Tunable chiral and nematic states in the triple-Q antiferromagnet Co_1/3TaS_2

Domain: neuroscience
Published: 2026-01-31
Editorial outcome: minor_revision

Abstract
Complex spin configurations in magnetic materials, ranging from collinear single-Q to non-coplanar multi-Q states, exhibit rich symmetry and chiral properties. However, their detailed characterization is often hindered by the limited spatial resolution of neutron diffraction techniques. Here we employ magnetic circular dichroism and magnetic linear dichroism to investigate the triangular lattice antiferromagnet Co1/3TaS2, revealing three-state (Z3) nematicity and also spin chirality across its multi-Q magnetic phases. At intermediate temperatures, the presence of linear dichroism identifies nematicity arising from a single-Q stripe phase, while at high magnetic fields and low temperatures, a phase characterized solely by circular dichroism emerges, signifying a purely chiral non-coplanar triple-Q state. Notably, at low temperatures and small fields, we discover a unique phase where both chirality and nematicity coexist. A theoretical analysis based on a continuous multi-Q manifold captures the emergence of these distinct magnetic phases, as a result of the interplay between four-spin interactions and weak magnetic anisotropy. Additionally, both circular and linear dichroism microscopy spatially resolves the chiral and nematic domains. Our findings establish Co1/3TaS2 as a rare platform hosting diverse multi-Q states with distinct combinations of spin chirality and nematicity while demonstrating the effectiveness of polarized optical techniques in characterizing complex magnetic textures.

Introduction
Antiferromagnets, long overshadowed by their ferromagnetic counterparts, are rapidly emerging as a new frontier in condensed matter physics1–4. Their complex spin orders offer deep insights into magnetic symmetries, chiralities, and associated emergent phenomena5–8. Antiferromagnets (AFMs) can exhibit diverse spin configurations, ranging from collinear structures described by a single ordering wave vector Q, to complex non-collinear and non-coplanar multi-Q spin configurations arising from the superposition of multiple Fourier components with distinct wave vectors Qν (ν=1,2,3..,). Notably, multi-Q states can give rise to topologically non-trivial spin textures9–12 that manifest unique phenomena13–17. Understanding these AFM orders requires precise identification of their magnetic symmetries and potential topological properties, both of which are central themes in modern magnetism research.

The triangular lattice antiferromagnetism in Co1/3TaS2, an intercalated metallic van der Waals system with Co2+ spins on 2D triangular lattices (depicted in Fig. 1a), provides an excellent platform for exploring the rich physics associated with competing AFM orders. In this system, conduction-electron-mediated two-spin and four-spin interactions stabilize a non-coplanar triple-Q ground state below 26.5 K18,19. This state, characterized by ordering wave vectors at the three M points (edges) of the hexagonal Brillouin zone, represents the shortest-wavelength limit of a magnetic skyrmion crystal20. The resulting chiral spin texture can generate a pronounced topological Hall effect (without requiring relativistic spin-orbit coupling), leading to a large Hall conductivity σxy despite vanishing magnetization, and even in the absence of applied magnetic field18,19. At intermediate temperatures between 26.5 K and 38 K, Co1/3TaS2 transitions to a stripe-like single-Q antiferromagnetic phase19,21, which has been proposed as a promising candidate for realizing discrete three-state (Z3) electronic nematicity22–24, arising from rotational symmetry breaking in a hexagonal lattice. Beyond these zero-field chiral and nematic phases, out-of-plane magnetic fields H≳3.5 T induce two additional AFM phases, whose spin configurations and ordering mechanisms remain unknown18,19,25.

These recent discoveries raise fundamental questions about how magnetic chirality and nematicity evolve across these temperature- and field-induced phases. Prior studies of similar phenomena are scarce26,27, and neutron diffraction methods applied to date struggle to fully distinguish and characterize magnetic chirality and nematicity, due to a lack of spatial resolution and inability to resolve magnetic domains18,19. This challenge highlights the need for complementary approaches to investigate the rich (H, T) phase diagram of Co1/3TaS2, ideally with experimental probes that can illuminate the interplay and potential coexistence of single-Q and triple-Q orderings within the same system.

Results
Here we demonstrate that optical methods for magnetic circular dichroism (MCD) and magnetic linear dichroism (MLD) directly provide spatially-resolved measurements of chirality and nematic symmetries, respectively, of the different antiferromagnetic orders in Co1/3TaS2. These techniques reveal stripe-like (Z3 nematic) antiferromagnetism arising from single-Q order at intermediate T, purely chiral order (“three-fold symmetric triple-Q”) at low T and large H, and – crucially – the coexistence of both chiral and nematic (“distorted triple-Q”) order at low T and H. Our findings establish Co1/3TaS2 as a rare material hosting diverse and tunable multi-Q states, and highlight the power of magneto-optical techniques for characterizing complex magnetic textures.

Antiferromagnetic phases in Co1/3TaS2
Co1/3TaS2 consists of 2D triangular lattices of Co spins, intercalated within the van der Waals gaps of 2H-TaS2 (see Fig. 1a). Nearest-neighbor intra- and inter-layer couplings and AB stacking create an effective tetrahedral network of Co, which orders antiferromagnetically. Figure 1c, d briefly review its diverse AFM phases, revealed by recent transport, neutron scattering, and magnetization studies18,19,25. When cooled in H=0, AFM order first appears below TN1 = 38 K, where neutron studies identify collinear single-Q (stripe) order21 with Co spins aligned out-of-plane18,19. Upon further cooling, a new AFM ground state emerges below TN2 = 26.5 K that is characterized by a large spontaneous Hall conductivity σxy, despite vanishingly small zero-field magnetization (Mz = 0.01 μB/Co2+). This low-temperature phase is identified as a chiral non-coplanar “tetrahedral” triple-Q order. Electrons moving within this chiral spin texture accumulate a geometric (Berry) phase that generates an emergent magnetic field, resulting in substantial σxy despite tiny Mz – generally referred to as a topological Hall effect13,28–30. These triple-Q and single-Q phases are designated Phases I and III, respectively18.

Crucially, applying out-of-plane H larger than Hm ≈ 3.5 T) induces additional metamagnetic transitions, observed as small jumps in σxy and Mz (see Fig. 1d)18,25. These transitions lead to two additional phases — Phase II and Phase IV — presumably with distinct spin configurations and ordering mechanisms that remain unresolved (note that the data shown in Fig. 1c, d do not explicitly reveal the transition to Phase IV; however, this transition is shown in refs. 18, 19, 25, and is also evident in Fig. 2 below).

To directly probe the magnetic order of these phases, we employ MCD and MLD – optical techniques that can selectively measure chiral and nematic properties, respectively. The experiment is depicted in Fig. 1b (see caption for details, and Methods). Traditionally, MCD has been used to detect ferromagnetic order, as it probes the nonzero off-diagonal optical conductivity σxy(ω), which also underpins the dc Hall effect at ω ≈ 0. However, recent discoveries of antiferromagnets with non-trivial spin orders whose (reduced) symmetry allows for σxy(ω) – such as non-collinear Mn3Sn31–34 or here for non-coplanar Co1/3TaS2 – suggests that MCD can directly probe such AFM order35–39.

Complementing MCD, MLD has recently been demonstrated as an effective probe of stripe-like single-Q AFM order in 2D hexagonal magnets22–24. Such order breaks the rotational symmetry of the underlying crystal, inducing in-plane (C2z) anisotropy of the optical conductivity and giving rise to three-state Z3 nematic order (note: by C2z we mean the subgroup consisting of the identity and the transformation (x,y,z) → (-x, -y, z+1/2), which is the product of a π rotation about the c-axis and translation along the same axis). Together, MLD and MCD provide a powerful framework for characterizing multi-Q magnetic states in Co1/3TaS2.

Detecting chiral and nematic AFM order with MCD and MLD
To establish MCD as a probe of the chiral triple-Q order, we measure its temperature and field dependence. Figure 1e shows that strong MCD signals emerge below TN2, closely following the behavior of σxy(T) from transport measurements (Fig. 1c). Furthermore, Fig. 1f shows that MCD exhibits a marked hysteresis with H, in agreement with prior transport studies shown in Fig. 1d18,19,25. These findings confirm that MCD is sensitive to chiral AFM order in Co1/3TaS2. The scalar spin chirality of this triple-Q state, defined as χijk = Si · (Sj × Sk) where i, j, k label spins around any triangular plaquette in the lattice, can be switched between time-reversed configurations by H (≈ ±1 T at 20 K), giving opposite σxy(ω) and MCD. The sharp switching behavior indicates high sample quality and uniform reversal of the chiral order. Additionally, MCD signals reveal the meta-magnetic transition at Hm ≈ ±3.5 T, in close correspondence with transport and magnetization measurements (Fig. 1d; see also Supplementary Figs. S1 and S2). The correspondence of MCD and Hall conductivity suggests their common origin: the real-space Berry curvature generated by chiral triple-Q magnetism18,19.

In marked contrast, Fig. 1g shows that MLD emerges below TN1 = 38 K, coinciding with the onset of stripe-like single-Q (nematic) AFM order. As shown in recent studies of 2D hexagonal antiferromagnets FePS3 and Fe1/3NbS222–24, collinear single-Q stripe order can generate substantial MLD due to rotational symmetry breaking and consequent asymmetry of the in-plane optical conductivity. The MLD in Co1/3TaS2 saturates at a large value below TN2, indicating the persistence of nematicity in the low-temperature phase. The MLD does not vary significantly when cooled in small ±H, or in H=0. Importantly, the simultaneous presence of both MLD and MCD at T < TN2 (Fig. 1e, g) provides direct evidence that Phase I exhibits both chiral and nematic order.

Field-dependent MLD further clarifies the nature of the puzzling metamagnetic phase transition at Hm ≈ ±3.5 T. As shown in Fig. 1h, the MLD is large at low H but vanishes when |H| > Hm, indicating the disappearance of nematicity. Meanwhile, the MCD remains large (see Fig. 1f), demonstrating that the Hm separates a nematic-chiral phase (Phase I) from a purely chiral phase (Phase II). Notably, small jumps in the MLD signal coincide with chirality reversals, likely arising in part from cross-talk between linear and circular dichroism signals. This cross-talk also generates the very small MCD signal between TN1 and TN2 in Fig. 1e (see Supplementary Fig. S2).

Mapping the magnetic phase diagram
Figure 2a, b show field-dependent MCD and MLD at different temperatures. As previously observed in σxy18,25, both the chiral switching field Hc and the amplitude of the hysteresis loops increase rapidly below TN2. Importantly, the disappearance of MLD at large H persists for all T < TN1, reinforcing the phase distinction [see also MLD(T) scans at fixed H in Supplementary Fig. S3]. Using these magneto-optical measurements, we construct the (H, T) phase diagram in Fig. 2c, defining Phases I-IV based on the presence or absence of spontaneous topological MCD (chiral AFM order), and MLD (nematic, stripe-like AFM order).

The MCD results confirm that Phases I and II exhibit chiral triple-Q order, while the high temperature Phases III and IV do not. More importantly, MLD offers new insights into the symmetries of Phases I, II, and IV. The absence of rotational C3z symmetry in Phase I, revealed by MLD, contradicts the recently-proposed threefold-symmetric triple-Q ground state18. Instead, the presence of both MCD and MLD in Phase I suggests a triple-Q state with broken C3z rotational symmetry – that is, with both chiral and nematic AFM order. As described in the next section, such novel states can be realized through intermediate spin configurations involving coexisting single-Q and triple-Q components.

Continuous multi-Q manifold model
These observations motivate a continuous multi-Q manifold model that smoothly interpolates between single-Q and three-fold symmetric triple-Q states, and which naturally captures the evolution of spin chirality and nematicity in Co1/3TaS2. For M-ordering wave vectors, this continuous interpolation can be modeled by the following manifold (see Fig. 3a):

S(r) = Σν=1^3 Sν cos(Qν · r), (1)

where r is a lattice vector on the 2D triangular lattice, and S(r) denotes a classical spin vector. The derivation of Eq. (1) and its validity for Co1/3TaS2 is described in Supplementary Section 2A. By adjusting the relative magnitudes of the three Fourier components (|Sν| ≡ Δν) while maintaining their orthogonality and a constant total magnitude Δ1^2 + Δ2^2 + Δ3^2, this ansatz spans a spherical surface in (Δ1, Δ2, Δ3) phase space (see Fig. 3a), which interpolates between all possible single-, double-, and triple-Q spin configurations with fixed |S(r)|. We also note that ref. 26 considered a 1D sub-manifold of the 2D manifold of degenerate ground states. However, Phase IV is not captured by Eq. (1) and will be discussed separately.

