Discovery of van Hove singularities: electronic fingerprints of 3Q magnetic order in a van der Waals quantum magnet

Domain: physics
Published: 2026-03-07
Editorial outcome: major_revision

Abstract
Magnetically intercalated transition metal dichalcogenides are emerging as a rich platform for exploring exotic quantum states in van der Waals magnets. Among them, Co TaS has attracted intense interest following the recent discovery of a distinctive 3Q magnetic ground state and a pronounced anomalous Hall effect below a critical doping of x 1/3, both intimately tied to cobalt concentration. To date, direct signatures of this enigmatic 3Q magnetic order in the electronic structure remain elusive. Here we report a comprehensive doping dependent angle resolved photoemission spectroscopy study that unveils these long-sought fingerprints. Our data reveal an unexpected inverse-Mexican-hat dispersion along the K-M-K ′ direction, accompanied by two van Hove singularities. These features are consistent with theoretical predictions for a 3Q magnetic order near three-quarters band filling on a cobalt triangular lattice. These results provide evidence of 3Q magnetic order in the electronic structure, establishing TMD van der Waals magnets as tunable materials to explore the interplay between magnetism and topology.

Introduction
The family of transition metal dichalcogenides (TMDs) MX (M:transition-metal atoms, 2 X:chalcogen atoms) host layered structures with van der Waals (vdW) gaps and exhibit a plethora of exotic properties, such as superconductivity[1], charge density waves (CDW)[2], and topological electronic states[3]. Magnetically intercalated TMDs, as an important class of van der Waals magnets, open a new paradigm in TMD research and provide a versatile platform to investigate the interplay between magnetic order and electronic properties. This newly discovered variant of TMDs is formed by inserting magnetic ions into the vdW gaps of 2H-TMDs. The intercalated magnetic layers can induce magnetic moments that couple with the host conducting layers, giving rise to a wide class of interesting magnetic orders and electronic properties. One class of these new materials is TM (Ta,Nb)S (TM: 3d 1/3 2 transition-metal atoms), where the TM ions form a triangular lattices with a √3 √3 su- × perlattice with respect to the unit cell of the host compounds (Ta,Nb)S . Depending on the 2 host TMD layers and intercalants, highly tunable magnetic orders and interesting physical phenomena have been reported[4]. For example, V or Ni intercalation leads to a ferromag- netic (FM) order within the V or Ni plane and an antiferromagnetic (AFM) order along the out-of-planedirection[5–7]; theintercalationofCrorMnleadstoapredominantFMorder[8], characterized by a chiral helimagnetic structure in (Cr,Mn) NbS [8–10]; Fe NbS [11–13] 1/3 2 1/3 2 shows AFM order with the magnetic moments predominantly aligned along the c-axis, while Fe TaS shows FM order along the c-axis[14]; in the case of Co intercalation, Co NbS 1/3 2 1/3 2 shows collinear in-plane AFM order and weak c-axis FM order[15, 16]. Moreover, it is observed that slight changes in the stoichiometry of the intercalated ions can modify the magnetic order and transport properties. For example, in Co NbS , the Co content can x 2 tune the magnitude of the anomalous Hall effect in the proximity of x=1/3[17].

The report of the unusual magnetic order with three non-coplanar ordering wavevector (the 3Q antiferromagnetic order)[18, 19] coexisting with a pronounced doping-dependent anomalous Hall effect in Co TaS [20] has spurred a lot of interest in the study of the Co- x 2 intercalated TaS family. It is believed that the anomalous Hall effect has a topological 2 origin, namely, the Berry curvature induced by the scalar spin chirality in the non-coplanar magnetic order[18]. Importantly, slight variations in cobalt doping across the critical point x=1/3 has dramatic effects on the magnetic order and the anomalous Hall conductance of Co TaS [20]. As of today, very little is known of the fingerprints of this peculiar magnetic x 2 order on the electronic structure of this material, and more importantly, on the potential topological nature of such order.

Here, we present a comprehensive angle-resolved photoemission spectroscopy (ARPES) study of the Co doping-dependent electronic structure of Co TaS in the range of x =0.29 x 2 ∼ 0.36 across the transition from 3Q to helical AFM order and compare the results with the pristine 2H-TaS . The data reveal that, in addition to electron-doping the TaS -derived 2 2 bands, the Co intercalation induces new shallow bands originating from Co-orbitals. Furthermore, within the studied x range, we find the Co intercalants mainly dope the TaS - 2 derived bands and have little effect on the Co-derived bands. Combined with potassium deposition experiments, we identify a high density of states in the Co-derived bands. Calculations on a triangular lattice, both without and with 3Q order, reveal that the Co-derived bands with 3/4-filling host van Hove singularities with divergent density of states near the Fermi level, consistent with the doping-dependent results. Furthermore, the Fermi surface and band structure are modified by the presence of the 3Q order, with corresponding signatures observed in our data.

