or its symmetrically equivalent wave vectors.It is worth noting that this observation is inconsistent with the previously reported wave vector qm=(1/3, 1/3, 0) 25 ; i.e., Co1/3TaS2 does not possess a 120° magnetic ordering.The magnetic Bragg peaks at the three different M points connected by the three-fold rotation along the c-axis (C3z) have equivalent intensities within the experimental error (Fig. 2c).This result suggests either single-Q or double-Q ordering with three equally weighted magnetic domains or triple-Q ordering.and  z ( = 0 ) due to its residual symmetry relevant to chirality cancellation 19 .Triple-Q ordering is, therefore, the only possible scenario that can resolve this contradiction since it allows for finite  xy ( = 0) and  z ( = 0) due to broken τ 1  symmetry [19][20][21] .In general, determining whether the magnetic structure of a system is a single/double-Q phase with three magnetic domains or a triple-Q ordering requires advanced experiments.However, qm = (1/2, 0, 0) is a special case where a triple-Q state can be easily distinguished from the other possibilities by probing non-zero TRS-odd quantities incompatible with the symmetry of single/double-Q structures.
The most symmetric triple-Q ordering that produces the same neutron diffraction pattern as that of Fig. 2h is illustrated in Fig. 2i.This is precisely the four-sublattice tetrahedral ordering shown in Fig. 1c, except that Co1/3TaS2 has an additional 3D structure with an AB stacking pattern.Such a triple-Q counterpart can be obtained through a linear combination of three symmetrically-equivalent single-Q states (    =   cos(   •   ) with  = 1, 2, 3) connected by the three-fold rotation about the c-axis 22 (see Supplementary Notes for more explanation).However, an arbitrary linear combination of these three components yields a triple-Q spin configuration with a site-dependent magnitude of ordered moments on the Co In the high-temperature ordered phase at TN2 < T < TN1, the triple-Q ordering that yields the same neutron diffraction pattern as that of the single-Q ordering shown in Fig. 2g is collinear, giving rise to highly nonuniform |  tri | (see Supplementary Fig. 7).Such a strong modulation of |  tri | is unlikely for S = 3/2 moments weakly coupled to conduction electrons (see discussion below).In addition, the collinear triple-Q ordering allows for finite Mz(H = 0) and  xy ( = 0) [19][20][21][22] , while they are precisely zero within our measurement error for TN2 < T < TN1.On the other hand, the single-Q ordering shown in Fig. 2g is more consistent with Mz(H = 0) =  xy ( = 0) = 0 in the temperature range TN2 < T < TN1 due to its τ 1  symmetry.
Therefore, the combined neutron diffraction and anomalous transport data indicate that a transition from a collinear single-Q to non-coplanar triple-Q ordering occurs at TN2.
Interestingly, as we will see below, our theoretical analysis captures this two-step transition process.