The case Δ1 = Δ2 = Δ3 (red circle in Fig. 3a) yields a chiral and three-fold symmetric ground state, where the four spin sublattices align along the principal axes of a regular tetrahedron with 109.5° mutual angles. Phase II corresponds to this case, but with the addition of a small net out-of-plane magnetization arising from spin canting induced by the applied magnetic field. This canted configuration is naturally described by including a constant (ferromagnetic) Fourier component S0 to the three-dimensional manifold in Eq. (1); for additional details see Supplementary Section 2A and Eq. (S7). Importantly, even with canting, the three-fold symmetry remains intact (see Supplementary Fig. S6d) and therefore we refer to Phase II as a “three-fold symmetric triple-Q” state throughout this work, to emphasize its rotational symmetry (though not implying a perfect 109.5° tetrahedral configuration.). In contrast, a single non-zero Δν (blue circles in Fig. 3a) corresponds to the single-Q stripe order of Phase III (Fig. 3e). Intermediate states arise when 0 < Δi = Δj < Δk for indices i, j, k ∈ 1, 2, 3 (green lines throughout Fig. 3). The broken C3z symmetry is evident from the unequal Δν magnitudes (Fig. 3c), resulting in the four spin sublattices spanning a distorted tetrahedron, which we term “distorted triple-Q”. From this perspective, thermal fluctuations and H cause the magnetic ground state to evolve smoothly within this multi-Q manifold, effectively controlling both the spin chirality and Z3 nematicity in Co1/3TaS2.

While the emergence of a continuous multi-Q manifold is rarely observed in real systems, our theoretical analysis adds strong evidence for its feasibility based on a realistic spin Hamiltonian for Co1/3TaS2. The isotropic low-energy Hamiltonian introduced in previous studies19,21, which incorporates bilinear Heisenberg and four-spin interactions, fails to fully capture the field- and temperature-driven multi-Q manifold. While a more generalized model could include additional four-spin terms with varying forms40, we find that a simple real-space biquadratic (i.e., four-spin) term, Hbq, captures the most general classical ground state of the multi-Q M-ordering and its long-wavelength fluctuations:

Hbq = K Σ<i,j> (Si · Sj)^2, (2)

where <i,j> is the vector connecting nearest-neighbor sites (derived in Supplementary Sections 2B and C). Notably, Hbq with K > 0 successfully captures the single-Q to triple-Q transition at TN219,21. However, it incorrectly predicts a C3z-symmetric triple-Q ground state at T=H=0 (Phase I), and fails to explain any field-induced metamagnetic transition.

Magnetic anisotropy in Co1/3TaS2, though smaller than K in magnitude, is suggested by the finite magnon energy gap in Phase I and by the out-of-plane spin configuration in Phase III19,21 (see Supplementary Section 2D). To capture the field-induced transitions and the C3z symmetry breaking in Phase I, we include an anisotropy term, Hani, and a field-dependent term, Hfield:

Hani = A Σi (Siz)^2, (3)

Hfield = - Σi H · Si,

where <i,j> runs over nearest-neighbor bonds, x ∥ a-axis, and ϕα ∈ {0, 2π/3, 4π/3} is the angle between a bond vector (i→j) and the a-axis. The resultant phase diagram at T=H=0 spanned by J±± and A is shown in Supplementary Figs. S7b and S8, where we effectively visualize it through a quantity rΔ ≡ Δ2 + Δ3 / Δ1, where Δ3 ≤ Δ2 ≤ Δ1. Introducing a non-zero A immediately changes the ground state from a three-fold symmetric triple-Q (rΔ=2, Fig. 3b) to a distorted triple-Q (1 < rΔ < 2, Fig. 3c), consistent with the observed MLD in Phase I. Thus, an intermediate triple-Q ground state on the continuous manifold is a natural explanation for Phase I under the presence of magnetic anisotropy.

Further exploration of temperature- and field-dependent ground states was performed using classical Monte Carlo simulations (see Methods, and Supplementary Section 2D). The results in Fig. 3f–h, presented as Δν and rΔ, reveal: (i) a distorted triple-Q ground state (1 < rΔ < 2) in Phase I, (ii) a field-induced three-fold symmetric triple-Q state (rΔ=2) in T < TN2 (Phase II), and (iii) a single-Q state (rΔ=0) in TN2 < T < TN1 (Phase III). The consistency with Fig. 2 strongly supports the continuous multi-Q manifold model. The field-induced Phase II is stabilized by the larger residual magnetic moment of the three-fold symmetric triple-Q phase (see the magnetization data in Fig. 1d), favoring a state in which one of the four spin sublattices aligns along H. A residual net moment exists in both Phases II and I (even at zero field) due to the anisotropy terms in our model, and its direction (±bc) is linked to the sign of the underlying spin chirality.

We now discuss Phase IV and the limitations of our model. The absence of both MLD and MCD in Phase IV implies preserved C3z symmetry, suggesting a non-chiral triple-Q state. A likely configuration is the up-up-up-down (UUUD) structure, a collinear triple-Q ordering (Fig. 3d) observed in triangular lattice antiferromagnets under a magnetic field41. In classical simulations, the UUUD phase requires a magnetization of half the saturation value (M = 1.5 μB/Co2+), thereby appearing at higher fields in our model compared to Hm ~ 3.5 T (Phase IV exhibits M = 0.2 μB/Co2+ at 7 T). We attribute this discrepancy to longitudinal spin fluctuations not captured in our classical model. Indeed, the ordered magnetic moment of 1.3 μB/Co2+ observed at 3 K – much smaller than the classical 3 μB/Co2+ – indicates substantial spin fluctuations19. A possible consequence of strong fluctuations in triangular lattice antiferromagnets is a site-dependent renormalization of ordered moments42, e.g., the three “up” moments could become shorter than the single “down” moment (Fig. 3e), reducing net magnetization. Precise determination of Phase IV requires future neutron diffraction measurements under magnetic field.

Imaging nematic and chiral antiferromagnetic domains
A notable benefit of optical methods is the ability to directly image real-space profiles of different magnetic states and their domains. In Figs. 4 and 5, we employ MLD and MCD microscopy (see Methods) to spatially resolve, respectively, the discrete Z3 nematic domains arising from single-Q AFM order, and the binary domains associated with positive and negative spin chirality. Figure 4a shows three MLD images of the same area, acquired in Phase III (single-Q), using probe light linearly polarized at ϕ = 0°, 120°, and 240° with respect to the crystal a-axis. The images reveal distinct regions with strong positive MLD (red) and weaker negative MLD (light blue). Importantly, the red regions are different in each image. Largest MLD occurs when ϕ aligns parallel to the stripes of single-Q magnetic order. The images therefore indicate three different single-Q domains where the stripes align along 0°, 120°, and 240°, as expected from a triangular lattice. The C2z symmetry of the domains is demonstrated in Fig. 4b, where MLD at the three indicated locations is measured versus ϕ. Each polar plot exhibits C2z in-plane anisotropy, oriented approximately along 0°, 120°, and 240° (see also Supplementary Fig. S4). These images confirm Z3 nematic order in Co1/3TaS2, induced by single-Q AFM order, with different nematic domains having ordering wave vectors Qν related by ±120° rotation. Analyzing the MLD at each position yields a spatial map of the nematic director (Fig. 4c). The directors indicate the angle ϕ giving maximum MLD signal, and have lengths proportional to the MLD magnitude. For clarity, blue, red, and green directors represent orientations closest to 0°, 120°, and 240°. This classification scheme remains consistent at different locations on the sample (see Supplementary Fig. S5). Notably, the nematic domains can be quite large, extending nearly 1 mm. Figure 4d shows the director map overlaid with the MLD map of Fig. 4a.

Interestingly, repeated thermal cycling into the paramagnetic state, even staying at 300 K for days or weeks, did not affect the nematic domain patterns. It suggests that they are pinned by extrinsic factors that locally break the underlying hexagonal crystal symmetry, such as local strain. Similarly, Fe1/3NbS2 and FePSe3 have also shown biasing of an underlying three-state AFM nematicity by uniaxial strain22,24. Furthermore, the MLD signals remain essentially unchanged below TN2 where spin chirality also emerges (cf. Fig. 1g), suggesting a smooth transition between the nematic order in Phase III and Phase I. This implies that at TN2, single-Q domains with Δi ≠ 0 and Δj,k = 0 (i, j, k ∈ {1, 2, 3}) transform continuously into distorted triple-Q domains with Δi > Δj = Δk > 0.

Finally, we use MCD microscopy to study spontaneous formation of chiral AFM domains (Fig. 5). MCD images in the paramagnetic state (40 K, Fig. 5a) show no signal, as expected. However, Fig. 5b and d show that Co1/3TaS2 can be completely poled to a positive or negative chiral state by cooling into Phase I in applied H as small as ±70 mT. Most importantly, Fig. 5c shows that small and irregularly-shaped chiral domains spontaneously form when cooled in H=0. Their characteristic size is much smaller than the nematic domains. Also in marked contrast to the nematic domains, the chiral domain pattern changes randomly after thermally cycling above TN2, suggesting that the underlying nematicity of the distorted triple-Q order does not bias the handedness of spontaneously-forming chiral domains. Figure 5e shows chiral domains at a different location, again showing a random pattern. These images confirm that, to within our microscope’s 1 micron spatial resolution, Phase I is not a phase-separated mixture of coexisting single-Q and triple-Q ground states. Finally, Fig. 5f shows that, starting from a random configuration, the chiral domains are readily poled by fields much less than the coercive field required to switch a fully polarized sample, suggesting weak pinning.

Methods
Samples synthesis and characterization
Single-crystal Co1/3TaS2 was synthesized using a standard chemical vapor transport method applied to polycrystalline Co1/3TaS2, as detailed in refs. 19, 25, 43. Since the magnetic properties of Co1/3TaS2 are sensitive to Co compositions43, careful assessments of the Co composition were conducted. The TN2 value, which reaches a maximum of 26.5 K with a minimized extent of Co vacancies43, serves as a reliable indicator of sample quality. Only samples exhibiting TN2 = 26.5 K were used for this study, with an estimated vacancy concentration of less than 3%.

MCD and MLD measurements
The experimental setup, depicted in Fig. 1b, used wavelength-tunable probe light (typically 650 nm or 700 nm) derived from a white light source (xenon lamp) spectrally filtered through a 300 mm spectrometer. The probe light was mechanically chopped at 137 Hz, linearly polarized, and then polarization-modulated by a photoelastic modulator (PEM). For MCD, the polarization was modulated between right- and left-circular (±quarter-wave modulation) at 50 kHz. For MLD, the polarization was modulated between linear and cross-linear (±half-wave modulation) at 100 kHz. The Co1/3TaS2 samples were mounted in helium vapor in the variable-temperature (2–300 K) insert of a 7 T split-coil magnet with direct optical access. The probe light was weakly focused on the sample (≈1 mm spot size) at near-normal incidence, and the reflected light intensity was measured by an avalanche photodiode, amplified, and demodulated using two lock-in amplifiers. We confirmed that perfectly normal incidence (H ∥ k ∥ c) gave the same results. The MCD experiment measured the normalized difference between right- and left-circularly polarized reflected intensities, (IR - IL)/(IR + IL). Similarly, MLD measured (Iϕ - Iϕ+90)/(Iϕ + Iϕ+90), where ϕ is the angle of the probe’s linear polarization. We note that MCD is a closer relative of the magneto-optical Kerr effect (MOKE), and in particular the Kerr ellipticity, and is sensitive to magnetic order(s) that generate non-zero off-diagonal conductivity σxy(ω). Complementing MCD, MLD is sensitive to in-plane anisotropy of the optical conductivity [e.g., σxx(ω) - σyy(ω)], which can arise from single-Q (stripe-like) AFM order.

For imaging experiments requiring high spatial resolution, the light source was a 650 nm superluminescent diode, and the samples were instead mounted on the vacuum cold finger of a small optical cryostat. A high numerical aperture (NA = 0.55) microscope objective was used to focus the probe light at normal incidence down to ≈1 μm spot, that could be raster-scanned across the sample surface. Small out-of-plane magnetic fields up to 200 mT were applied using external permanent NdFeB magnets.