Results
Structure and physical property of Co TaS . The crystal symmetry of Co TaS 1/3 2 1/3 2 is the non-centrosymmetric space group P6 22 (No.182) sketched in Fig. 1a. Magnetic Co 3 atoms are intercalated into the van der Waals gap in the layered compound 2H-TaS . The 2 Co intercalants form a √3 √3 triangular lattice rotated by 30 with respect to the 1 1 ◦ × × unit cell of TaS (Fig. 1b). The top view of the crystal structure in Fig. 1b indicates that Co 2 lies underneath or above the Ta atoms occupying the 2c Wyckoff positions. In the nearly- stoichiometric samples with x =1/3, the 2c sites are fully occupied; for lower/higher doping, vacancies at the 2c/occupancy at the 2b sites are observed[21]. Experimentally, tuning the Co doping across x=1/3 can drastically change the bulk properties of Co TaS . Specifically, x 2 as shown in Fig. 1c, Co TaS with x 0.325 exhibits two magnetic phases, single-Q AFM x 2 ≤ order at higher T and triple-Q (3Q) A FM order at lower T (see 3Q magnetic order in Fig. 1d), where the latter coexists with a pronounced anomalous Hall effect (AHE)[19]. For x>0.325, a coplanar helical AFM order with zero Hall conductivity emerges[20]. From here onward we will refer to x=0.33 as the critical doping x .

Comparative Analysis of the Electronic Structures of Undoped and Cobalt- Doped 2H-TaS . Figure 1e and 1f compare the low-temperature (7K) Fermi surfaces 2 of the undoped compound 2H-TaS with that of Co TaS . The experimentally extracted 2 0.32 2 Fermisurfacesheetsareplottedintheupper-leftquadrant. For2H-TaS (Fig.1e), theFermi 2 surfaces consist of two circular hole-like pockets centered at Γ and K , and a dumbbell- 0 0 shaped electron pocket around M . These Fermi surfaces originate from the bands labeled 0 α and β, respectively, which are a pair of spin-split bands due to broken inversion symmetry[22]. The charge-density-wave (CDW) order with a 3 3 superlattice modulation below × T 75K[23, 24] opensgapson theβFermi surface(marked bythearrow), in agreement CDW ≈ with density functional theory[2, 25–27]. In the case of Co TaS , two Fermi surfaces are 0.32 2 observed (see Fig. 1f and Supplementary Fig. S1), and are associated with different surface terminations (TaS - and Co-terminations). We differentiate these two terminations with 2 spatially resolved core-level spectroscopy and the respective band structures along the high- symmetry directions as shown in Supplementary Fig. S1. This is similar to the V atoms doped NbS , where the V-terminations leads to more electron-doped band structures than 2 the TaS -termination[28]. All band structures presented here are from the TaS -terminated 2 2 surface, as it provides much sharper band structures. We next describe the evolution of the electronic structure upon Co intercalation. In detail, the α hole-like pocket centered at Γ 0 is significantly reduced in size compared to that of the undoped sample. In contrast, the α pocket at K exhibits a much smaller shrinkage from 2H-TaS to Co TaS . This suggests 0 2 0.32 2 that the intercalation causes pocket-selective doping of the Fermi surfaces. In addition, the Fermi surface associated with the β band evolves from a dumbbell-shaped electron pocket around M 0 in 2H-TaS 2 to two rounded hole-like pockets centered at Γ 0 and K 0 . Most interestingly, upon Co intercalation, a shallow trigonally-warped electron-like pocket γ appears at the corners of a 1 1 hexagonal Brillouin zone (dashed lines in Fig. 1f), whose ori- √3 × √3 entation is rotated by 30 relative to that of the original 1 1 Brillouin zone of 2H-TaS . ◦ 2 N × This pocket likely originates from Co-derived bands, as it is centered at the corners of the Co-lattice Brillouin zone. For simplicity, we refer to these bands as the Co-derived bands hereafter. However, the γ pocket is absent in the Co-terminated surface (see Supplementary Fig. S1). One might wonder why the Co-terminated surface did not show the Co-derived band. A scanning tunneling microscopy (STM) study on a similar material, Cr NbS , 1/3 2 provided a plausible explanation: an ideal cleavage is expected to split the Co intercalants equally between the two cleaved surfaces, resulting in a disordered Co configuration, which renders the Co-derived bands ill-defined[29].

Fig. 1g and 1h compare the band structure of undoped and Co-intercalated samples along the high-symmetry direction M -Γ -K . The direct comparison reveals two striking 0 0 0 differences: (i) The α and β bands are significantly electron-doped in Co TaS , signifying 0.32 2 that Co acts as electron dopant by transferring some of its 3d electrons to the TaS layers. 2 (ii) A shallow electron-like band emerges at K (γ ) together with a near-E hole-like band K F at M (γ ), giving rise to the triangular Fermi pockets (γ).

