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1
Tetrahedral triple-Q magnetic ordering and large spontaneous Hall
conductivity in the metallic triangular antiferromagnet Co1/3TaS2
Pyeongjae Park1,2, Woonghee Cho1,2, Chaebin Kim1,2, Yeochan An1,2, Yoon-Gu Kang3, Maxim
Avdeev4,5, Romain Sibille6, Kazuki Iida7, Ryoichi Kajimoto8, Ki Hoon Lee9, Woori Ju10, En-Jin
Cho10, Han-Jin Noh10, Myung Joon Han3, Shang-Shun Zhang11, Cristian D. Batista12,13*, and Je-
Geun Park1,2,14*
1Center for Quantum Materials, Seoul National University; Seoul 08826, Republic of Korea
2Department of Physics & Astronomy, Seoul National University; Seoul 08826, Republic of Korea
3Department of Physics, KAIST; Daejeon 34141, Republic of Korea
4Australian Nuclear Science and Technology Organisation (ANSTO); New Illawarra Road, Lucas
Heights, NSW 2234, Australia
5School of Chemistry, The University of Sydney; Sydney, NSW 2006, Australia
6Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut; 5232 Villigen, Switzerland
7Comprehensive Research Organization for Science and Society (CROSS); Tokai, Ibaraki 319-1106,
Japan
8Materials and Life Science Division, J-PARC Center, Japan Atomic Energy Agency; Tokai, Ibaraki 319-
1195, Japan
9Department of Physics, Incheon National University; Incheon, 22012, Republic of Korea
10Department of Physics, Chonnam National University; Gwangju 61186, Republic of Korea
11School of Physics and Astronomy and William I. Fine Theoretical Physics Institute, University of
Minnesota, Minneapolis, MN 55455, USA
12Department of Physics and Astronomy, The University of Tennessee; Knoxville, Tennessee 37996, USA
13Quantum Condensed Matter Division and Shull-Wollan Center, Oak Ridge National Laboratory; Oak
Ridge, Tennessee 37831, USA
14Institute of Applied Physics, Seoul National University; Seoul 08826, Republic of Korea
* Corresponding author: cbatist2@utk.edu & jgpark10@snu.ac.kr
Abstract
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 Co1/3TaS 2 as the first example of
tetrahedral triple-Q ordering with the associated topological Hall effect (non-zero
σxy(H=0)). Our measurements of the inelastic neutron scattering cross section are also
consistent with the calculated dynamical structure factor of the tetrahedral triple-Q
state.
2
Discovering magnetic orderings with novel properties and functionalities is one of the
primary goals 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 components1,2. 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 electrons3,4. When conduction electrons propagate through a skyrmion spin texture,
they exhibit spontaneous Hall effect. The origin of this effective spin-orbital coupling 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
3
Brillouin zone (see Fig. 1e). Remarkably, for S = ½, these two phases seem to be separated by
a quantum spin liquid state5-11 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 freedom12,13. 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
microscopy15,16. 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 crystal17. 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
4
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 a large effective spin-orbit coupling). 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 ((= 0) ) below TN2, which rules out the possibility of single-Q and double-Q
ordering19-22. 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 angle-
resolved 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.
Co1/3TaS2 is a Co-intercalated metal comprising triangular layers of magnetic Co2+ 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 d7 configuration of Co2+),23-25 including a neutron
diffraction study that reported an ordering wave vector qm = (1/3, 1/3, 0) characteristic of a
120° ordering25. More recently, an experimental study on single-crystal Co1/3TaS 2 observed a
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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, (=
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). This new observation forced us
to conduct a more extensive investigation and come up with a scenario consistent with all the
experimental facts.
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.
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 3–4). As a result, we found
6
that the spin configurations for T < TN2 and TN2 < T < TN1 correspond to Figs. 2g and 2h,
respectively, where each configuration belongs to the + (
+
, see
Supplementary Table 4) and (
) representations – see Supplementary Notes. The
refinement result yielded an ordered moment of 1.27(1)/Co2+ at 3 K.