We now examine the feasibility of the tetrahedral triple-Q ground state in Co1/3TaS2.Notably, in contrast to typical triple-Q orderings reported in other materials [27][28][29] , this state emerges spontaneously in Co1/3TaS2 without requiring an external magnetic field.It was proposed on theoretical grounds that this state could arise in a 2D metallic TLAF formulated by the Kondo lattice model 12,13,18,30 : From a crystal structure perspective, Co1/3TaS2 is an ideal candidate to be described by this model.The nearest Co-Co distance (5.74 Å) is well above Hill's limit, and the CoS6 octahedra are fully isolated ((Fig.3a).This observation suggests that the Co 3d bands would retain their localized character, while itinerant electrons mainly arise from the Ta 5d bands.Our density functional theory (DFT) calculations confirm this picture and reveal that the density of states near the Fermi energy has a dominant Ta 5d orbital character (Supplementary Fig. 11).In this situation, a magnetic Co 2+ ion can interact with another Co 2+ ion only via the conduction electrons in the Ta 5d bands.Thus, in a first approximation, the Co 2+ 3d electrons can be treated as localized magnetic moments interacting via exchange with the Ta 5d itinerant electrons 23,25 (see Fig. 3b).Indeed, the Curie-Weiss behaviour observed in Co1/3TaS2 provides additional support for this picture; the magnitude of the fitted effective magnetic moment indicates S ~ Tetrahedral ordering can naturally emerge when the Fermi surface (FS) is threequarters (3/4) filled 12,13,18,31 because the shape of the FS is a regular hexagon (for a tight-binding model with nearest-neighbour hopping), whose vertices touch the M-points of the first Brillouin zone (Fig. 3c).In this case, there are three nesting wave vectors connecting the edges of the regular hexagon and the van Hove singularities at different M points, leading to magnetic susceptibility of the conduction electrons (χ(,  F )) that diverges as log 2 | −  m  | at  ∈ Mpoints 30 .This naturally results in a magnetic state with ordering wave vectors corresponding to the three M points (i.e., triple-Q).While perfect nesting conditions are not expected to hold for real materials, χ(,  F ) is still expected to have a global maximum at the three M points for Fermi surfaces with the above-mentioned hexagonal shape 18 .As shown on the right side of Fig. 3c, our ARPES measurements reveal that this is indeed the case of the FS of Co1/3TaS2, indicating that the filling fraction is close to 3/4 and that the effective interaction between the Co 2+ magnetic ions is mediated by the conduction electrons.
The above-described stabilization mechanism based on the shape of the Fermi surface suggests that the effective exchange interaction between the Co 2+ magnetic ions and the conduction electrons is weak compared to the Fermi energy.Under these conditions, it is possible to derive an effective RKKY spin Hamiltonian by applying degenerate second-order perturbation theory in J/t 30,31 .Since the RKKY model includes only bilinear spin-spin interactions, the single-Q (stripe) and triple-Q (tetrahedral) orderings remain degenerate in the classical limit.Moreover, by tuning the mutually orthogonal vector amplitudes   while preserving the norm , it is possible to continuously connect the single-Q (stripe) and triple-Q (tetrahedral) orderings via a continuous manifold of degenerate multi-Q orderings.The classical spin model's accidental ground state degeneracy leads to a gapless magnon mode with quadratic dispersion and linear Goldstone modes.The accidental degeneracy is broken by effective four-spin exchange interactions that gap out the quadratic magnon mode, which naturally arise from Eq. ( 1) when effective interactions beyond the RKKY level are taken into consideration 14,17,30,31 .The simplest example of a four-spin interaction favouring the triple-Q ordering is the bi-quadratic term  bq (  •   ) 2 with  bq > 0 .This was explicitly demonstrated using classical Monte-Carlo simulations with a simplified phenomenological J1-J2-Jc-Kbq model (see Figs. 3a and 3e), as shown in Supplementary Fig. 8.This model successfully captures the tetrahedral ground state and manifests a two-step transition process at TN2 and TN1 with an intermediate single-Q ordering, which is consistent with our conclusion based on experimental observations.The latter outcome can be attributed to thermal fluctuations favouring the collinear spin ordering 32,33 .In addition, previous DFT studies on TM1/3NbS2 (TM = Fe, Co, Ni), which is isostructural to Co1/3TaS2, also revealed that a noncoplanar state could be energetically more favourable than collinear and co-planar states 34 .
Before analyzing the magnon spectra in detail, it is worth considering the inter-layer network of Co1/3TaS2 with AB stacking.Along with the refined magnetic structure (Fig. 2g-i), the steep magnon dispersion shown in Fig. 3d indicates non-negligible antiferromagnetic NN interlayer exchange:  c  2 ~2.95 meV (see Fig. 3e).However, the finite value of Jc does not change the competition between stripe and tetrahedral orderings analyzed in the 2D limit.More importantly, the antiferromagnetic exchange Jc forces the tetrahedral spin configuration of the B layer to be the same as that of the A layer (see Supplementary Fig. 8a).Therefore, all triangular layers have the same sign of the scalar chirality   (or skyrmion charge), resulting in the realization of 3D ferro-chiral ordering.In this 3D structure, each magnetic unit cell of Co1/3TaS2 includes four skyrmions.