These findings are quantified in Fig. 1j and 1k, where the experimental band dispersions of 2H-TaS and Co TaS were obtained by fitting the positions of the Lorentzian-shaped 2 0.32 2 peaks of the momentum distribution curves (MDCs) and energy distribution curves (EDCs), as shown in Supplementary Fig. S2. The fitting results reveal the following effects of Co intercalation: (i) Co not only electron-dopes the spin-orbit-split α and β bands overall, but also modifies their dispersions anisotropically, indicating non-rigid shifts. Quantitatively, along the Γ -K direction, the shift of the α band minimum is 260meV, while 0 0 ≈ along the Γ -M direction the shift of the α and β bands is much larger ( 340meV at 0 0 ≈ the band bottom and 360meV at M ), as extracted from EDC spectra in Supplementary 0 ≈ Fig. S3. Density Functional Theory (DFT) calculations for 2H-TaS reveal that the ob- 2 served 300meV band shift can be reproduced by adding one electron per TaS unit cell, 2 ∼ in good agreement with a simple electron counting picture of Co TaS consisting of alter- 1/3 2 nating [Co ]+ and [TaS ] layers (see Supplementary Fig. S4). (ii) Spin-orbit coupling is 1/3 2 − enhanced, as supported by the enlarged momentum splitting between the α and β bands from 0.06 to 0.18˚A (see MDCs at E in Fig. 1j and 1k, and Supplementary Fig. S3)[32]. F (iii) A shallow electron-like band (γ ) appears at K within 40meV of E (Fig. 1h and K F 1k). (iv) A hole-like feature (γ ) is observed around M within 30meV of E (see also M F 1h and 1k). This result is reminiscent of the magnetically intercalated NbS compound, 2 where a shallow electron pocket develops due to the Co interstitials[33, 34]. While it may be tempting to attribute the γ and γ bands to unoccupied states of pristine TaS , band- K M 2 structure calculations for 2H-TaS show no evidence of such states (see Fig. 1l). We further 2 performed DFT calculations for Co TaS and identified highly-dispersive bands primarily 1/3 2 derived from Ta orbitals, together with near-E flat bands mainly derived from Co orbitals F (see Supplementary Fig. S5). This result strongly supports the conclusion that the γ and K γ bands arise from the Co intercalants. (v) In addition to these near E bands, a highly M F dispersive band around Γ (ϵ) emerges.

To gain further insights into the orbital characters of these bands, Fig. 1i presents the band dispersions of Co TaS measured using linearly vertically (LV) polarized light. Com- 0.32 2 pared with the linearly horizontally (LH) data (Fig. 1h), the LV spectra show three main differences: the disappearance of the near-E γ and γ bands, a pronounced reduction of F K M spectral weight in the highly-dispersive α and β bands, and a significant enhancement of the spectral intensity of the ϵ band. Based on the matrix-element analysis and the DFT- calculated orbital projected band structures (see Supplementary Note 1 and Supplementary Fig. S6-S7)[30, 31], we infer that the near-E γ and γ bands mainly originate from F K M Co 3d orbitals, the highly-dispersive α and β bands are dominated by Ta 5d and Ta z2 z2 5d /d orbitals, and the ϵ band is primarily contributed by d /d and d /d xy x2 y2 xz yz xy x2 y2 − − states. In addition, the LV spectra reveal a non-dispersive band near -0.75eV (see the arrow in Fig. 1i), which could be associated with an impurity-like localized state.

Evolution of Band Structure with Co Doping and Potassium Deposition. Figure 2 summarizestheelectron-dopingeffectproducedbyvaryingCocompositionandbysuccessive potassium depositions. Figure 2a presents the Co-doping-dependent spectra along the high symmetry direction M -Γ -K , revealing a continuous downward shift of the α and β 0 0 0 bands with increasing doping. Consistent with this observation (Fig. 2c), the peak positions of the raw EDCs at M, which mark the bottoms of the α and β bands, shift toward higher binding energy as the doping increases from x=0.29 to 0.36. The overall energy shift of the α and β bands is 50meV. The observed doping dependence of the α-band minimum ∼ is well reproduced by a rigid-band-shift approximation applied to the TaS band structure 2 (see Supplementary Fig. S4). In contrast, the extrema of the γ and γ bands exhibit K M only weak doping dependence (Fig. 2a). In line with this, the near-E EDC peaks of the F γ and γ bands move only slightly (Fig. 2c and Supplementary Fig. S8). Specifically, as M K indicated by the black curves in Fig. 2e, small upturns ( 3 and 8meV for the γ and γ ∼ ∼ M K bands, respectively) are observed at x=0.36. Taken together, these results indicate that Co doping induces a much larger electron-doping effect on the TaS -derived bands than on the 2 Co-derived bands.

Tovalidatetheintrinsicnatureoftheobserveddopingdependenceandeliminateextrinsic sources, such as inhomogeneous doping level or variations in sample quality across different doping levels, we report in Fig. 2b the evolution of the band structure upon tuning the chemical potential on the same sample. This is achieved via in situ doping by depositing of alkali-metal atoms, a technique widely used in ARPES to study doping-dependent electronic structures in a variety of materials[35]. The alkali metal atoms donate electrons due to their low ionization potential, causing the exposed surface to be electron-doped. The EDCs obtained at M for Co TaS before and after multiple rounds of potassium(K) depositions 0.29 2 are shown in Fig. 2d, and the band minima as a function of K dosing are represented by the orange curves in Fig. 2e. Specifically, after four rounds of K deposition, we observe a downshiftoftheαandβbandscomparabletothatinducedbyincreasingtheCodopingfrom x=0.29 to 0.36. Moreover, the γ band remains nearly unchanged within the experimental M resolution.