However, a single-Q spin configuration with qm = (1/2, 0, 0) possesses time-reversal
symmetry (TRS) combined with lattice translation ( , see Fig. 2g or 2h), which strictly
forbids the finite (= 0 ) and (= 0 ) observed at T < TN2 (Fig. 2e–f). Similarly,
double-Q spin configurations in a triangular lattice are not compatible with finite (= 0)
and (= 0 ) due to its residual symmetry relevant to chirality cancellation19. Triple-Q
ordering is, therefore, the only possible scenario that can resolve this contradiction since it
allows for finite (= 0) and (= 0) due to broken symmetry19-21. 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-axis22 (see Supplementary Notes for more
explanation). However, an arbitrary linear combination of these three components yields a
7
triple-Q spin configuration with a site-dependent magnitude of ordered moments on the Co
sites, i.e., |
| depends on i (see Supplementary Fig. 7). A uniform |
| is obtained only
when the Fourier components of the three wave vectors
are orthogonal to each
other12, i.e.,
=cos(
)
, with for . We note, however,
that the magnitude of the ordered moments (|
|) 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 non-
equilateral ( | |||) tetrahedral configurations are consistent with our Rietveld
refinement of neutron diffraction data.
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 |
| (see Supplementary Fig. 7). Such a strong
modulation of |
| 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 (= 0) 19-22, 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) = (= 0) = 0 in the temperature range TN2 < T < TN1 due to its 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.
8
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 materials27-29, 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:
=
†
,
†
. (1)
From a crystal structure perspective, Co1/3TaS 2 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 Co2+ ion can interact with another Co2+ ion only via the conduction
electrons in the Ta 5d bands. Thus, in a first approximation, the Co2+ 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 ~
1.35, close to the single-ion limit of Co2+ 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
9
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 spin-
spin interactions.
Tetrahedral ordering can naturally emerge when the Fermi surface (FS) is three-
quarters (3/4) filled12,13,18,31 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 ((,) ) that diverges as log|
| at
M-points30. 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, (,) is still expected to have a global maximum at the three M points
for Fermi surfaces with the above-mentioned hexagonal shape18. 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
Co2+ 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 Co2+ magnetic ions and the conduction
electrons is weak compared to the Fermi energy. Under these conditions, it is possible to derive
10
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
consideration14,17,30,31. The simplest example of a four-spin interaction favouring the triple-Q
ordering is the bi-quadratic term (SS) with > 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 ordering32,33. 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 states34.
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
11
magnon dispersion shown in Fig. 3d indicates non-negligible antiferromagnetic NN interlayer
exchange: ~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 line-shaped 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
12
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 out-
of-plane and in-plane magnetic fields. Fig. 4c illustrates the field dependence of the measured
anomalous Hall conductivity (
) and Mz for H // c, showing evident hysteresis with a sign
change at ±. Indeed, as already discussed in Ref. 26, Mz cannot characterize the observed
. Instead, as explained in the introduction, it is that characterizes
for the
tetrahedral ordering. Therefore, based on our triple-Q scenario (see Figs. 4a–b), the sign change
at ± 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
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,
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
13
due to a meta-magnetic transition at ± , 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 //
= (1/2, 0, 0)) intensities of the three
magnetic Bragg peaks, each originating from three different
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 27-29.
In summary, we have reported a tetrahedral triple-Q ordering in Co1/3TaS2, as the only magnetic
ground state consistent with our bulk properties and neutron scattering data. Our study provides
a complete picture of how this exotic phase can be stabilized in the triangular metallic magnet
Co1/3TaS2 and opens avenues for exploring chiral magnetic orderings with the potential for
spontaneous integer quantum Hall effect30.
14
Methods
Sample preparation and structure characterization. Polycrystalline Co1/3TaS2 was synthesized by the solid-
state reaction. A well-ground mixture of Co (Alfa Aesar, > 99.99 %), Ta (Sigma Aldrich, > 99.99%), and S
(Sigma Aldrich, > 99.999%) was sealed in an evacuated quartz ampoule and then sintered at 900 for 10 days.