The low-energy magnon spectra (< 3 meV) of Co1/3TaS2 measured by INS are presented in Fig. 3f.In addition to the linear (Goldstone) magnon modes, which appear as bright circular signals centred at the M points, a ring-like hexagonal signal connecting six M points was additionally observed with weaker intensity.This can be interpreted as the trace of the quadratic modes since they should be present together with linear modes at low energy (see Supplementary Fig. 9).Since the signal is only present for E > 1.5 meV, we infer that the quadratic mode is slightly gapped.We used linear spin-wave theory to compare the measured INS spectra with the theoretical spectra of both single-Q and triple-Q orderings.Despite the simplicity of the J1-J2-Jc-Kbq model, the calculated magnon spectra of the tetrahedral ordering (Fig. 3g) successfully describe the measured INS spectra.The magnon spectra of the stripe order are also presented in Fig. 3h for comparison.The intensity of the quadratic magnon mode is much stronger than that of the triple-Q spectra, in apparent disagreement with our INS data.
A complete comparison between our data and the two calculations is shown in Supplementary Fig. 10.Additionally, the linear magnon modes of Co1/3TaS2 are also slightly gapped (~ 0.5 meV, see Fig. 3i).This feature can be explained for tetrahedral ordering only by considering both exchange anisotropy and higher order corrections in the 1/S expansion (see Supplementary Notes).
Finally, we discuss the behaviour of the tetrahedral ordering in Co1/3TaS2 in response to outof-plane and in-plane magnetic fields.Fig. 4c illustrates the field dependence of the measured anomalous Hall conductivity (  AHE ) and Mz for H // c, showing evident hysteresis with a sign change at ± c1 .Indeed, as already discussed in Ref. 26 , Mz cannot characterize the observed   AHE .Instead, as explained in the introduction, it is   that characterizes   AHE for the tetrahedral ordering.Therefore, based on our triple-Q scenario (see Figs. 4a-b), the sign change at ± c1 can be interpreted as the transition between tetrahedral orderings with positive and negative values of the scalar chirality   .In addition, the coexistence of weak ferromagnetic moment and large   ( = 0) is expected because, as we explained before, the real-space Berry curvature of the tetrahedral triple-Q ordering generates both the orbital ferromagnetic moment (of conduction electrons) and spontaneous Hall conductivity.However, the experimental techniques used in this study cannot discriminate between the spin and orbital contributions to Mz(H=0).While it will be interesting to identify the nature of this weak ferromagnetic moment, this is left for future studies.
We also compared the field dependence of the measured   AHE and   calculated from our Mz(H) data based on the canting expected in the tetrahedral order of Co1/3TaS2 (blue and orange arrows in Fig. 4).As shown in Fig. 4d,   AHE decreases slightly in response to both positive and negative magnetic fields, consistent with the calculated   of the tetrahedral ordering with mild canting.However, the model cannot capture the sudden decrease of   AHE due to a meta-magnetic transition at ± c2 , indicating that this transition changes the tetrahedral spin configuration.Additional neutron diffraction is required to identify the new ordering for |H| > Hc2.
The effect of an in-plane magnetic field was investigated using single-crystal neutron diffraction.Fig. 4e shows field-dependent (H //  m 1 = (1/2, 0, 0)) intensities of the three magnetic Bragg peaks, each originating from three different  m  of the tetrahedral ordering.
Interestingly, the equal intensity of the three peaks remains almost unchanged by a magnetic field up to 10 T, indicating robustness of the tetrahedral ordering against an in-plane magnetic field.This should be contrasted with triple-Q states found in other materials, which are induced by a finite magnetic field and occupy narrow regions of the phase diagram due to the ferromagnetic nature of the dominant exchange interaction [27][28][29] .