Classical Monte Carlo simulations
In addition to the zero-temperature magnetic phase diagram calculation (described in Supplementary Section 2E), the phase diagram as a function of temperature and out-of-plane magnetic field (H) was obtained using classical Monte Carlo simulations. The simulations employed a combination of the Langevin dynamics algorithm and simulated annealing. All calculations were conducted with the Sunny software package44,45. To achieve statistically well-averaged results, 20 replicas of 36×36×6 sized Co1/3TaS2 supercell (15,552 Co sites) were prepared and simulated in parallel. The spin systems were initialized by field cooling from 50 K to ensure a uniform alignment of the scalar spin chirality sign, which was necessary to achieve consistent results due to the intertwined nature of the transition between Phases I and II and the chirality sign. At a given temperature and field, we sampled the time evolution of each spin system using the Langevin dynamics after 5000 Langevin steps for initial thermalization. The Langevin time step and damping constant were set to 0.05 (meV-1) and 0.1, respectively, where S=3/2, J=1.212 meV21, and H expressed in units of Tesla. From the collected samples, we calculated the magnitudes of the three Fourier components (|Sν| ≡ Δν, see Eq. (1) or Supplementary Section 2A) via Fourier transformation. The results are presented in Fig. 3f, g.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This study investigates the complex multi-Q magnetic textures of the triangular antiferromagnet Co₁/₂TaS₂ using optical techniques, namely magnetic circular dichroism (MCD) and magnetic linear dichroism (MLD). By combining MCD and MLD, the authors are able to distinguish between chiral order and nematic order, thereby corroborating and complementing the results of neutron experiments. Moreover, owing to the excellent real-space resolution of MCD and MLD, the authors have resolved the real-space magnetic domain information and further determined the magnetic structure of this system. In addition, this technique has generality and is expected to be extended to other complex systems as one of the technical means for identifying intricate magnetic textures.\nOverall, this is a thorough work and is worthy of publication. I have only a few minor comments to address:\n1. In the Introduction, the authors state that antiferromagnets can exhibit a variety of complex spin structures, whereas ferromagnets cannot. This seems to involve some ambiguity in definitions. For example, MnSi, a famous skyrmion host, is believed to have a weak itinerant ferromagnetic ground state.\n2. In Fig. 1(e), the MCD signal between TN2 and TN1 remains visible (albeit weaker compared to the signal below TN1). Does this imply the presence of a potential chiral magnetic structure in this temperature range or other possible effects? The authors should clarify this point.", "reviewer_2": "Erik Kirstein and coworkers have investigated the magnetic phases of the van der Waals material CoTa3S6, using optical microscopy (MLD/MCD), transport measurements and magnetic simulations including Monte Carlo methods. The manuscript is well-written and the figures are clear. The new finding seems to be a mixed low T zero field phase, where 1Q and 3Q magnetic order coexists. The authors show laterally resolved optical data with a resolution of about 1 micron, where the MLD and MCD signals are used as a measure of 1Q and 3Q magnetic character, respectively.\nA main weakness of the manuscript is that the authors do not properly cite and relate their work to the existing literature, making it unnecessarily difficult for the reader to discriminate possibly new results from already established ideas and facts. For instance, the concept of a continuum of states between 1Q, 2Q and 3Q, has already been put forward in Ref. 38, however, a comparison between the proposed two-dimensional \"multi-q manifold\" and the one-dimensional mixing scheme of Ref. 38 is not made. Moreover, part of the phase diagram (Fig. 2c) is based on results from a previous publication of the authors (Ref. 19) and results from the Tokyo group (Ref. 18).\nThe proposed distorted 3Q state in phase I -- predicted for hcp Mn/Re(0001) in Ref. 38 (without experimental proof) and there explained based on higher than 4th-order exchange interactions (beyond 4-spin or biquadratic exchange) is here argued to be the result of crystalline anisotropy without taking other possibilities into account. Moreover, I don't find the argument for such a state convincing: If phase I constitutes a 3Q state with some 1Q component, why is then the MLD signal (which is introduced by the authors to be a measure of 1Qness) at a maximum in phase I, higher than in the supposedly pure 1Q phase III?\nOverall, the manuscript does not clearly offer substantially new results which are backed by experiments to warrant publication in Nature Communications.\nIn addition, I would like to raise more specific criticism:\n1. In contrast to \"chirality\", the authors use the term \"nematicity\" without properly defining it. For that reason, stating that the 1Q AFM order gives rise to nemetic domains does not explain nor add anything.\n2. What is the exact relationship between 1Q AFM order (row-wise AFM) and the MLD and MCD signal? What is the exact relationship between 3Q order (109° between all neighboring spins) and the MCD and MLD signal?\n3. Which MLD and MCD signals are expected for the coplanar 120° Neel state proposed for CoTa3S6 in Ref. 25?\n4. page 2, \"coexistence of single-Q and triple-Q orderings within the same system – an uncommon feature in real materials.\":\nFirstly, a distorted 3Q state has been proposed already in Ref. 38. Secondly, 1Q, 3Q and transient 2Q states co-exist in the Mn/Re(0001) system due to different stacking order and inside domain walls, respectively.\n5. page 2, \"effective tetrahedral network of Co\":\nThis sentence implies that the magnetic coupling between Co atoms is identical within and across layers, which is certainly not the case.\n6. \"Continuous multi-Q manifold model\": The mixed magnetic states in Ref. 38 are constructed in a one-dimensional fashion on a path on which the standard Heisenberg exchange is constant. Is this also the case for the authors' \"manifold\"?\n7. page 4, \"successfully captures the single-Q to triple-Q transition at TN2\":\nIs it essential which 4th-order exchange term (biquadratic, 3spin, 4spin) is used for the transition to occur? Is the driving mechanism for this transition entropy?\n8. page 5, \" field-induced equilateral triple-Q state\": It is not straightforward to make a distorted 3Q state more symmetric by applying an external B field. There is an energy term which might do the trick, but it is not mentioned in the manuscript.\n9. page 7, \" can be completely poled to a positive or negative chiral state\": Is this behavior reproduced by simulations?\nWhich energy term is responsible for the coupling of chirality and external magnetic field?\n10. J. Spethmann et al. try to determine the orientation of their 3Q state in hcp Mn/Re(0001) and the accompanied DFT theory predicts a spin orientation in the 1Q state along the parallel spin-rows, driven by the magnetic dipolar interaction and anisotropic symmetric exchange. Is it possible to tackle these questions also for CoTa3S6.\n11. Compared to SP-STM, where these states are imaged directly with atomic resolution, MLD and MCD are indirect measures, averaging over large surface areas. The effect of domain sizes and chemical inhomogeneity with a lateral size below the resolution limit is not discussed in the manuscript.\n12. What is the information depth of the used techniques? Can a scenario be ruled out where the magnetic state at the surface is different from the bulk?\n13. What can be concluded for the magnetic coupling between the layers experimentally? Is the proposed 109° coupling in phase I an assumption or a conclusion?\n14. Finally, can the proposed uuud state (phase IV) be reproduced in the simulation?", "reviewer_3": "The work by Kirstein et al. presents a comprehensive study of the chiral and nematic states in the antiferromagnet Co₁/₃TaS₂. They mapped out the antiferromagnetic phase diagram under different magnetic fields and temperature ranges using MLD and MCD measurements. A continuous multi-Q manifold model is proposed to understand the phase diagram. Finally, they presented spatially resolved chiral and nematic domains using MCD and MLD microscopy.\nI think this paper is well written, and the results are scientifically sound. This work not only demonstrates that Co₁/₃TaS₂ hosts rich nematic and chiral multi-Q states, but also represents a high-quality study of antiferromagnetic textures using optical techniques. I therefore recommend its publication in Nature Communications. Before publication, I have a few comments for the authors to improve the presentation of their manuscript.\nComments:\n1. Phase I exhibits both MCD and MLD signals. The authors propose that this phase corresponds to a non-equilateral triple-Q state. Could the authors clarify how they rule out the possibility that Phase I arises from the coexistence of spatially separated single-Q and triple-Q domains? In such a case, both MCD and MLD signals could also be expected.\n2. Do the authors have an intuitive physical picture or a simple explanation for why the magnetic textures in Phase I exhibit both nematicity and chirality? Although the authors attribute this to magnetic anisotropy, the explanation remains unclear. Magnetic anisotropy is relatively common, but the coexistence of nematic and chiral textures is much less so. What is special about Co₁/₃TaS₂ in this context?\n3. The MCD curve versus magnetic field in Fig. 2a shows a tilt, suggesting that the field induces some MCD even at high temperatures such as 30 K or 40 K. In contrast, at high temperatures (e.g., 30 K), the MCD is labeled to vanish at large fields in Fig. 2c. The authors should clarify this apparent inconsistency.\n4. I have some questions regarding Phase IV: (i) In Section A, the authors state that magnetic fields induce two new phases, Phase II and Phase IV. However, this is not evident from Fig. 1. (ii) There is no visible sharp transition in Fig. 2b at high temperatures. How do the authors determine the boundary between Phase III and Phase IV in Fig.2c?\n5. Figures 4(c) and 4(d) appear repetitive. Moreover, there is no description of Fig. 4(d) in the main text. The authors should address this omission."}, "author_response": "Reply to Reviewers [manuscript NCOMMS-25-30582-T, “Tunable chiral and nematic states in the\ntriple-Q antiferromagnet Co1/3TaS2”]\nWe are grateful to all three Reviewers for their careful reading of the manuscript, and we sincerely\nappreciate the positive assessments of our work from Reviewers #1 and #3 and for their recommendations\nto publish. The revised and resubmitted manuscript incorporates the suggested changes for improvement;\nsee our point-by-point response to all Reviewer feedback below, and Changes to the Manuscript listed at\nthe end. For ease of reading, all Reviewers’ original remarks are reprinted in blue italics, and changes to\nthe manuscript appear in red.\nYours sincerely, the authors\nReply to Reviewer 1\nThis study investigates the complex multi-Q magnetic textures of the triangular antiferromagnet Co₁/₂TaS₂\nusing optical techniques, namely magnetic circular dichroism (MCD) and magnetic linear dichroism\n(MLD). By combining MCD and MLD, the authors are able to distinguish between chiral order and\nnematic order, thereby corroborating and complementing the results of neutron experiments. Moreover,\nowing to the excellent real-space resolution of MCD and MLD, the authors have resolved the real-space\nmagnetic domain information and further determined the magnetic structure of this system. In addition,\nthis technique has generality and is expected to be extended to other complex systems as one of the\ntechnical means for identifying intricate magnetic textures. Overall, this is a thorough work and is worthy\nof publication. I have only a few minor comments to address:\nReply: We appreciate the Reviewer’s positive assessment of the manuscript and recommendation to\npublish (after addressing the Reviewer’s feedback in the revised manuscript), and especially for noting\nthe general applicability of these techniques to other complex magnetic textures.\n1. In the Introduction, the authors state that antiferromagnets can exhibit a variety of complex spin\nstructures, whereas ferromagnets cannot. This seems to involve some ambiguity in definitions. For\nexample, MnSi, a famous skyrmion host, is believed to have a weak itinerant ferromagnetic ground state.\nReply: We thank the Reviewer for pointing out that our phrasing in the Introduction could be interpreted\nas dismissive of non-trivial ferromagnetic orders. This was not our intent; rather we meant only that\nantiferromagnets can generally be described by one or more finite-Q orderings, whereas ferromagnets\ngenerally exhibit uniform net magnetization (Q=0). But as the Reviewer correctly notes, skyrmion phases\nin otherwise ferromagnetic materials are obviously not trivial. To avoid any misinterpretation, and\nwithout loss of generality, the revised manuscript now simply omits the phrase “Unlike ferromagnets…”\nso that the third sentence now begins “Antiferromagnets can exhibit diverse spin configurations…”\n2. In Fig. 1(e), the MCD signal between TN2 and TN1 remains visible (albeit weaker compared to the\nsignal below TN1). Does this imply the presence of a potential chiral magnetic structure in this\ntemperature range or other possible effects? The authors should clarify this point.\nReply: The very small MCD signal that can be seen in Fig. 1e (between TN2 and TN1) arises from a\nsmall amount of cross-talk between MCD and MLD signals, and not from any chiral magnetic structure.\nThis cross-talk was discussed in detail in Supplementary Section I.B. and in Supplemental Fig. S2 of the\noriginal manuscript (a more vivid example is shown in panel S2c). The revised manuscript makes this\nconnection more explicitly clear, at the end of section I B where this cross-talk effect is first noted and\nwhere Fig. S2 is first referenced, by adding “This cross-talk also generates the very small MCD signal\nbetween TN1 and TN2 in Fig. 1e (see Supplementary Fig. S2)”\nReply to Reviewer 2\nErik Kirstein and coworkers have investigated the magnetic phases of the van der Waals material\nCoTa3S6, using optical microscopy (MLD/MCD), transport measurements and magnetic simulations\nincluding Monte Carlo methods. The manuscript is well-written and the figures are clear. The new finding\nseems to be a mixed low T zero field phase, where 1Q and 3Q magnetic order coexists. The authors show\nlaterally resolved optical data with a resolution of about 1 micron, where the MLD and MCD signals are\nused as a measure of 1Q and 3Q magnetic character, respectively.