A molar ratio of the three raw materials was :3:6 with 1.05 < < 1.1, which is necessary to obtain the resultant
stoichiometric ratio of Co close to 1/3. Single-crystal Co1/3TaS2 was grown by the chemical vapour transport
method with an I2 transport agent (4.5 mg I2 /cm3). The pre-reacted polycrystalline precursor and I2 were placed
in an evacuated quartz tube and then were heated in a two-zone furnace with a temperature gradient from 940 to
860 for 10~14 days.
We measured the powder X-ray diffraction (XRD) pattern of Co1/3TaS2 using a high-resolution (Smartlab, Rigaku
Japan) diffractometer, which confirmed the desired crystal structure without any noticeable disorder
(Supplementary Fig. 1). In particular, the intercalation profile of Co atoms was carefully checked by the (10L)
superlattice peak pattern in the low- region. See Supplementary Table 1 for the refinement results. The powder
neutron diffraction experiment further corroborated this result (see the relevant subsection below). Finally, single
crystals were examined by Raman spectroscopy (XperRam Compact, Nanobase Korea), confirming their high
quality based on the sharp Raman peak at 137 cm-1 26.
The composition x of CoxTaS2 single crystal was confirmed primarily by energy-dispersive X-ray (EDX)
spectroscopy (Quantax 100, Bruker USA & EM-30, Coxem Korea). We measured 12 square areas of 300 m ×
300 m wide for every single piece, yielding homogeneous x centred at ~0.320 with a standard deviation of
~0.004. The homogeneity of x was further verified by the spatial profile (~1.5 m resolution) of the EDX
spectra, which is very uniform, as shown in Supplementary Fig. 2. The obtained x from EDX was again cross-
checked by the composition measured by inductively coupled plasma (ICP) spectroscopy (OPTIMA 8300,
Perkin-Elmer USA), which is almost the same within a measurement error bar26.
Bulk property measurements. We measured the magnetic properties of Co1/3TaS2 using MPMS-XL5 and
PPMS-14 with the VSM option (Quantum Design USA). To measure the spontaneous magnetic moment (Figs.
3d and 5a), a sample was field-cooled under 5 T and then measured without a magnetic field. Transport properties
of Co1/3TaS2 were measured by using four systems: our home-built set-up, PPMS-9 (Quantum Design, USA),
PPMS-14 (Quantum Design, USA), and CFMS-9T (Cryogenic Ltd, UK). To observe the temperature dependence
of the anomalous Hall effect (Figs. 3d and 5b), we field-cooled the sample under ± 9 T and then measured the
Hall voltage without a magnetic field. The measured Hall voltage was anti-symmetrized to remove any
longitudinal components. Hall conductivity () was derived using the following formula:
=
+. (2)
15
Anomalous Hall conductivity
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 Co1/3TaS2 using the
ECHIDNA high-resolution powder diffractometer ( = 2.4395 Å) at ANSTO, Australia. To acquire clear
magnetic signals of Co1/3TaS2, which are weak due to the small content of Co and its small ordered moment, we
used 20 g of powder Co1/3TaS2. 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 nuclear Bragg peaks (Supplementary Fig. 4). We
also performed Rietveld refinement and magnetic symmetry analysis using Fullprof software35. The results are
summarized in Supplementary Table 2-5 and Supplementary Fig. 3-4.
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 single-
crystal 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 = 1.383 Å with less than 1% of λ/2
contamination. Bragg intensities as a function of temperature and magnetic field were measured using a single
3He-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 spectrometer36. 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 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 in the measurements.
Single-crystal inelastic neutron scattering. We performed single-crystal inelastic neutron scattering of
Co1/3TaS2 using the 4SEASONS time-of-flight spectrometer at J-PARC, Japan37. 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
16
(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 technique38. We used the Utsusemi39 and Horace40 software for the
data analysis. Based on the crystalline symmetry of Co1/3TaS2, 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 41 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/(J1S2). 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 (Mstagg), and total scalar spin chirality () using the
following equations:
=
, (3)
=
(1)(
)
, (4)
=
(×)
, (5)
where is an ensemble average of A estimated by the sampling, is the total energy per Co2+ ion, g is the
Lande g-factor,
(= 1, 2, 3) is the ordering wave vector of two-sublattice stripe order (see Supplementary
Text), is the index for a single triangular plaquette on a Co triangular lattice consisting of three sites (, ,
17
), and N (Nt) is the total number of Co2+ ions (triangular plaquettes). When calculating Mstagg, one should first
identify which ordering wave vector the spin system chose among the three
(= 1, 2, 3), and then use proper
to calculate M. For all the simulations, we used S = 3/2. The results of the simulations are shown in
Supplementary Fig. 8.