These findings lead to two key insights: (i) The fact that the Co-derived γ band barely M shifts upon K dosing suggests a high density of states. (ii) Increasing the Co doping to x=0.36 causes a small upward shift in the γ and γ bands, in contrast to the nearly K M unchanged band extrema observed under K deposition. This suggests that Co doping affects the Co-derived γ and γ bands beyond simple electron doping. This motivates the K M question: could this be a manifestation of changes in magnetic order as a function of Co doping?

Signature of 3Q magnetic order on the Fermi surface. A tempting assertion is that the effects on the γ and γ bands stem from changes in the non-coplanar (3Q) K M magnetic order. Since DFT struggles to accurately describe the dispersion of the near-E F flat bands, primarily due to strong electron-electron correlations and the presence of local magnetic moments, it cannot provide a reliable description of our system. At the same time, the near-E flat bands are dominated by Co 3d orbitals, whereas the dispersive α F and β bands mainly originate from TaS -derived states (see Supplementary Fig. S5). Given 2 both the limitations of DFT and the distinct orbital characters of these states, we adopt a phenomenologicaltight-bindingmodelonaCotriangularlatticetoaccountfortheflatbands and their coupling to local magnetic moments. We simulate the Co-derived bands using a √3 √3 triangular lattice model formed by the Co atoms, as illustrated in Fig. 1. In the × absence of the 3Q magnetic order—that is, in the nonmagnetic state (exchange coupling constant J =0)—the Fermi surface of the 3/4-filled band with nearest-neighbor hopping is shown in Fig. 3a. Here, the triangular electron-like pockets centered at K touch each other at M, where a van Hove singularity with a divergent density of states is located. The existence of a VHS can naturally explain the nearly electron-doping-independent nature of the γ and γ bands, observed in Fig. 2. At 3/4-filling, the 3Q magnetic order is predicted K M to emerge due to Fermi surface nesting[36, 37]. The nesting wave vectors are shown by the black solid arrows in Fig. 3a and 3c (see Supplementary Fig. S9). In the presence of the 3Q magnetic order, two significant modifications to the Fermi surface occur (Fig. 3b): (i) When the Fermi level passes through the band top, the triangular Fermi surface shows a notable shrinkage, as a single VHS at M evolves to two VHSs located away from M. (ii) The magnetic order enlarges the unit cell and shrinks the Brillouin zone, inducing band folding from M to Γ (see Supplementary Fig. S9).

Figures 3d-3f show the experimental Fermi surfaces for three doping levels: x=0.29, 0.32(both below x ), and x=0.34(above x=x ). Additional Fermi surfaces for other doping c c levels are provided in Supplementary Fig. S10. In samples with x<x , i.e., in the presence of c the 3Q magnetic order, no signatures of band folding are observed. This absence could be attributed to the coherence factor, which dramatically weakens the intensity of the folded bands—a phenomenon commonly seen in many reconstructed Fermi surfaces[38]. The comparison shows a clear enlargement of the triangular Fermi pockets (γ) as the doping increases. This observation is illustrated in Fig. 3g, where the γ pockets are extracted from the experimental Fermi surfaces for all doping levels. Such enlargement is evident in the MDCs at E along the K-M direction, where the peak-to-peak distance clearly increases as F the doping rises from x=0.29 to x=0.36.This trend is further evidenced by the variation in the relative spectral weight on the Fermi surface between the tip of the γ pocket and the M point, as shown in Fig. 3h, where a transition across x is observed. We attribute the c observed enlargement of the γ pockets to the weakening of the 3Q magnetic order above x , c rather than to increased Co doping. This assignment is supported by the observation that the γ band bottom does not shift downward with doping (see Fig. 2), which is inconsistent with a simple band-filling picture. Instead, the weakening of the 3Q order alters the Fermi surface topology, naturally leading to the enlarged γ Fermi pockets.