\nReply: We thank the Reviewer for their thorough reading of the manuscript. We’re glad to know that the\nReviewer finds the manuscript well-written and the data clear.\nMoreover, we respectfully emphasize that the new findings include not only the experimental\nidentification of a distorted triple-Q antiferromagnetic phase with co-existing chiral and C_2z symmetries\n(noted by the Reviewer), but also 1) the demonstration that magneto-optical methods for circular and\nlinear dichroism can function as incisive reporters of these antiferromagnetic symmetries, 2) experimental\ndemonstration that a three-state (Z3) nematic order parameter is what distinguishes the various B- and T-\ndependent phases in this archetypal triangular-lattice antiferromagnet, and 3) the experimental\ndemonstration that the chiral and Z3 nematic domains can be directly imaged via microscopy. Moreover,\nthe data are supported by a consistent theoretical framework (multi-Q manifold) that incorporates\nmagnetic anisotropy.\nA main weakness of the manuscript is that the authors do not properly cite and relate their work to the\nexisting literature, making it unnecessarily difficult for the reader to discriminate possibly new results\nfrom already established ideas and facts. For instance, the concept of a continuum of states between 1Q,\n2Q and 3Q, has already been put forward in Ref. 38, however, a comparison between the proposed two-\ndimensional \"multi-q manifold\" and the one-dimensional mixing scheme of Ref. 38 is not made.\nReply: We would like to emphasize that the primary focus and achievement of this work lies in the\nexperimental characterization of multiple antiferromagnetic phases in Co TaS , and not in the\n1/3 2\ndevelopment of a comprehensive theoretical model to explain every nuance of these new data. We did\nof course carefully study the theoretical model of Ref. [38] (by Haldar et al), and we recognized its\nimportance and relevance to our work – which is precisely why Ref. [38] was already cited in our original\nmanuscript. However, we understand the Reviewer’s point of view, and agree that it could be more\nappropriate to cite this work sooner and more prominently. Our revised manuscript now also cites Haldar\net al in the Introduction section, as an example of the most relevant prior study (to our knowledge) of\nsuch evolving and tunable multi-Q antiferromagnetic orders. Moreover, we have expanded the discussion\nin Section II C and also in Section II D to let readers know how the theory of Haldar et al relates to our\nexperimental findings (see Changes to Manuscript below).\nThat said, we stress that the main message of our manuscript is based on experimental discoveries.\nFor completeness our manuscript includes a new theoretical model that successfully captures our data,\nincluding the importance of magnetic anisotropy. But it is outside the scope of our experimental paper to\ncompare and contrast the details of our model to the details of other theoretical models (such as the 6-spin\ninteractions considered by Haldar et al). Others may do this if they wish, and indeed we hope that our\nresults will motivate more theoretical work along these lines. We hope that the Reviewer understands and\nappreciates our rationale and approach.\nTo reiterate: Through the combined use of two magneto-optical probes (MLD and MCD), supported by\nprevious characterization results, we were able to determine the chiral and, crucially, nematic magnetic\nsymmetries of these distinct phases. The identification of both the three-fold symmetric and distorted\ntriple-Q states—based solely on experimentally inferred magnetic symmetry and spin chirality—is a key\nresult of this study. These findings stand independently of which theoretical modelling we adopted and\nrepresent a significant new advance in the experimental investigation of complex spin textures.\nMoreover, part of the phase diagram (Fig. 2c) is based on results from a previous publication of the\nauthors (Ref. 19) and results from the Tokyo group (Ref. 18).\nReply: Yes, and as the manuscript explicitly states, the general structure of the magnetic phase diagram of\nFig. 2c is already suggested by the recent magnetization, neutron, electrical transport, and susceptibility\nmeasurements of Refs. [18, 19, 25] (indeed, the manuscript clearly states that data shown in Figs. 1c,d is\nadapted from [19]). We reiterate that the primary goal and main message of our manuscript is to\nexperimentally elucidate the underlying nature and magnetic symmetries of these different magnetic\nphases, using optical methods based on circular and linear dichroism, which allows to classify the\nvarious phases by the presence/absence of chirality and C_2z symmetry (stripe-like order).\nThe proposed distorted 3Q state in phase I -- predicted for hcp Mn/Re(0001) in Ref. 38 (without\nexperimental proof) and there explained based on higher than 4th-order exchange interactions (beyond 4-\nspin or biquadratic exchange) is here argued to be the result of crystalline anisotropy without taking\nother possibilities into account. Moreover, I don't find the argument for such a state convincing: If phase I\nconstitutes a 3Q state with some 1Q component, why is then the MLD signal (which is introduced by the\nauthors to be a measure of 1Qness) at a maximum in phase I, higher than in the supposedly pure 1Q\nphase III?\nReply: Again we emphasize that we had carefully read Ref. 38 (Haldar et al) and recognized its relevance\nto the present work—indeed, this is why it was explicitly cited in the relevant sections of the main text.\nRegarding the sixth-order exchange interaction proposed in Ref. 38, we did attempt to incorporate such\nterms into our momentum-space modeling framework. However, this higher-order term failed to capture\nthe experimental features of Co TaS . Specifically: 1) Increasing its strength resulted only in a first-order\n1/3 2\ntransition between the three-fold symmetric 3Q and 2Q states, without stabilizing an intermediate\ndistorted 3Q state as a distinct ground state, and 2) While the observation above might still suggest room\nfor a continuous manifold between 3Q and 2Q states (that is, Δ < Δ = Δ) around the transition, our\ni j k\nexperimental observations instead point to a manifold connecting 1Q and 3Q phases (that is, Δ > Δ = Δ )\ni j k\nin Co TaS , an opposite side to this. For these reasons, we concluded that the sixth-order interaction\n1/3 2\nproposed in Ref. 38 is unlikely to be responsible for our observations in Co TaS .\n1/3 2\nOur theoretical model instead attributes this phase to the very common (and thus more plausible)\nmagnetic anisotropy, guided by a broad set of experimental constraints. Notably, neutron diffraction has\nidentified an out-of-plane spin orientation in the single-Q phase and specific high-symmetry directions in\nPhase I. Additionally, the observed finite magnon gap in Phase I indicates full breaking of spin-rotational\nsymmetry. The anisotropy terms included in our model—namely, the single-ion anisotropy (A) and the\nbond-dependent exchange anisotropy (J ) in Eq. (3) of the main text—were chosen carefully to cover all\n±±\nthese findings. We would like to emphasize that, unlike four-spin or six-spin interaction terms, these\nsecond-order anisotropy terms are common in many magnetic materials.\nGiven (i) the inconsistency of the Ref. 38 model with our experimental results and (ii) the ability of our\nminimal spin model, based on well-established and commonly found anisotropy terms, to quantitatively\ncapture the key features of Co TaS , we believe it offers a natural and sufficient explanation without the\n1/3 2\nneed to invoke higher-order interactions beyond four-spin terms.\nFinally, we note that the MLD signal (a reporter of 1Q order parameter) emerges and grows throughout\nPhase III, achieving a maximum at the Phase I – Phase III boundary, which remains approximately constant\nas the sample is cooled throughout Phase I (i.e., the MLD is not larger in Phase I as the Reviewer suggests).\nFigure 1g shows this reasonably well, and Supplemental Fig. S2d shows this extremely clearly.\nOverall, the manuscript does not clearly offer substantially new results which are backed by experiments\nto warrant publication in Nature Communications.\nReply: Again, we respectfully remind the Reviewer that our manuscript describes new experimental\nresults (Figs. 1, 2, 4, & 5), which are backed by theory -- not the other way around as the Reviewer’s\ncomment implies. Moreover, the theoretical framework that we developed fully captures the experiments,\nby incorporating common magnetic phenomena (such as anisotropy).\nIn addition, I would like to raise more specific criticism:\n1. In contrast to \"chirality\", the authors use the term \"nematicity\" without properly defining it. For that\nreason, stating that the 1Q AFM order gives rise to nemetic domains does not explain nor add anything.\nReply: By “nematicity” we assume the usual definition: an order parameter that breaks rotational\nsymmetry. Nematicity in crystalline solids (as opposed to, say, liquid crystals), is technically a ‘discrete\nnematicity’, as it does not break a continuous rotational symmetry but rather a discrete lattice rotational\nsymmetry. For a 2D triangular lattice, “nematic” therefore means a broken discrete rotational symmetry\nfrom C6 to C2. In our manuscript we followed recent literature on single-Q (stripe-like)\nantiferromagnetism in hexagonal magnets [Nature Physics 20, 1888 (2024); Nature Materials 19, 1062\n(2020)], and explicitly refer to the single-Q stripe-like AFM order in CoTaS as “discrete three-state” or\n“Z3” nematic order, since there are three possible choices of direction (ie, along the three M-ordering\nwavevectors). This definition is already explicitly stated in the manuscript, for example in the abstract\nand in the 2nd paragraph of the Introduction section. Nonetheless, for added clarity we have added\nphrasing throughout the paper to emphasize the connection between C_2z symmetry, stripe-like single-Q\nantiferromagnetic order, and three-state (Z3) nematicity.\nWe note that the concept of nematic order in solid crystals has a long history, for example nematic order\nin iron-based superconductors, where C4 symmetry becomes C2 (called “Ising-like” or “Z2” nematic\norder); see for example [Fernandes et al., Nature Physics 10, 97 (2014)].\n2. What is the exact relationship between 1Q AFM order (row-wise AFM) and the MLD and MCD signal?\nWhat is the exact relationship between 3Q order (109° between all neighboring spins) and the MCD and\nMLD signal?\nReply: We thank the Reviewer for this question, which has alerted us to a potential point of\nmisunderstanding. We do *not* mean to imply that the tetrahedral 3Q order has perfect 109-degree\nangles between neighboring spins (the manuscript never said this). Firstly, the tetrahedral lattice defined\nby the Co atoms is itself not geometrically perfect (there are different inter- and intra-plane Co-Co\ndistances), and moreover the intra- and inter-plane exchange couplings J and J are different. Thanks to\n1 c1\nthe Reviewer, we recognize that the words “equilateral” and “non-equilateral” might incorrectly suggest\nthe presence/absence of geometrically perfect tetrahedra and perfect 109 degree spin order. Rather, what\nwe mean is that our data reveal phases that are “three-fold symmetric triple-Q” (defined in our manuscript\nas  =  =  , with chirality but without nematicity, such as Phase II), and phases that are “distorted\n1 2 3\ntriple-Q” (where  are not equal; these exhibit both chirality and nematicity such as Phase I). To avoid\n1,2,3\nany misinterpretation, we have updated our terminology, replacing “equilateral 3Q” with “three-fold\nsymmetric 3Q”, and “nonequilateral 3Q” with “distorted 3Q”. Moreover, Fig. 1 and its caption now\nexplicitly show/mention the different intra- and inter-plane exchange couplings J and J .\n1 c1\nTo answer the Reviewer’s question: The single-Q “stripe” AFM order discussed in the manuscript (where\nQ is half a reciprocal lattice vector, or at the M-points of the hexagonal Brillouin zone) generates non-\nzero MLD because it breaks the rotational symmetry of the crystal (reduction to C_2z symmetry). Related\nphenomena were recently observed in hexagonal antiferromagnets magnets [e.g., Nature Physics 20, 1888\n(2024); Nature Materials 19, 1062 (2020); Nano Letters 21, 6938 (2021)] and arises from the resulting in-\nplane anisotropy of the optical conductivity. Such purely single-Q AFM order does not possess chirality,\ndoes not generate any Hall conductivity  , and therefore does not generate MCD.\nxy\nIn contrast, “three-fold symmetric 3Q order” (defined in our manuscript as  =  =  ) does possess\n1 2 3\nchirality and therefore does generate  and MCD, but does not generate MLD because it does not break\nxy\nthe discrete rotational symmetry of the crystal (i.e., does not possess C_2z symmetry). These concepts\nand definitions are described and clearly defined in the manuscript.\n3. Which MLD and MCD signals are expected for the coplanar 120° Neel state proposed for CoTa3S6 in\nRef. 25?\nReply: The coplanar Neel state proposed in Ref. [25] would be expected to give MCD (since\n\\sigma_{xy} is allowed), but would not be expected to give MLD (since its symmetry is not C_2z).\n4. page 2, \"coexistence of single-Q and triple-Q orderings within the same system – an uncommon feature\nin real materials.\": Firstly, a distorted 3Q state has been proposed already in Ref. 38. Secondly, 1Q, 3Q\nand transient 2Q states co-exist in the Mn/Re(0001) system due to different stacking order and inside\ndomain walls, respectively.\nReply: Please see the discussion above relating to Ref. [38] by Haldar et al., which is now cited earlier\nand more prominently in the revised manuscript. Again, we stress that our manuscript already cited Ref.\n[38] -- now Ref. [26] in the revised manuscript -- precisely *because* of its relevance to this distorted\ntriple-Q state. As the authors of [38] will surely agree, such an unusual distorted 3Q state is indeed not\ncommon – we carefully searched through the existing literature for relevant precedent, and found that\nMn/Re was one of only a very few examples containing related physics.\n5. page 2, \"effective tetrahedral network of Co\": This sentence implies that the magnetic coupling\nbetween Co atoms is identical within and across layers, which is certainly not the case.\nReply: We did not assume identical intra- and inter-layer magnetic couplings (as the theory section\nalready made clear, via use of different couplings J_1 and J_c1). Nonetheless, as discussed above in our\nreply to Comment #2, to make this even more explicitly clear and avoid any confusion, Figure 1a is now\nupdated with the intra-layer and inter-layer magnetic couplings (J_1 and J_c1) now labeled.