Spin-wave calculations. We calculated the magnon dispersion and INS cross-section of Co1/3TaS2 using linear
spin-wave theory. For this calculation, we used the SpinW library.42 Since we did not align positive and negative
domains of the tetrahedral ordering in our INS experiments, we averaged the INS cross-section of both cases.
Density functional theory (DFT) calculations. We performed first-principles calculations using ‘Vienna ab
initio simulation package (VASP)’43-45 based on projector augmented wave (PAW) potential46 and within
Perdew-Burke-Ernzerhof (PBE) type of GGA functional47 (see Supplementary Fig. 11). DFT+U method48,49 was
adopted to take into account localized Co-3d orbitals properly, where U = 4.1 eV and JHund = 0.8 eV were used
as obtained by the constrained RPA for CoO and Co50. For the 2 × 2 magnetic unit cell, we used the Γ-centered
6 × 6 × 6 k-grid. We obtained and used the optimised crystal structure with the force criterion for the relaxation
fixed at 1 meV/Å. The plane-wave energy cutoff was set to 500 eV.
Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding
author upon request.
Code Availability
Custom codes used in this article are available from the corresponding author upon request.
Acknowledgements
We acknowledge S. H. Lee, S. S. Lee, Y. Noda, and M. Mostovoy for their helpful discussions and M.
Kenzelmann for his help with the experiments at SINQ. The Samsung Science & Technology Foundation
supported this work (Grant No. SSTF-BA2101-05). The neutron scattering experiment at the Japan Proton
Accelerator Research Complex (J-PARC) was performed under the user program (Proposal No. 2021B0049). One
of the authors (J.-G.P.) is partly funded by the Leading Researcher Program of the National Research Foundation
of Korea (Grant No. 2020R1A3B2079375). This work is based on experiments performed at the Swiss spallation
neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland. C.D.B. acknowledge support from the U.S.
18
Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0022311.
Author contributions
J.-G.P. initiated and supervised the project. P.P. synthesized the polycrystalline and single-crystal samples. P.P.
performed all the bulk characterizations. M.A. carried out the powder neutron diffraction experiment. P.P. and
R.S. performed the single-crystal neutron diffraction experiment at ZEBRA. P.P. analyzed the neutron diffraction
data together with M.A.. P.P., C.K., Y.A., K.I., and R.K. conducted the single-crystal inelastic neutron scattering
experiment at 4SEASONS. W.J., E.-J.C. and H.-J.N. conducted the ARPES experiment. Y.-G.K. and M.J.H.
performed the DFT calculations. P.P., W.H.C., S.-S.Z., and C.D.B. conducted spin model calculations. P.P.,
W.H.C., S.-S.Z., K.H.L., Y.-G.K., M.J.H, H.-J.N., C.D.B., and J.-G.P. contributed to the theoretical analysis and
discussion. P.P. C.D.B. and J.-G.P. wrote the manuscript with contributions from all authors.
Competing Interests
The authors declare no competing financial or non-financial interests.
Supplementary Information is available for this paper at [website url].
19
Fig. 1 | The Tetrahedral triple-Q state and crystal structure of Co1/3TaS2. 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
Co1/3TaS2. h, The temperature-dependent magnetization of single-crystal Co1/3TaS2 with H//c.
20
Fig. 2 | Magnetic ground state of Co1/3TaS2 revealed by neutron diffraction. a, The powder
neutron diffraction pattern of Co1/3TaS 2, measured at 60 K (Grey squares) and 3 K (Red circles).