In summary, we have reported a tetrahedral triple-Q ordering in Co1/3TaS2, as the only magnetic ground state compatible with our bulk properties (non-zero  xy ( = 0) and Mz(H = 0)) and neutron diffraction data (qm = (1/2, 0, 0) or its symmetry-equivalent pairs).Moreover, we provide a complete theoretical picture of how this exotic phase can be stabilized in the triangular metallic magnet Co1/3TaS2, which is further corroborated by our measurements of electronic structure and long-wavelength magnetic excitations using ARPES and INS.We further investigated the effect of external magnetic fields on this triple-Q ordering, which demonstrates its resilience against the fields.Our study opens avenues for exploring chiral magnetic orderings with the potential for spontaneous integer quantum Hall effect 30 .
Note added.After the submission of this work, we came across another neutron diffraction work on Co1/3TaS2 published recently 35 .While they have drawn the same conclusion of magnetic structure as our tetrahedral triple-Q magnetic ordering, our study further delves into the microscopic aspects of this magnetic ground state, specifically in relation to electronic structure and the properties of a geometrically frustrated triangular lattice.We also substantiate our findings through the ARPES and INS measurements.
Anomalous Hall conductivity   AHE was derived by first subtracting a normal Hall effect from measured   and then using Eq. 2.
Powder neutron diffraction.We carried out powder neutron diffraction experiments of Co 1/3 TaS 2 using the ECHIDNA high-resolution powder diffractometer ( = 2.4395 Å) at ANSTO, Australia.To acquire clear magnetic signals of Co 1/3 TaS 2 , which are weak due to the small content of Co and its small ordered moment, we used 20 g of powder Co 1/3 TaS 2 .The quality of the sample was checked before the diffraction experiment by measuring its magnetic susceptibility and high-resolution powder XRD.The neutron beam at ECHIDNA contains weak /2 harmonics (~ 0.3 %), yielding additional Bragg peaks (Supplementary Fig. 3).We also performed Rietveld refinement and magnetic symmetry analysis using Fullprof software 36 .The results are summarized in Supplementary Tables 1-3 and Supplementary Figs.2-4.
Single-crystal neutron diffraction.We carried out single-crystal neutron diffraction under a magnetic field using ZEBRA thermal neutron diffractometer at the Swiss spallation neutron source SINQ.We used one singlecrystal piece (~ 16 mg) for the experiment, which was aligned in a 10 T vertical magnet (Oxford instruments) with (HHL) horizontal (Supplementary Fig. 5a).The beam of thermal neutrons was monochromatic using the (220) reflection of germanium crystals, yielding a neutron wavelength of  = 1.383Å with less than 1% of λ/2 contamination.Bragg intensities as a function of temperature and magnetic field were measured using a single 3 He-tube detector in front of which slits were appropriately adjusted.
Angle-resolved photoemission spectroscopy (ARPES) measurements.The ARPES measurements were performed at the 4A1 beamline of the Pohang Light Source with a Scienta R4000 spectrometer 37 .Single crystalline samples with a large AHE were introduced into an ultra-high vacuum chamber and were cleaved in situ by a top post method at the sample temperature of ~20 K under the chamber pressure of ~5.0 × 10 −11 Torr.
A liquid helium cryostat maintained the low sample temperature.The photon energy was set to 90 eV to obtain a high photoelectron intensity for the states of Co 3d characters.The total energy resolution was ~30 meV, and the momentum resolution was ~0.025 Å −1 in the measurements.
Single-crystal inelastic neutron scattering.We performed single-crystal inelastic neutron scattering of Co 1/3 TaS 2 using the 4SEASONS time-of-flight spectrometer at J-PARC, Japan 38 .For the experiment, we used nearly 60 pieces of the single crystal with a total mass of 2.2 g, which were co-aligned on the Al sample holder with an overall mosaicity of ~ 1.5° (Supplementary Fig. 5b).We mounted the sample with the geometry of the (H0L) plane horizontal and performed sample rotation during the measurement.The data were collected with multiple incident neutron energies (3.5, 5.0, 7.9, 14.0, and 31.6 meV) and the Fermi chopper frequency of 150 Hz using the repetition-rate-multiplication technique 39 .We used the Utsusemi 40 and Horace 41 software for the data analysis.Based on the crystalline symmetry of Co 1/3 TaS 2 , the data were symmetrized into the irreducible Brillouin zone to enhance statistics.