\n6. \"Continuous multi-Q manifold model\": The mixed magnetic states in Ref. 38 are constructed in a one-\ndimensional fashion on a path on which the standard Heisenberg exchange is constant. Is this also the\ncase for the authors' \"manifold\"?\nReply: Thank you for the question. We consider the full two-dimensional (not one-dimensional) degenerate\nground state manifold of the Heisenberg model (ie, constant Heisenberg exchange). This manifold\nobviously connects all possible single-, double-, and triple-Q states and the degeneracy persists regardless\nof the specific values of the Heisenberg exchange interactions, provided that their Fourier-transform, J(q),\nis minimized at the M-point—consistent with the ordering wave vector observed in Co TaS .\n1/3 2\n7. page 4, \"successfully captures the single-Q to triple-Q transition at TN2\": Is it essential which 4th-\norder exchange term (biquadratic, 3spin, 4spin) is used for the transition to occur? Is the driving\nmechanism for this transition entropy?\nReply: It is not essential. As along as the chosen fourth-order interaction favors the triple-Q ground state\nover the single-Q state, its specific form does not affect the thermally-induced transition to single-Q. As\nthe Reviewer correctly notes, this transition is entropy-driven: thermal fluctuations favor collinear (single-\nQ) states over non-collinear ones [Villain, J. et al., Journal de Physique 41, 1263 (1980)], thereby\ngenerating the transition regardless of the detailed form of the real-space four-spin interaction.\nOur work does not attempt to identify the specific microscopic real-space four-spin interactions, which is\na complex task for metallic materials like Co TaS . Instead, we formulate the problem in momentum\n1/3 2\nspace and retain only the uniform (Q =0) and ordering wave vector components (Q , Q , Q )of the spin\n0 1 2 3\nfield (see Supp. Note II. B). A given four-spin interaction in momentum space can arise from multiple\ndistinct combinations of four-spin terms in real space. As a result, our analysis does not distinguish\nbetween biquadratic and more general four-spin interactions in real space. The simple scalar biquadratic\nform is chosen only for Monte Carlo simulations that require a concrete real-space model that reproduces\nthe momentum space Hamiltonian restricted to the above-mentioned wave vectors. This choice of real\nspace interactions is not relevant for the long wavelength static and low-energy properties of M-ordered\nsystems. That said, it can affect the nature of field and temperature induced phases, such as phase IV.\n8. page 5, \" field-induced equilateral triple-Q state\": It is not straightforward to make a distorted 3Q\nstate more symmetric by applying an external B field. There is an energy term which might do the trick,\nbut it is not mentioned in the manuscript.\n9. page 7, \"can be completely poled to a positive or negative chiral state\": Is this behavior reproduced by\nsimulations? Which energy term is responsible for the coupling of chirality and external magnetic field?\nReply: We thank the Reviewer for raising these questions. Since they are related, we provide here a\nsingle Reply that addresses both. The key mechanism behind the field-induced three-fold symmetric\ntriple-Q state (Phase II) is its higher magnetic susceptibility. This is strongly supported by the M(H)\nmagnetization curve of Co TaS (see Fig. 1d), which shows an immediate gain of additional\n1/3 2\nmagnetization right after passing the metamagnetic transition field (H > H ). Applying a large magnetic\nm\nfield along the c-axis restores the C_3z symmetry by stabilizing a state in which one of the four spin\nsublattices aligns with the field (as depicted in Fig. 3); this state has larger net residual moment.\nYes, the behavior is reproduced by our simulations (see Figs. 3f and 3g, for example). In our spin model,\nthe coupling arises from the weak, residual net magnetic moment (present even at zero field; see Fig. 1d),\nwhich couples the external magnetic field to the sign of the scalar spin chirality. Such a moment in\ncomplex antiferromagnetic orders originates from either slight spin canting (due to magnetic anisotropy –\nspecifically, to the A and J anisotropy terms in our Equation 3), or to orbital magnetization, and is\n±±\ncommonly found in nontrivial antiferromagnetic spin orders that exhibit a spontaneous Hall conductivity\n(e.g., Mn Sn). The direction of the net moment is linked to the sign of the scalar spin chirality: domains\n3\nwith positive (negative) scalar spin chirality have net magnetization along +z (−z). Coupling of this\nresidual moment, and therefore chirality, to H is via Zeeman terms (see Eq. S20). This weak net moment\nand the resulting field-tunability of complex order parameters has been widely discussed in prior studies\nof antiferromagnets with spontaneous Hall effects: for instance, see Šmejkal et al. Nature Reviews\nMaterials 7, 482–496 (2022) and Chen et al., Phys. Rev. B 101, 104418 (2020).\nWe agree with the Reviewer that this coupling is an important aspect of the model and underlying\nphysics, and the revised manuscript now more completely describes this coupling mechanism and the role\nof the anisotropy terms, in the second-to-last paragraph of Section II D.\n10. J. Spethmann et al. try to determine the orientation of their 3Q state in hcp Mn/Re(0001) and the\naccompanied DFT theory predicts a spin orientation in the 1Q state along the parallel spin-rows, driven\nby the magnetic dipolar interaction and anisotropic symmetric exchange. Is it possible to tackle these\nquestions also for CoTa3S6.\nReply: We emphasize that the magnetic structure of bulk Co TaS has already been determined (or at least\n1/3 2\nstrongly constrained) by neutron diffraction–the most powerful technique to determine/constrain\nmicroscopic spin orientations (Refs. 18–19). Our theoretical model was constructed to be fully consistent\nwith these experimental findings. In particular, the magnetic anisotropy terms used in our spin Hamiltonian\nwere chosen to reflect the experimentally observed spin orientations—such as the out-of-plane alignment\nof moments in the single-Q phase. These orientation profiles predicted by our model for each phase are\nexplicitly shown in Figs. 3b–e. Indeed, spin orientations are constrained to certain high-symmetry directions\nand agree with the information imposed by the diffraction results from prior studies (Refs. 18–19).\n11. Compared to SP-STM, where these states are imaged directly with atomic resolution, MLD and MCD\nare indirect measures, averaging over large surface areas. The effect of domain sizes and chemical\ninhomogeneity with a lateral size below the resolution limit is not discussed in the manuscript.\nReply: All spatially-resolved experiments have a resolution limit, and the spatial resolution of optical\nmicroscopy is approximately given by the wavelength of light (of order 1 micron) – which is small, but is\nobviously larger than the atomic-scale resolution of a tunneling microscope that can see individual atoms.\nHappily, our images in Figs. 4 and 5 show clear – and very different— characteristic lengthscales for\nchiral and Z3 nematic domains in Co TaS that exceed this resolution limit (a few microns, and 10s-100s\n1/3 2\nof microns, respectively). Moreover, we point out that if the material hosted equal distributions of up- or\ndown- chiral domains (or 0/ 120 /240 degree nematic domains) on lengthscales significantly smaller than\nour 1 micron spatial resolution, then MCD and MLD images would be expected to show uniformly zero\nsignal, which is clearly not the case.\n12. What is the information depth of the used techniques? Can a scenario be ruled out where the magnetic\nstate at the surface is different from the bulk?\nReply: The effective penetration depth of visible light in this material is of order 50-100 nm (based on the\nmaterial’s conductivity). Experiments of this type therefore effectively constitute a bulk probe, not a\nsurface probe. (Moreover, freshly-exfoliated surfaces showed similar behavior as surfaces exposed for\nweeks to ambient conditions).\n13. What can be concluded for the magnetic coupling between the layers experimentally? Is the proposed\n109° coupling in phase I an assumption or a conclusion?\nReply: Firstly, we stress that our manuscript *does not* propose 109 degree coupling in Phase I (as the\nReviewer suggests), nor do we propose 109 degree spin order in any other phase. Please see our Reply to\nthe Reviewer’s comment #2 above, where we address this point of potential misunderstanding. Rather,\nFigures 2 and 3 emphasize that the magnetic configuration in Phase I is a distorted tetrahedron. This is a\ncentral finding of our work, discussed throughout the main text and Supplemental Material. A key aspect\nof the spin configuration in Phase I is that it not only possesses chirality, but also breaks in-plane\nsymmetry (ie, has “Z_3 nematicity”). This is a conclusion based on the experimental presence of both\nMCD and MLD. The balance and interplay of the 4-spin interactions and magnetic anisotropy that can\ngive rise to such a state are discussed in detail in the main text and in the Supplemental Material.\n14. Finally, can the proposed uuud state (phase IV) be reproduced in the simulation?\nReply: The proposed uuud state cannot be captured by our Monte Carlo simulations, since we consider a\nclassical spin model. This outcome is expected, as stabilizing this phase requires strong longitudinal\nfluctuations that permit unequal moment lengths across the four sublattices. We emphasize that this uuud\nstate is distinct from the conventional field-induced uuud phase with half-saturated magnetization, which\ndoes emerge in our simulations at very high magnetic fields (>10 T).\nNonetheless, as discussed in the final paragraph of Section II.D of the main text, the proposed uuud state\nremains the most plausible scenario consistent with the constraints imposed by magnetization, MLD, and\nMCD measurements. We regard this experimentally-driven inference as a significant outcome of the\npresent study, irrespective of the current limitations of theoretical modeling.\nReply to Reviewer 3\nThe work by Kirstein et al. presents a comprehensive study of the chiral and nematic states in the\nantiferromagnet Co₁/₃TaS₂. They mapped out the antiferromagnetic phase diagram under different\nmagnetic fields and temperature ranges using MLD and MCD measurements. A continuous multi-Q\nmanifold model is proposed to understand the phase diagram. Finally, they presented spatially resolved\nchiral and nematic domains using MCD and MLD microscopy.\nI think this paper is well written, and the results are scientifically sound. This work not only demonstrates\nthat Co₁/₃TaS₂ hosts rich nematic and chiral multi-Q states, but also represents a high-quality study of\nantiferromagnetic textures using optical techniques. I therefore recommend its publication in Nature\nCommunications. Before publication, I have a few comments for the authors to improve the presentation\nof their manuscript.\nReply: We sincerely appreciate the Reviewer’s very positive assessment of the manuscript and\nrecommendation to publish in Nature Communications (after addressing the Reviewer’s comments and\nsuggestions for the revised manuscript). We are happy to address and accommodate these suggestions;\nplease see point-by-point reply below.\nComments: 1. Phase I exhibits both MCD and MLD signals. The authors propose that this phase\ncorresponds to a non-equilateral triple-Q state. Could the authors clarify how they rule out the possibility\nthat Phase I arises from the coexistence of spatially separated single-Q and triple-Q domains? In such a\ncase, both MCD and MLD signals could also be expected.\nReply: Excellent question: The spatially-resolved MCD and MLD images indicate that Phase I is not\ncomposed of spatially-separated single-Q and triple-Q domains. For example, Fig. 4a shows that large\nnematic domains are everywhere present in this region (and every other region we measured). That is, to\nwithin the experiment’s 1-micron spatial resolution, there are no regions without an underlying 0, 120, or\n240-degree nematic order. Simultaneously, imaging the same region with MCD microscopy shows that\ndomains with non-zero chirality are everywhere present (that is, there are no regions without an\nunderlying positive or negative chirality; see Fig 5). Every spatial region we have studied exhibits both\n(large) nematic domains and (small) chiral domains in Phase I. Phase I is therefore consistent with a\nmagnetic state possessing both chiral and nematic (C ) order – a distorted triple-Q phase -- rather than\n2z\nspatially separated phases of purely 1Q and 3Q order. As another example, Figs. 5b and 5d show that the\nhandedness of the chirality can be uniformly poled (the images are uniformly red and blue, respectively),\nhowever when these same regions are imaged using MLD, large domains with nematic (C_2z) character\ncompletely populate the image (similar to those shown in Fig. 4). The last paragraph of the manuscript\ndoes already make this important point, where it states “…the images confirm that Phase I cannot be a\nphase-separated mixture of coexisting single-Q and triple-Q ground states”.\n2. Do the authors have an intuitive physical picture or a simple explanation for why the magnetic textures\nin Phase I exhibit both nematicity and chirality? Although the authors attribute this to magnetic\nanisotropy, the explanation remains unclear. Magnetic anisotropy is relatively common, but the\ncoexistence of nematic and chiral textures is much less so. What is special about Co₁/₃TaS₂ in this\ncontext?\nReply: We appreciate the incisive question. The short answer / “intuitive” physical picture is that Phase I\narises from the balance between i) the strength of magnetic anisotropy (which favors 1Q order) and ii) the\nstrength of 4-spin interactions that exist in Co TaS (which favors threefold-symmetric chiral 3Q order).\n1/3 2\nIn Co TaS , what is special is that the anisotropy is apparently sufficiently weak that it competes\n1/3 2\neffectively with 4-spin interactions, leading to a situation where the magnetic ground state at low\ntemperatures and low fields is noncoplanar triple-Q (which gives chirality) but with unequal Fourier\ncomponents  ≠  (which leads to nematic character as well). Otherwise, if the anisotropy were\n1 2,3\nextremely strong, it would dominate and give single-Q stripe order everywhere (with, say,  =0).