A weak λ/2 signal is highlighted explicitly (see Methods). The solid black line is the diffraction
pattern of Co1/3TaS 2 simulated with the spin configuration shown in h, or equivalently i. The
full diffraction data can be found in Supplementary Fig. 4. b, The single-crystal neutron
diffraction pattern at 5 K, demonstrating magnetic Bragg peaks located at the M points. c,. The
three magnetic Bragg peaks' intensities originate from three different ordering wave vectors.
d, The temperature-dependent intensities of some magnetic Bragg peaks in single-crystal
Co1/3Ta S 2. e–f, The temperature dependence of (H = 0) and (H = 0) in Co1/3TaS2,
measured after field cooling under 5 T. g–h, The refined magnetic structures for g 26.5 K < T
< 38 K and h T < 26.5 K. i, The triple-Q counterpart (tetrahedral) of the single-Q (stripe)
ordering shown in h. Note that the spin configurations in h and I give the same powder
diffraction pattern.
21
Fig. 3 | Stabilization mechanism and dynamical properties of the tetrahedral order in
Co1/3TaS2. a, The in-plane crystal structure of Co1/3TaS 2 demonstrates isolated CoS6
octahedrons (purple-coloured) and a long NN Co-Co distance. b, The exchange interaction
between Co local moments and conduction electrons from TaS2 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 Co1/3Ta S 2 measured by ARPES. d, The
magnon spectra of Co1/3Ta S 2 at 5 K along the (00L) direction. Ei = 7.9 and 14 meV data are
plotted. e, Antiferromagnetic NN interlayer coupling (Jc) of Co1/3Ta S 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 (< TN2). 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 Ei = 5 (7.9) meV data. In addition
to bright circular spots centred at six M points (=linear modes), a weak line-shaped signal
connecting the six M points (= the quadratic mode) appears for E > 1.5 meV. g, The calculated
INS cross-section of the tetrahedral triple-Q ordering with J1S2 = 3.92 meV, JcS2 = 2.95 meV,
J2/J1 = 0.19, and Kbq/J1 = 0.02 (see Supplementary Materials). h, The calculated INS cross-
section of the single-Q ordering with three domains, using J1S2 = 3.92 meV, JcS2 = 2.95 meV,
J2/J1 = 0.1, and Kbq/J1 = 0. The line-shaped signal in h has a much higher intensity than in f or
g. The simulations in g–h include proper resolution convolution (see Supplementary Fig. 9),
and their momentum and energy integration are the same as f. i, Low energy magnon spectra
measured with Ei = 3.5 meV at 5 K, showing the energy gap of the linear magnon mode.
22
Fig. 4 | The effect of the out-of-plane and in-plane magnetic field on the tetrahedral
triple-Q ordering in Co1/3TaS2. 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 Co1/3TaS2 by an out-of-plane magnetic field. C,
Comparison between the measured
(orange) and Mz (red) under the out-of-plane field
at 3 K. d, Comparison between the measured
(orange) at 3 K and (red) calculated
from the Mz data in c. e, Intensities of the three magnetic Bragg peaks in Fig. 2c under an
external magnetic field along the a* direction.
23
References
1 Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. Nature
Nanotechnology 11, 231-241 (2016).
2 Baltz, V. et al. Antiferromagnetic spintronics. Reviews of Modern Physics 90, 015005 (2018).
3 Nagaosa, N. Anomalous Hall Effect—A New Perspective—. Journal of the Physical Society of
Japan 75, 042001-042001 (2006).
4 Zhang, S.-S., Ishizuka, H., Zhang, H., Halász, G. B. & Batista, C. D. Real-space Berry curvature
of itinerant electron systems with spin-orbit interaction. Physical Review B 101, 024420 (2020).
5 Zhu, Z. & White, S. R. Spin liquid phase of the S= 1/2 J1−J2 Heisenberg model on the triangular
lattice. Physical Review B 92, 041105 (2015).
6 Hu, W.-J., Gong, S.-S., Zhu, W. & Sheng, D. Competing spin-liquid states in the spin-1/2
Heisenberg model on the triangular lattice. Physical Review B 92, 140403 (2015).