Signature of 3Q magnetic order on the band dispersions. Further evidence of the 3Q magnetic order is provided by the near-E band dispersion along the K-M-K direction. F ′ The second derivative images of the experimental spectra for different doping levels across x are shown in Fig. 4a. Two clear qualitative changes are observed near the M point as the c doping increases: (i) an increase in the spectral weight at the M point relative to the Fermi momentumk , asindicatedbytheredandbluearrowsintheleftmostpanel; (ii)achangeof F curvature around M from electron-like to hole-like. To quantify the first feature, we extract the momentum-resolved EDCs, as shown in Supplementary Fig. S11. In Fig. 4b, we plot the doping-dependent ratio of EDC peak intensities at k and at M. An obvious decrease in the F ratio across x suggests a spectral weight transfer from the tip of the γ pocket to the M point. c To quantify the second feature, in Fig. 4c-d and 4e-f we show zoom-in views of the raw spectra and representative EDCs for x=0.29 and 0.36, respectively. For x=0.29, a shallow but distinct dip in the dispersion is observed at M, accompanied by two band tops nearby. This is even more evident in the EDCs where the peak position at the M point appears at a lower energy than in the EDCs away from M (Fig. 4e), in contrast to the spectra for x=0.36, where the EDC at M has the same peak position as other EDCs(Fig. 4f). The extracted dispersions in Fig. 4g-h clearly summarize these observations, showing an evolution from an inverse Mexican-hat-like dispersion in x=0.29 to a hole-like dispersion in x=0.36. To understand the origin of such evolution, in Fig. 4i-j we report the calculated band dispersion along K-M-K with and without the 3Q order, respectively. The calculation reveals that, ′ in the absence of the 3Q order, the dispersion is hole-like, and hence there is a single VHS at the M point (Fig. 4j). In contrast, with the 3Q order, the dispersion evolves into an inverse Mexican-hat-like shape, characterized by two VHSs positioned away from M along the K-M-K direction(Fig. 4i). We further tested the robustness of this result by varying ′ hopping amplitude t and coupling constant J, and found that the inverse Mexican-hat-like dispersion persists as long as the 3Q order is present (see Supplementary Fig. S12). Our experimental band dispersion along K-M-K for samples with x<x closely resembles the ′ c inverse Mexican-hat-like dispersion predicted in Fig. 4i, strongly suggesting that the band modification is induced by the 3Q magnetic order.

It is noteworthy that the band dispersions for samples with x>x (Fig. 4h) cannot be c simply interpreted using the simulated band structure without the 3Q order(Fig. 4j), as a reliable prediction of the electronic structure in this region is hindered by the uncertainty in the moment arrangement of the “helical” magnetic order. Nonetheless, several key features serve as circumstantial evidence for the alleged transition from 3Q to helical magnetic order. First, the dispersion dip at M is completely suppressed in x=0.34 and 0.36 (see Supplementary Fig. S11 for raw spectra and EDC stacks at all dopings). Second, the spectral-weight transfer, as summarized in Fig. 3h and 4b, from the tip of the triangular pocket for x<x , c to the M point for x>x , is suggestive of the displacement of the VHSs from Fig. 4i to c Fig. 4j. Although the electronic structure in samples with x>x can not be adequately c described by the model without incorporating 3Q order, our results, when taken at face value, qualitatively align with the doping-tuning experiments conducted via Co composition variation[20] and ionic gating[39]. Namely, x=0.29 and x=0.33 mark the onset of two distinct magnetic ground states: a tetrahedral 3Q magnetic structure accompanied by a pronounced anomalous Hall effect, and a helical magnetic order that lacks the anomalous Hall effect, respectively.

Temperature-dependent measurements provide further evidence for the role of 3Q magnetic order. As shown in Supplementary Fig. S14 and discussed in Supplementary Notes 2 and 3, the inverse Mexican-hat-like dispersion is observed only in the 3Q ordered phase at low temperature. At higher temperatures, corresponding to the 1Q ordered and paramagnetic states, the dispersion evolves into a simple hole-like band along the K-M-K direction. ′ These observations are in close agreement with tight-binding calculations, which show that only the 3Q ordered state gives rise to the inverse Mexican-hat-like dispersion (Fig. 4 and Supplementary Fig. S15). Together, these results demonstrate that the reconstruction of the near-E bands is a distinctive fingerprint of the 3Q order.

In summary, we investigated the electronic structures of 2H-TaS and Co TaS as a 2 x 2 function of Co doping within the range of x=0.29 0.36. A comparison between 2H-TaS ∼ 2 and Co TaS shows that Co intercalation injects electrons into the parent TaS layers 0.32 2 2 and leads to the emergence of Co-derived bands near the Fermi level. In addition, our experiment provides vivid evidence of the 3Q magnetic order, namely an inverse Mexican- hat-like dispersion around M, and the associated high density of states regions in momentum space. Moreover, the spectral-weight transfer is suggestive of a transition from 3Q to helical magnetic order. Key open questions include the precise arrangement of magnetic moments in the helical magnetic order and the corresponding predictions for the electronic structure, as well as the discrepancy between experiment and theory in the dispersion of the γ M band (see Supplementary Fig. S16). Nevertheless, the present results illustrate the effect of an underlying 3Q magnetic order, motivating the exploration of this class of systems as potential hosts for a quantized quantum anomalous Hall effect at 3/4-filling.