\n2,3\nThis interplay between anisotropy and 4-spin interactions in Co TaS – and how it can lead to the “distorted\n1/3 2\ntriple-Q” order in Phase I, is discussed in section II D of the main text (after Equation 3), and in detail in\nthe Supplementary Information (see section 2D, and Figs. S7 and S8 in particular).\n3. The MCD curve versus magnetic field in Fig. 2a shows a tilt, suggesting that the field induces some\nMCD even at high temperatures such as 30 K or 40 K. In contrast, at high temperatures (e.g., 30 K), the\nMCD is labeled to vanish at large fields in Fig. 2c. The authors should clarify this apparent inconsistency.\nReply: We thank the Reviewer for pointing out an apparent inconsistency. What we intended Fig. 2c to\nindicate is the presence or absence of any additional, spontaneous, topological MCD (due to chiral\nantiferromagnetic order) that exists on top of the trivial linear-in-field MCD that arises from ordinary Hall\neffects. This is now clarified in the Fig. 2c caption of the revised manuscript.\n4. I have some questions regarding Phase IV: (i) In Section A, the authors state that magnetic fields\ninduce two new phases, Phase II and Phase IV. However, this is not evident from Fig. 1. (ii) There is no\nvisible sharp transition in Fig. 2b at high temperatures. How do the authors determine the boundary\nbetween Phase III and Phase IV in Fig.2c?\nReply: Again, we are grateful for the opportunity to clarify an apparent inconsistency. The Reviewer\nnotes that the data in Figs 1c,d alone do not show any field-dependence at higher temperatures (27-38 K),\nand therefore do not explicitly show any phase III-IV boundary. Therefore, where the presence of Phase\nIV is first mentioned (after discussing Figs. 1c,d), the revised manuscript now clarifies “(Note that the\ndata shown in Figs. 1c,d do not explicitly reveal the transition to Phase IV; however this transition is\nshown in Refs. [18, 19, 25], and is also evident in Fig. 2 below).\nMore generally, we also note that the general outlines of the Co TaS phase diagram – including the\n1/3 2\npresence of high-field Phases II and IV – were recently postulated on the basis of the neutron diffraction,\nmagnetization, electrical transport, and susceptibility studies reported in Refs. [18, 19, 25] (and as our\nmanuscript notes, Figs. 1c and 1d are data from [19]). The primary goal of our present work is to\nelucidate the underlying nature and magnetic symmetries of these different magnetic phases, using optical\nmethods (we do not lay claim to inventing this notional phase diagram). While the data shown in Fig.1\nalone does not show any field-dependence at high temperatures (27-38 K), and therefore does not show\nthe phase III-IV boundary, we emphasize that our measurements shown in Fig. 2b do. In Fig. 2b, the\nPhase III-IV boundary is revealed by the steps in MLD vs B, which are indicated by arrows. The steps\nbecome broader and less sharp at higher temperatures --as is often the case for magnetic phase transitions\nat elevated temperatures-- but the position of the steps, accurately determined via the maximum of the\nslope d(MCD)/dB, allows us to identify this final phase boundary, which in turn is in good agreement\nwith the boundary identified in [18, 19, 25].\n5. Figures 4(c) and 4(d) appear repetitive. Moreover, there is no description of Fig. 4(d) in the main text.\nThe authors should address this omission.\nReply: Yes, as the caption states, panel (d) is just the director map of panel (c) overlaid with the first\nMLD map of panel (a). We agree that perhaps this is a bit repetitive, but during these studies we have\nfound it very helpful and of value to see the two maps directly overlaid, so that different nematic domains\ncan be clearly associated with particular MLD signals (otherwise it’s harder to determine whether features\nin two maps occur at exactly the same location). We anticipate that readers will also find this helpful.\nAnd finally, per the Reviewer’s suggestion, the revised manuscript now refers to Fig. 4d in the main text.\nLIST OF CHANGES TO MANUSCRIPT (changes appear in red color in the revised manuscript)\n• At the end of section I B where cross-talk effects are first noted and where Fig. S2 is first\nreferenced, we add “This cross-talk also generates the very small MCD signal between TN1 and\nTN2 in Fig. 1e (see Supplementary Fig. S2)” (per Reviewer #1)\n• The third sentence of the main text now begins “Antiferromagnets can exhibit diverse spin\nconfigurations…” (per Reviewer #1)\n• Updated Figure 1a, and its caption, now explicitly shows/discusses the different intra- and inter-\nlayer exchange constants J_1 and J_c1. (per Reviewer #2)\n• Throughout the manuscript (including Figures in the main text and Supplement), the terms\n“equilateral” and “non-equilateral” are replaced with the more explicitly descriptive phrases\n“three-fold symmetric” and “distorted” (please note that only the first few instances are\nhighlighted in red color in the revised manuscript). (per Reviewer #2)\n• The paper by Haldar et al [PRB 104, L180404 (2021)], formerly Ref. [38] in the original\nmanuscript but now Ref [26] in the revised manuscript, is now cited in the Introduction section of\nthe revised manuscript, and noted/described more prominently in Section II C “related AFM\nphases were studied theoretically for Mn monolayers~\\cite{haldar2021}”, and also in Section II\nD “We also note that Ref.~\\cite{haldar2021} considered a 1D sub-manifold of the 2D manifold of\ndegenerate ground states..” (per Reviewer #2)\n• The revised manuscript now includes (near the end of Section II D) a description of why\nmagnetic fields favor the threefold-symmetric 3Q state (Phase II), and what energy/anisotropy\nterms lead to the net residual moment that couples to applied fields. “The field-induced Phase II\nis stabilized by the larger residual magnetic moment of the three-fold symmetric triple-Q phase\n(see the magnetization data in Fig. 1d), favoring a state in which one of the four spin sublattices\naligns along $H$. A residual net moment exists in both Phases II and I (even at zero field) due to\nthe anisotropy terms in our model, and its direction ($\\pm \\hat{c}$) is linked to the sign of the\nunderlying spin chirality.” (per Reviewer #2)\n• The caption of Fig 2c is updated to indicate that the presence/absence of MCD refers to the\nspontaneous, topological MCD (due to chiral antiferromagnetic order) that exists on top of the\ntrivial linear-in-field MCD that arises from ordinary Hall effects. (per Reviewer #3)\n• Where Phase IV is first mentioned (after discussing Figs. 1c,d), the revised manuscript now\nclarifies “(Note that the data shown in Figs. 1c,d do not explicitly reveal the transition to Phase\nIV; however this transition is shown in Refs. [18, 19, 25], and is also evident in Fig. 2 below).”\n(per Reviewer #3)\n• Figure 4d is now referred to in the main text (per Reviewer #3)\n• At the end of Section II A we also explicitly define what we mean by “C ”.\n2z"}, {"round": 2, "reviewer_comments": {"reviewer_1": "publish as is.", "reviewer_2": "Despite the substantial and detailed criticism I have raised in the first review round, Kirstein and coworkers have resubmitted\na manuscript with only cosmetic changes. In their reply, most of my questions are evaded, some given answers are wrong\nand the clarifications and explanations which were indeed helpful to understand the work, did unfortunately not make it into\nthe new version of the manuscript. I do understand the tactics of the rebuttal, but this way of dealing with justified critique is\nrisky, and I remind the authors that two of them have already published a wrong conclusion about the magnetic state of the\nvery same system in Ref. [25]. Therefore, most of my critique is still valid and I come to the same conclusion, that I do not see\nsubstantially new physics within the manuscript which is conclusively backed up by experiment to warrant publication in\nNature Communication. In my opinion, certain inconsistencies of the manuscript need to be addressed before publication in\nany journal. Whether these problems can be fixed by simply writing more clearly or point toward deeper flaws of the work,\nI do not know.\nFor the future, I encourage the authors to refrain from the overuse of italic, bold and underlining in the review\ncorrespondence. Reviewers are quite capable of reading texts carefully without such \"help\".\nIn the following, I will not again comment on all problems of the manuscript and shortcomings of the rebuttal, but only\naddress the ones I deem most critical.\n1. Let us start with this claim from the rebuttal: \"Moreover, we respectfully emphasize that the new findings include not only\nthe experimental identification of a distorted triple-Q antiferromagnetic phase with co-existing chiral and C_2z symmetries\n(noted by the Reviewer), but also 1) the demonstration that magneto-optical methods for circular and linear dichroism can\nfunction as incisive reporters of these antiferromagnetic symmetries, ...\"\nAt face value, this sentence is circular: Fundamentally, you cannot establish a new magnetic technique using a specific spin\ntexture and then also claim that the observed spin texture is a new finding. Now, it seems that CoTa3S6 has four different\nmagnetic phases: which spin textures are used to calibrate MLD and MCD and which results are actually new?\n2. Another problem puts a main result of the work into question. Here, again a sentence from the rebuttal: \"Finally, we note\nthat the MLD signal (a reporter of 1Q order parameter) emerges and grows throughout Phase III, achieving a maximum at the\nPhase I – Phase III boundary, which remains approximately constant as the sample is cooled throughout Phase I (i.e., the\nMLD is not larger in Phase I as the Reviewer suggests). Figure 1g shows this reasonably well, and Supplemental Fig. S2d\nshows this extremely clearly.\"\nThis description is wrong! In both plots the highest MLD signal is found at the lowest temperature of T=4.2 K, not at T=26.5\nK. But let us assume for the sake of argument, that the MLD signal is constant in phase I; even then the problem arises that\nthe MLD signal interpreted as a direct measure of 1Q character is INCONSISTENT with the interpretation of phase III as a\npure 1Q state and phase I as a mixed 1Q-3Q state. When the state becomes more 3Q-like in phase I, the 1Q character needs\nto drop. There seems to be a further temperature dependence which is not discussed and not accounted for.\n3. As an answer to my comment #4 [page 2, \"coexistence of single-Q and triple-Q orderings within the same system – an\nuncommon feature in real materials.\": Firstly, a distorted 3Q state has been proposed already in Ref. 38. Secondly, 1Q, 3Q\nand transient 2Q states co-exist in the Mn/Re(0001) system due to different stacking order and inside domain walls, respectively]\nthe authors write:\n\"As the authors of [38] will surely agree, such an unusual distorted 3Q state is indeed not common – we carefully searched\nthrough the existing literature for relevant precedent, and found that Mn/Re was one of only a very few examples containing\nrelated physics.\"\nThis is a very misleading statement, and the term \"as the authors will surely agree\" is pure speculation. There are only a\nhandful of papers out there dealing with 3Q states at all (in the sense of nearest neighbor spins having angles close to\n109.5°, not in the sense of skyrmion lattices and such). As far as I know, the first experimental observation was in the\nMn/Re(0001) system by SP-STM, where 1Q and 3Q areas co-exist (due to different possible stacking order of the Mn layer)\nand in addition the 3Q state itself might be distorted (Haldar et al.). The former work is not even cited, although it is quite\nrelevant for the discussion.\nOne cannot say whether something is common or uncommon if there are only a handful of cases in total. Furthermore, one\nmight say that the ideal 109.5°-3Q state will never be observed in real systems, because there are always tiny interactions\npresent which distort it.\n4. The limited spatial resolution of MLD and MCD is an issue which is unfortunately not openly discussed in the manuscript.\nFor a single image data point, the information seems to come from a volume 50-100 nm in depth and about 1000 nm in\ndiameter. I agree with referee #3 that the authors do not convincingly exclude a scenario for phase I where separate 1Q and\n3Q domains co-exist beyond the resolution limit, compared to the proposed laterally homogenous and itself distorted 3Q\nstate (with 1Q component).\n5. Concerning the proposed states for phase I and II, the comments in the rebuttal were quite helpful. I try to be brief here, but\nI think the present manuscript is not clear enough at this point. The authors need to clearly discriminate at least these three\nstates: 1) the ideal, fully symmetric 109.5°-3Q state, 2) the state with one spin up and three spins symmetrically distorted,\nsometimes called the \"threefold symmetric 3Q\" and 3) the 3Q state which is distorted by crystal anisotropy, which looks like\na 2Q state seen from the top. There might be more. In any case, I was confused, because the authors mark the central (red)\npoint on the manifold in Fig. 3a as the three-fold symmetric 3Q. But at this point on the manifold is the ideal 109.5°-3Q. As\nfar as I understand, the state proposed for phase II is not even on the manifold, because it has a higher Heisenberg energy than\nthe ideal 109.5°-3Q.\nTwo more comments: starting from the ideal 109.5°-3Q state, uniaxial anisotropy alone does not produce a net moment.\nSecondly, the authors might want to consider the anisotropic symmetric exchange interaction, which can effectively couple\nthe 3Q state in the way they propose for phase II (1 spin up), see F. Nickel et al., new Ref. [51].\n6. Finally, as the authors are aware, higher-order interactions (HOI) such as the bi-quadratic one are needed to get a non-\ncoplanar state such as the 3Q state, because an interaction is needed which favors 90° angles. I have not found any\nreference or mention of HOIs in the main text of the manuscript. This is odd, because without HOI neither phase I and phase\nII, nor the T-driven transition from phase I to phase III can be understood.", "reviewer_3": "I think the authors have done a good job addressing the referees’ comments. The additional suggestions and remarks from\nReferee 2 are also very helpful. If possible, it would be great for the authors to incorporate Referee 2’s input into the\nmanuscript.\nReferee 2 also raises concern about the possible coexistence of 1Q and 3Q domains, particularly given the limited image\narea. If feasible, the authors could discuss the spatial resolution of their measurements and provide additional imaging data\nto further address this point.