Classical Monte-Carlo Simulations.To investigate the magnetic phase diagram of our spin model, we performed a classical Monte-Carlo (MC) simulation combined with simulated annealing.We used Langevin dynamics 42 for the sampling method of a spin system.To search for a zero-temperature ground state, a large spin system consisting of 30 × 30 × 4 unit cells (7200 spins) with periodic boundary conditions was slowly cooled down from 150 K to 0.004 K.The thermal equilibrium was reached at each temperature by evolving the system through 5000 Langevin time steps, with the length of each time-step defined as dt = 0.02/(J 1 S 2 ).Finally, we adopted the final spin configuration at 0.004 K as the magnetic ground state, which was further confirmed by checking whether the same result was reproduced in the second trial.
For finite-temperature phase diagrams, we used a 30 × 30 × 6 supercell (10,800 spins).After waiting for 600 ~ 2,000 Langevin time steps for equilibration, 600,000 ~ 2,000,000 Langevin time steps were used for the sampling.Since thermalization and decorrelation time strongly depend on the temperature, the length of equilibration and sampling time steps were set differently for different temperatures.From this result, we calculated heat capacity (CV), staggered magnetization (M stagg ), and total scalar spin chirality (  ) using the following equations: where 〈〉 is an ensemble average of A estimated by the sampling,  is the total energy per Co 2+ ion, g is the Lande g-factor,  m  ( = 1, 2, 3) is the ordering wave vector of two-sublattice stripe order (see Supplementary Note), Δ is the index for a single triangular plaquette on a Co triangular lattice consisting of three sites (Δ 1 , Δ 2 ,  ).An energy integration range for each plot is ±0.2 meV.The E = 1 and 1.5 meV (2.0 ~ 3.0 meV) plots are based on the E i = 5 (7.9) meV data.In addition to bright circular spots centred at six M points (=linear modes), a weak, diffuse ring-like scattering which we interpret as the quadratic mode predicted by spin-wave theory appears for E > 1.5 meV.g, The calculated INS cross-section of the tetrahedral triple-Q ordering with J 1 S 2 = 3.92 meV, J c S 2 = 2.95 meV, J 2 /J 1 = 0.19, and K bq /J 1 = 0.02 (see Supplementary Materials).h, The calculated INS cross-section of the single-Q ordering with three domains, using J 1 S 2 = 3.92 meV, J c S 2 = 2.95 meV, J 2 /J 1 = 0.1, and K bq /J 1 = 0.The line-shaped signal in h has a much higher intensity than in f or g.The simulations in g-h include resolution convolution (see Supplementary Fig. 9), and their momentum and energy integration range are the same as f.i, Low energy magnon spectra measured with E i = 3.5 meV at 5 K, showing the energy gap of the linear magnon mode.

1. 35 ,
close to the single-ion limit of Co 2+ in the high spin S = 3/2 configuration26 .However, our neutron diffraction measurement indicates a significant suppression of the ordered magnetic moment S ~ 0.64 (assuming that a g-factor is 2).Similar reductions of the ordered moment have been reported in other metallic magnets comprising 3d transition metal elements, and they are generally attributed to a partial delocalization of the magnetic moments.In addition, interactions between Co local moments and itinerant electrons from Ta 5d band can lead to partial screening of the local moments.Another possible origin of the reduction of the ordered moment is quantum spin fluctuations arising from the frustrated nature of the effective spinspin interactions.

Fig. 1 |Fig. 2 |
Fig. 1 | The Tetrahedral triple-Q state and crystal structure of Co 1/3 TaS 2 .a-c, Three fundamental antiferromagnetic orderings for a triangular lattice system.The red-shaded regions denote each magnetic unit cell.d-f, Positions of the magnetic Bragg peaks (red circles) in momentum space generated by a-c A black (red) hexagon corresponds to a crystallographic (magnetic) Brillouin zone.The green and blue circles in d-e denote the magnetic Bragg peaks from the other two magnetic domains.g, A crystallographic unit cell of Co 1/3 TaS 2 .h, The temperature-dependent magnetization of single-crystal Co 1/3 TaS 2 with H//c.