7 Iqbal, Y., Hu, W.-J., Thomale, R., Poilblanc, D. & Becca, F. Spin liquid nature in the Heisenberg
J1− J2 triangular antiferromagnet. Physical Review B 93, 144411 (2016).
8 Saadatmand, S. & McCulloch, I. Symmetry fractionalization in the topological phase of the spin-
1/2 J1− J2 triangular Heisenberg model. Physical Review B 94, 121111 (2016).
9 Wietek, A. & Läuchli, A. M. Chiral spin liquid and quantum criticality in extended S= 1/2
Heisenberg models on the triangular lattice. Physical Review B 95, 035141 (2017).
10 Gong, S.-S., Zhu, W., Zhu, J.-X., Sheng, D. N. & Yang, K. Global phase diagram and quantum spin
liquids in a spin-1/2 triangular antiferromagnet. Physical Review B 96, 075116 (2017).
11 Hu, S., Zhu, W., Eggert, S. & He, Y.-C. Dirac spin liquid on the spin-1/2 triangular Heisenberg
antiferromagnet. Physical review letters 123, 207203 (2019).
12 Martin, I. & Batista, C. D. Itinerant Electron-Driven Chiral Magnetic Ordering and Spontaneous
Quantum Hall Effect in Triangular Lattice Models. Physical Review Letters 101, 156402 (2008).
13 Kato, Y., Martin, I. & Batista, C. Stability of the spontaneous quantum Hall state in the triangular
Kondo-lattice model. Physical review letters 105, 266405 (2010).
14 Kurz, P., Bihlmayer, G., Hirai, K. & Blügel, S. Three-dimensional spin structure on a two-
dimensional lattice: Mn/Cu (111). Physical review letters 86, 1106 (2001).
15 Spethmann, J. et al. Discovery of magnetic single-and triple-q states in Mn/Re (0001). Physical
review letters 124, 227203 (2020).
16 Haldar, S., Meyer, S., Kubetzka, A. & Heinze, S. Distorted $3Q$ state driven by topological-chiral
magnetic interactions. Physical Review B 104, L180404 (2021).
17 Wang, Z. & Batista, C. D. Skyrmion crystals in the triangular kondo lattice model. arXiv preprint
arXiv:2111.13976 (2021).
24
18 Akagi, Y. & Motome, Y. Spin chirality ordering and anomalous Hall effect in the ferromagnetic
Kondo lattice model on a triangular lattice. Journal of the Physical Society of Japan 79, 083711
(2010).
19 Heinonen, O., Heinonen, R. A. & Park, H. Magnetic ground states of a model for MNb3S6 (M=Co,
Fe, Ni). Physical Review Materials 6, 024405 (2022).
20 Tenasini, G. et al. Giant anomalous Hall effect in quasi-two-dimensional layered antiferromagnet
Co1/3NbS2. Physical Review Research 2, 023051 (2020).
21 Ghimire, N. J. et al. Large anomalous Hall effect in the chiral-lattice antiferromagnet CoNb3S6.
Nature Communications 9, 3280 (2018).
22 Yanagi, Y., Kusunose, H., Nomoto, T., Arita, R. & Suzuki, M.-T. Generation of modulated magnetic
structure based on cluster multipole: Application to alpha-Mn and CoM3S6. arXiv preprint
arXiv:2201.07361 (2022).
23 Parkin, S. S. P. & Friend, R. H. 3d transition-metal intercalates of the niobium and tantalum
dichalcogenides. I. Magnetic properties. Philosophical Magazine B 41, 65-93 (1980).
24 Parkin, S. S. P. & Friend, R. H. 3d transition-metal intercalates of the niobium and tantalum
dichalcogenides. II. Transport properties. Philosophical Magazine B 41, 95-112 (1980).
25 Parkin, S. S. P., Marseglia, E. A. & Brown, P. J. Magnetic structure of Co1/3NbS2 and Co1/3TaS2.
Journal of Physics C: Solid State Physics 16, 2765-2778 (1983).
26 Pa r k, P. et al. Field-tunable toroidal moment and anomalous Hall effect in noncollinear
antiferromagnetic Weyl semimetal Co1/3TaS2. npj Quantum Materials 7, 42 (2022).