\nOverall, I believe this work remains a high-quality study of antiferromagnetic textures using optical techniques. I continue to\nsupport the publication of this paper in Nature Communications.", "reviewer_4": "The authors have provided thoughtful and scientifically grounded clarifications to the previously raised points, particularly\nregarding the distinctions among the triple-Q states, the interpretation of the MLD signal, and the treatment of higher-order\ninteractions.\nIn light of these revisions, I am satisfied that the manuscript has been strengthened, and I would be supportive of its\npublication."}, "author_response": "Reply to Reviewers [manuscript NCOMMS-25-30582A, “Tunable chiral and nematic states in the\ntriple-Q antiferromagnet Co1/3TaS2”]\nApologies to all Reviewers for our delay; travel schedules and experimental neutron beam and magnet\ntime, at Oak Ridge and the Magnet Lab respectively, regrettably delayed our reply. We appreciate the\nsecond reviews and the recommendation to publish in Nature Communications from Reviewers #1 and\n#3, and from the new Reviewer #4. Below we respond in detail to Reviewer #2. For ease of reading, all\noriginal remarks from the Reviewers are reprinted in blue italics. -Yours sincerely, the authors\nReply to Reviewer #1:\nReviewer #1 (Remarks to the Author): publish as is.\nReply: We appreciate the Reviewer’s second review and recommendation to publish in Nature Comm.\nReply to Reviewer #3:\nReviewer #3 (Remarks to the Author): The authors have satisfactorily addressed all my comments. I do\nrecommend the manuscript for publication in Nature Communications.\nReply: We thank the Reviewer for their second review of our manuscript, and for recommending\npublication in Nature Communications.\nReply to Reviewer #4:\nReviewer #4 (Remarks to the Author): The revised manuscript is much clearer and addresses the points\nraised in the first round. For Reviewer 1’s concerns, the phrasing in the Introduction has been adjusted\nand the small MCD signal between TN2 and TN1 is now properly explained as cross-talk.\nFor Reviewer 2’s main objections about Phase I, the authors have clarified why higher-order exchange\ninteractions cannot explain the data, and why an anisotropy-based model is more consistent with neutron\ndiffraction and magnon gap results. The clarification of the MLD signal behavior is also helpful. The\nimproved definitions and terminology (nematicity, distorted vs three-fold symmetric 3Q) make the\npresentation stronger.\nFor Reviewer 3’s questions, the authors have explained why Phase I is not phase coexistence, but a single\ndistorted 3Q state with both chirality and nematicity. The additional discussion of anisotropy, the field-\ninduced phases, and the clarifications in the figures all improve readability.\nOverall, the work convincingly demonstrates how combining MCD and MLD can disentangle chiral and\nnematic orders and provide real-space imaging of magnetic textures. This is a strong contribution, and I\nrecommend acceptance.\nReply: We appreciate Reviewer #4’s positive assessment of our revised manuscript, and for carefully\nevaluating our detailed responses to the original reviews of Reviewers 1-3. We are delighted that\nReviewer #4 finds that we addressed the original questions, concerns, and objections from Reviewers 1,\n2, and 3 in a satisfactory manner, and recommends acceptance in Nature Communications.\nOne small clarification is warranted here, because it comes up again in reviewer #2’s second report: In our\noriginal response to Reviewer #2, we discussed why higher-than-4th-order (namely, 6th-order) exchange\ninteractions did not capture our data. 4th-order interactions, of the type we already consider in our\nmanuscript (the biquadratic term H that is introduced in Eq. 2, in conjunction with single-ion anisotropy\nbq\nand bond-dependent exchange anisotropy of Eq. 3), are a crucial ingredient of our model.\nReply to Reviewer #2:\nReviewer #2 (Remarks to the Author): Despite the substantial and detailed criticism I have raised in the\nfirst review round, Kirstein and coworkers have resubmitted a manuscript with only cosmetic changes. In\ntheir reply, most of my questions are evaded, some given answers are wrong and the clarifications and\nexplanations which were indeed helpful to understand the work, did unfortunately not make it into the\nnew version of the manuscript. \nReply: We admit to being very puzzled by the Reviewer’s point of view, especially since our very\ndetailed (>6 pages) point-by-point response to their original report directly and (we believe) very clearly\naddressed every one of the Reviewer’s criticisms. Re-reading our original response, we maintain that it is\ncomprehensive and accurate (and certainly not “evasive”), and regret that the Reviewer feels otherwise.\nWe note that all other Reviewers (#1 and #3), as well as the new Reviewer #4 who reviewed the original\nreports and evaluated our Response, are satisfied with the revised manuscript and Response, and\nrecommend publication. Moreover, it can be easily confirmed that many substantial changes and\nimprovements to the original manuscript were made, in response to the questions and feedback from\nReviewer #2 (and #1 and #3). Not every single question or remark from the Reviewers resulted in a\nchange to the manuscript, particularly if the concern was already addressed (or the answer already\ncontained) in the original manuscript; in this case it is sufficient to point it out in the Response.\nI do understand the tactics of the rebuttal, but this way of dealing with justified critique is risky, and I\nremind the authors that two of them have already published a wrong conclusion about the magnetic state of the\nvery same system in Ref. [25]. \nReply: Regarding the Reviewer’s comment “...and I remind the authors that two of them have already\npublished a wrong conclusion about the magnetic state of the very same system in Ref [25]”, we\n(meaning all the authors of the current manuscript under consideration) would like to make very clear\nthat, in contrast to the Reviewer’s implication, the conclusion presented in Ref. [25] was not the result of\nunjustified assumptions or inconsistent interpretation. Two of us (PP, JGP) made an analysis in Ref. [25]\nbased on the best information available then – specifically, the 1983 neutron diffraction study of Parkin\nand Brown, which reported a magnetic structure with Q=(1/3, 1/3, 0), which is a coplanar 120° order.\nHowever, subsequent neutron diffraction measurements performed by the authors [PP, JGP, in Ref. 19]\nand by others [Takagi, Ref. 18], revealed a fundamentally different non-coplanar magnetic order\ncharacterized by Q=(1/2, 0, 0), which naturally requires an updated explanation (triple-Q order) based on\nthis new observation. Therefore, the discrepancy arises from the discovery that the original reference data\nused in Ref. [25] turned out to be inconsistent. Unless the Reviewer is suggesting that we, as scientists,\nare generally expected to automatically doubt the validity of peer-reviewed experimental results, this\ncriticism is potentially misleading and, we feel, quite unfair and misplaced.\nTherefore, most of my critique is still valid and I come to the same conclusion, that I do not see\nsubstantially new physics within the manuscript which is conclusively backed up by experiment to\nwarrant publication in Nature Communication. In my opinion, certain inconsistencies of the manuscript\nneed to be addressed before publication in any journal. Whether these problems can be fixed by simply\nwriting more clearly or point toward deeper flaws of the work, I do not know.\nReply: We are sorry to hear this view; we would like to note that all three other Reviewers (#1, #2, and\n#4) were satisfied with our revised manuscript and recommended it for publication. Below we respond,\npoint-by-point, to the Reviewer’s second report.\nFor the future, I encourage the authors to refrain from the overuse of italic, bold and underlining in the\nreview correspondence. Reviewers are quite capable of reading texts carefully without such \"help\".\nReply: OK.\nIn the following, I will not again comment on all problems of the manuscript and shortcomings of the\nrebuttal, but only address the ones I deem most critical.\n1. Let us start with this claim from the rebuttal: \"Moreover, we respectfully emphasize that the new\nfindings include not only the experimental identification of a distorted triple-Q antiferromagnetic phase\nwith co-existing chiral and C_2z symmetries (noted by the Reviewer), but also 1) the demonstration that\nmagneto-optical methods for circular and linear dichroism can function as incisive reporters of these\nantiferromagnetic symmetries, ...\"\nAt face value, this sentence is circular: Fundamentally, you cannot establish a new magnetic technique\nusing a specific spin texture and then also claim that the observed spin texture is a new finding. Now, it\nseems that CoTa3S6 has four different magnetic phases: which spin textures are used to calibrate MLD\nand MCD and which results are actually new?\nReply: We are having difficulty understanding this concern. Hopefully the Reviewer does not think we\nare claiming the invention of circular dichroism (CD) and linear dichroism (LD) methods, for detecting\nHall conductivity (~  ) and in-plane anisotropy (~  -  ), respectively. Definitely, we are not!\nxy xx yy\nBoth LD and CD are well-established optical techniques that have been used for over a century: LD is\nfamously sensitive to order parameters that break in-plane symmetry (such as crystal anisotropy, or\nnematicity), whereas CD is an established reporter of time-reversal-symmetry breaking magnetic\nphenomena that lead to nonzero  (such as ferromagnetism, or the chiral antiferromagnetism we\nxy\nconsider). Crucially, LD was also recently demonstrated to be an incisive probe of three-state magnetic\nnematicity resulting from stripe-like (1Q) order, in the closely-related triangular and hexagonal\nantiferromagnets Fe NbS , FePS , and FePSe [e.g., Nature Materials 19, 1062 (2020); Nano Letters 21,\n1/3 2 3 3\n6938 (2021); Nature Physics 20, 1888 (2024)]. Moreover, CD was shown to detect nontrivial\nantiferromagnetic orders that generate  [e.g., in Mn Sn: Nature Photonics 12,73 (2018); Appl. Phys.\nxy 3\nLett. 114, 032401 (2019)]. All these papers are cited in our manuscript.\nOur work applies these methods to the case of Co TaS , where the existence of four different competing\n1/3 2\nantiferromagnetic phases was recently postulated (e.g., Takagi, Ref. 18), but the underlying magnetic\nsymmetries of these four phases were unknown or incorrectly identified. The significant advance our\nwork makes is the new surprising finding and classification of these various phases based on the presence\nor absence of CD and LD – that is, by the presence or absence of Hall conductivity ( ) and in-plane\nxy\nanisotropy ( -  ), as shown in Figure 2. Let us state that these findings were never reported before\nxx yy\nand are very new. Together with neutron, transport, and magnetization data, and supported by theoretical\nmodeling, this allows us to identify Phase III as a purely single-Q phase (only nematicity, no chirality),\nPhase II as a threefold-symmetric triple-Q phase (only chirality, no nematicity), Phase I as a distorted\ntriple-Q phase (possessing both chirality and nematicity), and Phase VI as having neither chiral nor\nnematic character. This is a central new aspect of the work, not achieved previously. We note that the\nmagnetic structure of Phase I was incorrectly identified in Ref. [18] as being purely chiral, likely because\nRef. [18] was unable to test for nematicity. We also note that Phase III was independently and very\nrecently suggested to have purely single-Q character, based on inelastic neutron scattering studies of the\nmagnetic excitation spectrum [Phys. Rev. X 15, 031032 (2025)], consistent with our findings.\nWe do not see any circular logic here – we used well-established optical probes of underlying symmetry,\napplied them to an interesting new antiferromagnet, and have revealed the presence/absence of underlying\nsymmetries. Together with other data and modeling, this allows us to identify the nature of the various\nmagnetic states in these four phases, which had not previously been achieved. Moreover, being optical\nmethods, we have probed the various antiferromagnetic orders with micron spatial resolution to directly\nimage the chiral and nematic domains, which also had not previously been achieved. Finally, this work\ndemonstrates that in this class of antiferromagnets with chiral and/or nematic character, CD and LD\nfunction as incisive and flexible probes of the underlying magnetic order. We anticipate that many people\nworking in the broad fields of magnetism and optics will be very pleased (and likely surprised) to learn\nthat these optical methods can be used to study certain classes of antiferromagnets, because historically\nantiferromagnetism has evaded detection by such linear-optical means. This work suggests that the full\npower of spectrally-, spatially-, and temporally-resolved optical techniques can be brought to bear on such\nantiferromagnets, echoing decades of similar progress in optical studies of ferromagnets.\n2. Another problem puts a main result of the work into question. Here, again a sentence from the rebuttal:\n\"Finally, we note that the MLD signal (a reporter of 1Q order parameter) emerges and grows throughout\nPhase III, achieving a maximum at the Phase I – Phase III boundary, which remains approximately\nconstant as the sample is cooled throughout Phase I (i.e., the MLD is not larger in Phase I as the\nReviewer suggests). Figure 1g shows this reasonably well, and Supplemental Fig. S2d shows this\nextremely clearly.\"\nThis description is wrong! In both plots the highest MLD signal is found at the lowest temperature of\nT=4.2 K, not at T=26.5 K. But let us assume for the sake of argument, that the MLD signal is constant in\nphase I; even then the problem arises that the MLD signal interpreted as a direct measure of 1Q\ncharacter is INCONSISTENT with the interpretation of phase III as a pure 1Q state and phase I as a\nmixed 1Q-3Q state. When the state becomes more 3Q-like in phase I, the 1Q character needs to drop.\nThere seems to be a further temperature dependence which is not discussed and not accounted for.\nReply: Below we reprint Fig. 1g and Fig. S2d from the manuscript. As stated in our original response,\nthe experimentally-measured linear dichroism grows throughout Phase III (T=37 to 26.5K)... and remains\napproximately constant as the sample is cooled throughout Phase I (T<26.5K). We defer to the Editors\nand other Reviewers whether the exactness of our statement is actually an important point.