Fig. 3 |
Fig. 3 | Stabilization mechanism and dynamical properties of the tetrahedral order in Co 1/3 TaS 2 .a, The in-plane crystal structure of Co 1/3 TaS 2 demonstrates isolated CoS 6 octahedrons (purple-coloured) and a long NN Co-Co distance.b, The exchange interaction between Co local moments and conduction electrons from TaS 2 layers leads to an effective RKKY interaction between the local moments.c, The Fermi surface of a 2D TLAF with 3/4 filling (shaded hexagons) and the Fermi surface of Co 1/3 TaS 2 measured by ARPES.d, The magnon spectra of Co 1/3 TaS 2 at 5 K along the (00L) direction.E i = 7.9 and 14 meV data are plotted.e, Antiferromagnetic NN interlayer coupling (J c ) of Co 1/3 TaS 2 , which is necessary for explaining the data in d and the refined spin configuration (Fig. 2g-i).f, Const-E cuts of the INS data measured at 5 K (< T N2).An energy integration range for each plot is ±0.2 meV.The E = 1 and 1.5 meV (2.0 ~ 3.0 meV) plots are based on the E i = 5 (7.9) meV data.In addition to bright circular spots centred at six M points (=linear modes), a weak, diffuse ring-like scattering which we interpret as the quadratic mode predicted by spin-wave theory appears for E > 1.5 meV.g, The calculated INS cross-section of the tetrahedral triple-Q ordering with J 1 S 2 = 3.92 meV, J c S 2 = 2.95 meV, J 2 /J 1 = 0.19, and K bq /J 1 = 0.02 (see Supplementary Materials).h, The calculated INS cross-section of the single-Q ordering with three domains, using J 1 S 2 = 3.92 meV, J c S 2 = 2.95 meV, J 2 /J 1 = 0.1, and K bq /J 1 = 0.The line-shaped signal in

Fig. 4 |
Fig. 4 | The effect of the out-of-plane and in-plane magnetic field on the tetrahedral triple-Q ordering in Co 1/3 TaS 2 .a-b, A time-reversal pair of the tetrahedral spin configuration, having   opposite to each other.The blue and orange arrows in a depict the generic canting of the tetrahedral ordering in Co 1/3 TaS 2 by an out-of-plane magnetic field.c, Comparison between the measured   AHE (orange) and M z (red) under the out-of-plane field at 3 K. d, Comparison between the measured   AHE (orange) at 3 K and   (red) calculated from the M z data in c. e, Intensities of the three magnetic Bragg peaks in Fig. 2c under an external magnetic field along the a* direction.Error bars in e represent standard deviation of the measured intensity.
First, we analyzed the neutron diffraction data based on group representation theory and Rietveld refinement, assuming a single-Q magnetic structure:    =   cos( •   ) with  = 1, 2, and 3 (see Supplementary Text and Supplementary Table 2-3).As a result, we found that the spin configurations for T < TN2 and TN2 < T < TN1 correspond to Figs. 2g and 2h, respectively, where each configuration belongs to the  2 +  4 (  22  +  41  , see Supplementary Table 3) and  2 (  22  ) representations -see Supplementary Notes sites, i.e., |  tri | depends on i (see Supplementary Fig. 7).A uniform |  tri | is obtained only when the Fourier components   of the three wave vectors    are orthogonal to each with   ⊥  ′ for  ≠  ′ .We note, however, that the magnitude of the ordered moments(|  tri |) does not have to be the same for quantum mechanical spins.Nevertheless, as explained in detail in the Supplementary Information, only non-coplanar triple-Q orderings corresponding to equilateral (|  | = |  ′ |, Fig. 2h) or nonequilateral ( |  | ≠ |  ′ |) tetrahedral configurations are consistent with our Rietveld refinement of neutron diffraction data.