27 Mühlbauer, S. et al. Skyrmion Lattice in a Chiral Magnet. Science 323, 915-919 (2009).
28 Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901-
904 (2010).
29 Kurumaji, T. et al. Skyrmion lattice with a giant topological Hall effect in a frustrated triangular-
lattice magnet. Science 365, 914-918 (2019).
30 Batista, C. D., Lin, S.-Z., Hayami, S. & Kamiya, Y. Frustration and chiral orderings in correlated
electron systems. Reports on Progress in Physics 79, 084504 (2016).
31 Akagi, Y., Udagawa, M. & Motome, Y. Hidden Multiple-Spin Interactions as an Origin of Spin
Scalar Chiral Order in Frustrated Kondo Lattice Models. Physical Review Letters 108, 096401
(2012).
32 Villain, J., Bidaux, R., Carton, J.-P. & Conte, R. Order as an effect of disorder. Journal de Physique
41, 1263-1272 (1980).
33 Henley, C. L. Ordering due to disorder in a frustrated vector antiferromagnet. Physical Review
Letters 62, 2056-2059 (1989).
34 Park, H., Heinonen, O. & Martin, I. First-principles study of magnetic states and the anomalous
Hall conductivity of MNb3S6 (M =Co, Fe, Mn, and Ni). Physical Review Materials 6, 024201
25
(2022).
35 Rodríguez-Carvajal, J. Recent advances in magnetic structure determination by neutron powder
diffraction. Physica B: Condensed Matter 192, 55-69 (1993).
36 Kim, H. D. et al. Performance of a Micro‐Spot High‐Resolution Photoemission Beamline at PAL.
AIP Conference Proceedings 879, 477-480 (2007).
37 Kajimoto, R. et al. The Fermi Chopper Spectrometer 4SEASONS at J-PARC. Journal of the
Physical Society of Japan 80, SB025 (2011).
38 Nakamura, M. et al. First Demonstration of Novel Method for Inelastic Neutron Scattering
Measurement Utilizing Multiple Incident Energies. Journal of the Physical Society of Japan 78,
093002 (2009).
39 Inamura, Y., Nakatani, T., Suzuki, J. & Otomo, T. Development Status of Software “Utsusemi” for
Chopper Spectrometers at MLF, J-PARC. Journal of the Physical Society of Japan 82, SA031
(2013).
40 Ewings, R. A. et al. Horace: Software for the analysis of data from single crystal spectroscopy
experiments at time-of-flight neutron instruments. Nuclear Instruments and Methods in Physics
Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 834, 132-
142 (2016).
41 Dahlbom, D., Miles, C., Zhang, H., Batista, C. D. & Barros, K. Langevin dynamics of generalized
spins as SU(N) coherent states. Physical Review B 106, 235154 (2022).
42 Toth, S. & Lake, B. Linear spin wave theory for single-Q incommensurate magnetic structures.
Journal of Physics: Condensed Matter 27, 166002 (2015).
43 Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. Physical
Review B 48, 13115-13118 (1993).
44 Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using
a plane-wave basis set. Physical Review B 54, 11169-11186 (1996).
45 Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and
semiconductors using a plane-wave basis set. Computational Materials Science 6, 15-50 (1996).
46 Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method.
Physical Review B 59, 1758-1775 (1999).
47 Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple.
Physical Review Letters 77, 3865-3868 (1996).
48 Anisimov, V. I., Zaanen, J. & Andersen, O. K. Band theory and Mott insulators: Hubbard U instead
of Stoner I. Physical Review B 44, 943-954 (1991).
49 Liechtenstein, A. I., Anisimov, V. I. & Zaanen, J. Density-functional theory and strong interactions:
Orbital ordering in Mott-Hubbard insulators. Physical Review B 52, R5467-R5470 (1995).
50 Sakuma, R. & Aryasetiawan, F. First-principles calculations of dynamical screened interactions for
26
the transition metal oxides MO (M=Mn, Fe, Co, Ni). Physical Review B 87, 165118 (2013).