\nRegardless, this all seems somewhat beside the point in our view, because there is no law or requirement\nthat the linear dichroism signal – a reporter of broken in-plane symmetry and nematic character – must\ndrop (or change at all) upon transitioning from Phase III (single-Q) to Phase I (distorted triple-Q) at finite\ntemperature. (Note: this would not be true at T=0 where order parameters are maximal and saturated;\nperhaps the Reviewer has the T=0 case in mind). At T=26K (and below), the amplitude of the order\nparameters are still growing. Linear dichroism (LD) is simply a reporter of broken C symmetry, and will\n3\ntherefore be non-zero if the amplitudes of the three single-Q wave vectors ( ,  ,  ) are unequal -- for\n1 2 3\ndetails see Equation 1, and Fig. 3. Likely, the LD signal scales with their difference [e.g., as something\nlike  2 – ( 2 +  2)/2, although the actual relationship is not known]. If  ,  , and  grow with\n1 2 3 1 2 3\ndecreasing temperature approximately following the notional sketch below (left plot), then the LD signal\nwill indeed grow throughout Phase III and then remain approximately constant upon cooling into Phase I.\nCrucially, we emphasize that such a temperature dependence of  is, in fact, already strongly suggested\n1\nand motivated by the experimentally-measured neutron diffraction intensity that is shown in Fig. 1c of the\nmain text (right plot). The antiferromagnetic diffraction intensities should scale with ||2. Thus, the\ni\nmeasured LD – a measure of nematicity – is not required to drop when the magnetic order parameter\nacquires some triple-Q character upon cooling through T=26.5K and entering Phase I.\nWe do find it very interesting that the emergence and growth of the distorted triple-Q phase (Phase I) does\nseem to approximately maintain its degree of nematicity (as measured by MLD), but at the moment we do\nnot know whether this is coincidence or points to something deeper.\n3. As an answer to my comment #4 [page 2, \"coexistence of single-Q and triple-Q orderings within the\nsame system – an uncommon feature in real materials.\": Firstly, a distorted 3Q state has been proposed\nalready in Ref. 38. Secondly, 1Q, 3Q and transient 2Q states co-exist in the Mn/Re(0001) system due to\ndifferent stacking order and inside domain walls, respectively] the authors write:\n\"As the authors of [38] will surely agree, such an unusual distorted 3Q state is indeed not common – we\ncarefully searched through the existing literature for relevant precedent, and found that Mn/Re was one of\nonly a very few examples containing related physics.\"\nThis is a very misleading statement, and the term \"as the authors will surely agree\" is pure speculation.\nThere are only a handful of papers out there dealing with 3Q states at all (in the sense of nearest neighbor\nspins having angles close to 109.5°, not in the sense of skyrmion lattices and such). As far as I know, the\nfirst experimental observation was in the Mn/Re(0001) system by SP-STM, where 1Q and 3Q areas co-\nexist (due to different possible stacking order of the Mn layer) and in addition the 3Q state itself might be\ndistorted (Haldar et al.). The former work is not even cited, although it is quite relevant for the\ndiscussion.\nOne cannot say whether something is common or uncommon if there are only a handful of cases in total.\nFurthermore, one might say that the ideal 109.5°-3Q state will never be observed in real systems, because\nthere are always tiny interactions present which distort it.\nReply: This debate (see original Reviews and our Response) seems to center on a definition of the word\n“uncommon”. To avoid any misunderstanding, we’ve removed the phrase “...an uncommon feature in\nreal materials” from the Introduction section of the manuscript, with no loss of meaning or generality.\nFollowing the Reviewer’s comment, we also added the experimental work as a new reference.\n4. The limited spatial resolution of MLD and MCD is an issue which is unfortunately not openly discussed\nin the manuscript. For a single image data point, the information seems to come from a volume 50-100\nnm in depth and about 1000 nm in diameter. I agree with referee #3 that the authors do not convincingly\nexclude a scenario for phase I where separate 1Q and 3Q domains co-exist beyond the resolution limit,\ncompared to the proposed laterally homogenous and itself distorted 3Q state (with 1Q component).\nReply: We are puzzled why the Reviewer claims that the “the limited spatial resolution of MLD and\nMCD is an issue which is unfortunately not openly discussed in the manuscript”. The spatial resolution of\nour microscope (approx. 1 micrometer) is very explicitly stated in the Methods (Section IV.B). This is\nclose to the fundamental diffraction limit of red/infrared light (as is the case for most optical microscopy\nstudies with decent optics). Optical microscopy obviously cannot probe what is happening at nanometer\nor atomic length scales. But we can, and do, say that the images of Fig. 4 and S5 quite clearly show large\nhundreds-of-micron lengthscale spatial patterns having discrete C_2z (nematic) character, suggesting\nlarge domains with nematicity. Moreover, the images in Fig. 5 also show, in the exact same regions,\nsmaller few-micron scale patterns and domains having chiral character. As we have explained in detail\n(please see our original response to Reviewer #3, which we hope the Reviewer read), these experimental\nobservations, together with the fact that the images exhibit uniform chiral character when poled in tiny\nmagnetic fields (Fig. 5b,d), are consistent with a laterally-homogeneous distorted triple-Q state.\nTo restate our reasoning more clearly: if Phase I truly consisted of separate 1Q and 3Q domains, then the\nresulting nematic domain pattern–extending over several hundred micrometers–should have appeared in\nthe MCD images as regions with void signals. However, no such features are observed.\nWe note that both Reviewers #3 and #4 are satisfied with our response to Reviewer #3’s related question\nabout co-existing 1Q and 3Q domains. Nonetheless, to make it perfectly clear that optical microscopy\ndoes not probe below the diffraction limit of light, we amend the penultimate sentence of the manuscript’s\nlast paragraph to say “These images confirm that, to within our microscope’s 1 micron spatial resolution,\nPhase I is not a phase-separated mixture of coexisting single-Q and triple-Q ground states”.\n5. Concerning the proposed states for phase I and II, the comments in the rebuttal were quite helpful. I try\nto be brief here, but I think the present manuscript is not clear enough at this point. The authors need to\nclearly discriminate at least these three states: 1) the ideal, fully symmetric 109.5°-3Q state, 2) the state\nwith one spin up and three spins symmetrically distorted, sometimes called the \"threefold symmetric 3Q\"\nand 3) the 3Q state which is distorted by crystal anisotropy, which looks like a 2Q state seen from the top.\nThere might be more. In any case, I was confused, because the authors mark the central (red)\npoint on the manifold in Fig. 3a as the three-fold symmetric 3Q. But at this point on the manifold is the ideal 109.5°-3Q. As\nfar as I understand, the state proposed for phase II is not even on the manifold, because it has a higher Heisenberg energy than\nthe ideal 109.5°-3Q.\nReply: We thank the Reviewer for raising this point which, we now recognize, will benefit from\nadditional clarification in the paper. We agree that a clearer distinction is needed especially between case\n(2) and an idealized fully symmetric configuration with 109.5° angles. The revised and updated\nmanuscript now explicitly differentiates:\n(1) Ideal three-fold symmetric 3Q state. This state has equal Fourier amplitudes (Δ = Δ = Δ ),\n1 2 3\nmutual spin angles of 109.5°, and zero net magnetization. This state preserves three-fold rotational\nsymmetry and corresponds to the red central point on the multi-Q manifold in Fig. 3a.\n(2) Canted three-fold symmetric 3Q state (Phase II). This state also has Δ = Δ = Δ and therefore\n1 2 3\nretains three-fold rotational symmetry, but possesses an additional small net magnetization due to\nfield-induced (and/or anisotropy-induced) canting. This net moment is readily described by\nincluding a small ferromagnetic Fourier component 𝐒̃ to the three-dimensional manifold of\nstate is the Phase II ground state when a finite field or bond-dependent anisotropic exchange term\nH (which corresponds to the “anisotropic symmetric exchange” mentioned by the Reviewer in\n±±\nthe following comment) is included.\n(3) Distorted 3Q state (Phase I). This state breaks three-fold symmetry through unequal Fourier\namplitudes (Δ > Δ = Δ ), giving rise to nematicity (as well as chirality).\n1 2 3\nWe emphasize that cases (1) and (2) share a key property: both maintain Δ = Δ = Δ and preserve three-\n1 2 3\nfold symmetry, and are therefore chiral but not nematic, regardless of the precise values of their mutual\nspin angles. In this study, the presence or absence of three-fold symmetry –and hence nematicity, as\nmeasured by linear dichroism– is the primary focus of distinguishing criterion. We also stress that our\ntheoretical framework already systematically captures the evolution from case (1) to case (2) through the\ndevelopment of a net magnetic moment 𝐒̃ ; please see Eq. S7 in the Supplementary Information.\nexplicitly distinguish cases (1) and (2) (see new Fig. S6d and its caption), and provides a more detailed\ndiscussion in the main text:\n(Page 4) “The case Δ = Δ = Δ (red circle in Fig. 3a) yields a chiral and three-fold symmetric state,\n1 2 3\nwhere the four spin sublattices align along the principal axes of a regular tetrahedron with 109.5° mutual\nangles. Phase II corresponds to this case, but with the addition of a small net out-of-plane magnetization\narising from spin canting induced by the applied magnetic field. This canted configuration is naturally\ndescribed by including a constant (ferromagnetic) Fourier component 𝑺̃ to the three-dimensional\n0\nmanifold in Equation (1); for additional details see Supplemental Note II.A. (Equation S7). Importantly,\neven with canting, the three-fold symmetry remains intact (see Fig. S6d) and therefore we refer to Phase\nII as a “three-fold symmetric triple-Q” state through this work, to emphasize its rotational symmetry\n(though not implying a perfect 109.5° tetrahedral configuration.)”\nTwo more comments: starting from the ideal 109.5°-3Q state, uniaxial anisotropy alone does not produce\na net moment. Secondly, the authors might want to consider the anisotropic symmetric exchange\ninteraction, which can effectively couple the 3Q state in the way they propose for phase II (1 spin up),\nsee F. Nickel et al., new Ref. [51].\nReply: We appreciate the Reviewer’s suggestion, and we agree that uniaxial anisotropy alone does not\nproduce a net moment (our manuscript does not claim this). And we agree that anisotropic symmetric\nexchange can do so – in fact our model already does incorporate a term along these lines, as suggested by\nNickel et al [Ref 51]; this is precisely the “bond-dependent exchange anisotropy” term, H ,\n±±\nthat was introduced in Equation (3) of the main text (and discussed in more detail in the Supplement).\n6. Finally, as the authors are aware, higher-order interactions (HOI) such as the bi-quadratic one are\nneeded to get a non-coplanar state such as the 3Q state, because an interaction is needed which favors\n90° angles. I have not found any reference or mention of HOIs in the main text of the manuscript. This is\nodd, because without HOI neither phase I and phase II, nor the T-driven transition from phase I to phase\nIII can be understood.\nReply: Higher-order interactions of the type noted by the Reviewer (e.g., bi-quadratic) are already\nincorporated into our model from the very beginning. We refer the Reviewer to section II.D. of the\nmanuscript, Equations (2) and (3) in particular. Equation (2) and its preceding paragraph explicitly\nintroduce the bi-quadratic term H that is a crucial ingredient of our model. As the reviewer notes, this bi-\nbq\nquadratic term is a higher-order (four-spin) interaction, although the text does not explicitly refer to it as\nsuch. The revised manuscript now does. See also Supplemental Section II.B., which is entitled “Four-\nspin interactions”. Together with H , the single-ion anisotropy term H and the bond-dependent\nbq SI\nexchange anisotropy H (both defined in Equation 3) successfully capture the phase transitions between\n±±\nphase I and II, and the temperature-driven phase transitions between phase I and III – as already shown in\nFigure 3f and 3g of the main text and Figs. S7 and S8 of the Supplementary Material.\nChanges to manuscript (these new updates appear in orange color in the revised manuscript):\nAmend the penultimate sentence of the manuscript’s last paragraph to say “These images confirm that, to\nwithin our microscope’s 1 micron spatial resolution, Phase I is not a phase-separated mixture of\ncoexisting single-Q and triple-Q ground states”.\nTo avoid any potential misinterpretation, we’ve removed the phrase “...an uncommon feature in real\nmaterials” from the Introduction section of the manuscript, with no loss of generality.\nSentence preceding Equation (2): Add “...we find that a simple real-space biquadratic (i.e., four-spin)\nterm H_bq captures...”. And in the sentence preceding Supplemental Equation (S1): Add “...that includes\nbilinear and higher-order four-spin interactions:”\nAdded a new citation: J. Spethmann, S. Meyer, K. von Bergmann, R. Wiesendanger, S. Heinze, A.\nKubetzka, Discovery of Magnetic Single- and Triple-q States in Mn/Re(0001), Phys. Rev. Lett. 124,\n227203 (2020).\nNew Supplemental Figure S6d (and caption), which explicitly compares and discusses the threefold\nsymmetric 3Q ‘canted’ spin configuration of Phase II with the idealized and perfectly tetrahedral spin\nconfiguration.\nMain text, page 4, Section II.D., second paragraph is updated to read: “The case Δ = Δ = Δ (red circle\n1 2 3\nin Fig. 3a) yields a chiral and three-fold symmetric state, where the four spin sublattices align along the\nprincipal axes of a regular tetrahedron with 109.5° mutual angles. Phase II corresponds to this case, but\nwith the addition of a small net out-of-plane magnetization arising from spin canting induced by the\napplied magnetic field. This canted configuration is naturally described by including a constant\n(ferromagnetic) Fourier component 𝑺̃ to the three-dimensional manifold in Equation (1); for additional\n0\ndetails see Supplemental Note II.A. (Equation S7). Importantly, even with canting, the three-fold\nsymmetry remains intact (see Fig. S6d) and therefore we refer to Phase II as a “three-fold symmetric\ntriple-Q” state through this work, to emphasize its rotational symmetry (though not implying a perfect\n109.5° tetrahedral configuration.)”"}]

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