Methods
Growth and characterization of single crystals. High-quality single crystals of Co TaS were grown by a two-step procedure. First, a precursor was prepared. The x 2 elements were combined in a ratio of Co:Ta:S (x:1:2), where x=0.29 0.36 in different ∼ growth batches. These batches were separately loaded in alumina crucibles and sealed in quartz tubes under a partial pressure (200 torr) of Argon gas. The tube was heated to 900 C ◦ and kept there for 10 days. The furnace was then shut off and allowed to cool naturally. This reaction yields a free-flowing shiny powder of polycrystals that was ground separately for each batch. Second, the precursor was loaded with iodine in a quartz tube, evacuated, and placed in a horizontal two-zone furnace. The precursor and iodine were placed in zone 1 and the other end of the tube (the growth zone) was in zone 2. Both zones were heated to 850 C for 6 hours to encourage nucleation. Then, zone 1 was raised to 950 C while ◦ ◦ zone 2 was kept at 850 C. This condition was maintained for 10 days. The furnace was ◦ then shut off and allowed to cool naturally. Shiny hexagonal crystals up to 1cm in lateral length were collected from the cold zone. Energy dispersive spectroscopy (Xplore 30, Oxford Instruments) was used to determine the Co composition for each sample. For each single crystal, 10 different regions were measured, and their average value and standard deviation were used. 2H-TaS single crystals were purchased from 2D Semiconductors. 2

High resolution ARPES measurements. Angle-resolved photoemission measurements wereperformedatBeamline7.0.2(MAESTRO)oftheAdvancedLightSource,equippedwith an R4000 hemispherical electron analyzer (Scienta Omicron). Data for the Co-intercalated samples were taken with hν=79eV under two different light polarizations: linear horizontal (LH) and linear vertical (LV). For the pristine 2H-TaS compound, measurements were 2 conducted at hν=93eV and with LH-polarized light. All experiments were carried out at low temperature (T=7K). Preliminary data were also collected at Beamline 10.0.1.2 of the Advanced Light Source.

Theory. The Co-derived bands are modeled using the following Hamiltonian defined on the Co triangular superlattice.

H = − t c† iα c jα − µ c† iα c iα − J S i · c† iα σ αβ c iβ (1) ⟨ ij ⟩ ,α i,α i (cid:31) (cid:31) (cid:31)

Here, t represents the nearest-neighbor ij hopping matrix element. J denotes the coupling ⟨ ⟩ constant between the itinerant electrons (annihilated by c ) and the classical spins S asso- iα i ciated with the 3Q ordering. µ is the chemical potential. In the absence of the 3Q order, we set t = 0.062, µ = 2t, and J = 0. In the presence of the 3Q order, the parameters are − t = 0.062, µ = 2.545t, and J = 0.04. −

To examine the 1Q ordered state, we further constructed a tight-binding Hamiltonian on the 2 1 supercell. ×

H = − t 1 c† iα c jα − t 2 c† iα c jα − t 3 c† iα c jα − µ c† iα c iα − J S i · c† iα σ αβ c iβ (2) ⟨ i (cid:31) j ⟩ 1,α ⟨ i (cid:31) j ⟩ 2,α ⟨ i (cid:31) j ⟩ 3,α (cid:31) i,α (cid:31) i

The parameters are chosen as t =0.08, t =-0.04, t =-0.005, and µ=-2.85t . Calculations 1 2 3 1 show that the band dispersions remain hole-like along the K-M-K direction for all tested ′ J, indicating that the presence or absence of the 1Q magnetic order does not qualitatively alter the dispersion (see Supplementary Fig. S15).

DFT Calculations. We performed first-principles calculations using the Vienna ab initio simulation package (VASP)[40] within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation[41] for exchange-correlation and projector-augmented-wave (PAW) potentials. A plane-wave basis set with a cutoff energy of 400eV was utilized. Spin-orbit coupling was included via the second-variation method throughout. For Co TaS , we employ 0.33 2 a 2 2 1 supercell that hosts the 3Q non-coplanar antiferromagnetic order[19], with initial × × magnetic moment of 2.25µ for each Co atom. Electronic self-consistency used a Γ-centered B 7 7 7k-meshforthesupercell,anelectronicself-consistencyenergyconvergenceof10 6eV, − × × and Gaussian smearing with a broadening parameter of 0.04eV. To capture the strong electron correlation effects of localized Co 3d orbitals, we used the DFT+U approach[42] with an effective Hubbard U =4.1eV and J =0.8eV, as determined by the constrained RPA Hund method[43]. We obtained and used the optimized crystal structure with the force criterion ˚ for the relaxation fixed at 1meV/A with 3Q antiferromagnetic order. We performed post- processing including band plotting, band unfolding and orbital-resolved projections using VASPKIT tools[44]. To compute the density of states for 2H-TaS , we used a dense k-grid 2 of 21 21 10 in the primitive Brillouin zone and the tetrahedron method. × ×

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript reports the angle resolved photoemission spectroscopy (ARPES) study of CoxTaS2 providing the evidence of the electronic structure to support the sharp magnetic transition from a non-coplanar 3Q order to a helical order upon electron doping at low temperature. CoxTaS2 forms an intercalated Co triangular lattice between TaS2 layers and the doping of Co ions can change the electronic structure dramatically near x=0.33, possibly accompanied by the magnetic transition. The main conclusion of this manuscript is that the electron doping effect mainly changes Ta band minimum as evidenced from the ARPES measurements of both Co dopings and K depositions, while the changes of Co-derived bands are evident from only the Co doping effect which is attributed to the spin-exchange interaction change of itinerant bands with localized spins. The manuscript is clearly written and address an important and interesting topic in these Van-der-Waals quantum magnets. While these findings can be of great interests to other similar intercalated transition metal dichalcogenides, I am not convinced that a rather simple tight-binding model that they used can explain the essential physics of Co-derived bands near the Fermi energy and the parameters they used are not justified.\n1. To understand the ARPES band changes upon electron dopings, the authors used a rather simple tight-binding model including the nearest-neighbor hopping parameter t and the Kondo coupling J between itinerant electrons and the classical spins. However, there is no justification of the parameters they used. In my opinion, it would be beneficial to use some ab-initio calculations to verify the quantitative features of band structure changes near the Fermi energy. For example, the authors should at least check if the itinerant Co d band (J=0) is consistent with the Co projected DFT band structure of Co1/3TaS2. Moreover, these itinerant Co dz2 bands are covalently bonded with Ta dz2 bands. They should justify why these Ta dz2 orbitals can be neglected in this simple tight-binding model. They should also check how sensitively the band structure near the M point changes due to the change of t and J parameters.\n2. The authors argue that Fig.3a Fermi surface is obtained from the absence of the 3Q magnetic order- “namely, when the magnetic moments are randomly oriented”. However, it looks like the Fermi surface can be obtained using the tight binding model with itinerant bands (J=0) at the particular ¾ filling. Does this mean that the Fermi surface is obtained from the “non-magnetic” band structure? “Non-magnetic” and “Randomly oriented” are different. They should clarify this.\n3. The occupation of Co d bands at the ¾ filling seems to be a special point to realize the Van Hove singularity and the triple-Q nesting vectors. However, it is not clear how this particular occupation will be realized in the Co intercalated TaS2 structure. Is there any simple argument for this?\n4. The authors show the comparison of the Fermi surfaces with/without the 3Q order in Fig.3a and Fig.3b. What about the 1Q magnetic order Fermi surface using the same Hamiltonian? The experimental phase diagram shows the 3Q magnetic order changes to the 1Q order as temperature is elevated. Do the authors expect the Fermi surface or the band structure also changes qualitatively at higher temperatures? Do they have the experimental evidence of ARPES measurements?\n5. The authors argue that the Fermi surface nesting vectors at the ¾ filling are (pi,0)-G in Fig.3a. Are these vectors consistent with three magnetic ordering Q vectors in the non-coplanar magnetic structure measured experimentally when x<0.33?\n6. The experimental ARPES data shows that the Co band minimum with the Van Hove singularity shifts toward the Fermi energy upon electron doping. Can this be explained using the tight-binding model they used? Is this change originated from the reduction of the spin-exchange J or the hopping parameter t, or the chemical potential?\n7. It would be worthwhile to check if the change of Ta band minimums due to the doping effect makes sense quantitatively (Fig.2e). I would suggest performing DFT band calculation of TaS2 and plot the Fermi energy vs the doping level (total number of electrons) with the rigid-band shift approximation. One could check if the slope of the plot is consistent with the Ta band minimum slope in Fig.2e.\n8. I think that each Co ion obeys the trigonal crystal field symmetry (D3h) due to the surrounding S ions and, as a result, dxz and dyz orbitals should be degenerate and have higher energies as they are pointing toward S ion directions. At the same time, Co dz2 orbitals are covalently bonded with Ta dz2 orbitals and they form the bonding/anti-bonding states with partially filled bands near the Fermi energy. Based on the linearly polarized ARPES measurements, the authors argue that the partially filled Co bands are mostly dz2 orbitals and dxy/dyz orbitals are located ~0.75eV below the Fermi energy. Is the dxy orbital typo? I think it should be dxz orbital. Overall, the authors should extend their arguments about the orbital characters of measured bands to be consistent with the local symmetry and the crystal field splitting of Co ions. Moreover, they could compare the ARPES measured bands to the orbital-projected DFT band structure obtained from ab-initio calculations.\nAs a result, I would not recommend the current version of the manuscript to the publication in Nature Communications.", "reviewer_2": "The manuscript by Luo et al. reports electronic signature of the peculiar 3Q magnetic order in a Van der Waals quantum magnet CoTaS2 (x~0.33). The authors performed high-resolution ARPES measurements on samples with different dopings and revealed an “inverse Mexican hat” dispersion along the K-M-K direction, which was attributed to the van Hove singularity from the 3Q magnetic order. The key evidence for the 3Q magnetic order is the doping dependence of the van Hove singularity, where a change in the intensity ratio between the tip and M points and a variation of the band dispersion near the M point can be identified. In general, the experiments were carried out systematically on samples with different dopings and the experimental data were of good quality. The results are reasonably captured by model calculations. The paper also deals with an interesting topic of a novel magnetic order on a 2D triangular lattice, which can be of broad interest for the community. As far as I know, the paper also presents for the first time the electronic signature of the 3Q magnetic order, which is very timely and well suited for publication in Nature Communications.\nWhile the doping dependence provides electronic signature of the 3Q magnetic order, another key (probably more convincing) evidence should come from the temperature-dependent measurements, which unfortunately are not discussed at all in the current paper. This is my major concern for this paper. Considering that the transition temperatures (~40 K from PM to single Q, ~20 K from single Q to 3Q) are easily accessible for state-of-art ARPES measurements, results from temperature-dependent ARPES measurements are highly desirable and can be very important to confirm the origin. In particular, one would expect that the change of the electronic structure from the 3Q magnetic order can be verified by varying the temperature across the magnetic transition, similar to the change caused by doping. On the other hand, if no obvious temperature-driven change can be observed, the origin of the van Hove singularity might have to be reconsidered.\nTherefore, I would strongly recommend that the authors present the temperature-dependent ARPES data for samples with 3Q magnetic order and discuss the implications of the results.\nOther minor issues:\n1. The authors use two different approaches to understand the observed electronic states in CoTaS2. Namely, the dispersive bands are explained by electron doping of the pristine TaS2 bands (Fig. 1), while the flat bands near EF (from Co 3d) are compared with model calculations of a 2D triangular lattice (Fig. 4). Strictly speaking, the electronic states should be explained within the same framework, e.g., using DFT calculations for CoTaS2. I assume that DFT calculations for bulk CoTaS2 cannot account well for the observed bands. Some explanation on the rational for such separate treatment of electronic states might be helpful for the readers.\n2. Regarding the hole-like delta pocket at the M point: it spears to me that this band might simply be the tail of the gamma pocket centered at K. Why the authors specifically assign it as a separate delta band? Some explanation might be helpful.\n3. Page 8, third line from the bottom: “we attribute the observed enlargement of the gamma pocket to the weakening of the 3Q magnetic order above xc”. This is confusing: the enlargement of the gamma pocket should be simply caused by increased Co doping. Why is it related to the 3Q magnetic order directly?\n4. What is the difference between Fig. 3(h) and Fig. 4(b)? They look very similar and appear to be redundant.", "reviewer_3": "In the article “Discovery of Van Hove Singularities: Electronic Fingerprints of 3Q Magnetic Order in a Van der Waals Quantum Magnet” H.-L. Luo et al. report an indirect evidence of a 3Q magnetic order formation in Co-doped 2H-TaS . 2\nSingle crystals of pure 2H-TaS and Co-doped TaS , with doping levels ranging from 0.29 to 0.36, were investigated using 2 2\nangle-resolved photoemission spectroscopy (ARPES), with pure TaS₂ serving as the reference sample. A critical doping level of 0.33 was identified, above which the magnetic order in Co-doped TaS transitions from a 3Q antiferromagnetic 2\n(AFM) state (at low temperatures) to a coplanar helical AFM order. ARPES measurements reveal indirect signatures of the 3Q magnetic order, through an appearance of new bands whose dispersion cannot be explained in the absence of a 3Q order.\nWhile I find the study interesting and appreciate the careful and detailed experimental work, I have a few questions and concerns that I would like to see addressed before recommending this manuscript for publication.\n1. Page 2, lines 37-28: The authors write that “Magnetically intercalated TMD systems mark the emergence of a new frontier: van der Waals magnets”. Does this refer to an emergence of vdW magnets from a historic perspective, as today we have quite a few truly magnetic vdW materials that do not need any intercalation?\n2. What is the reason for majority of ARPES measurements to be done at doping of 0.32? This is very close to a critical doping of 0.325 where 3Q order is replaced by helical order.\n3. Page 5, lines 128-129: The authors write that the direct comparison of TaS and Co TaS reveals several striking 2 0.32 2\ndifferences, but only 2 are listed (significant electron-doping of α and β bands and appearance of a δ band). While two is technically several, it would be more correct to explicitly write “two striking differences”, except if this refers to also effects (i)-\n(iV) on page 6.\n4. Page 10, growth methods: Are 0.36 samples grown the same way as 0.29, 0.32 and 0.34? 0.36 samples are not listed in the growth methods.\n5. On page 10 it says that ARPES measurement temperature was 10 K, however, experimental methods list 7 K. What is the correct measurement temperature?\n6. Page 11, ARPES details: What is the reason for performing TaS and Co-doped TaS measurements at different photon 2 2\nenergies? Is it related to their matrix elements, or to a different size of the 3D unit cell?\n7. In the article authors write that there are 2 different terminations one can end up with after cleaving, TaS one and Co one.\nthe article.\n8. Related to my previous question, do authors observe the same type of behaviour (appearance of the same bands, as well as the same trends with doping) on the Co plane as well?\n9. Page 9 in the SI, Figure S8: The authors observe a Mexican hat-like dispersion for samples with 0.29, 0.32 and 0.33\ndoping, while at 0.34 and 0.36 this turns into a simple hole-like band without a dip at the M point. Can authors comment as to why they see the dip also at 0.33 doping since this is above the critical doping value (0.325 listed in the main text) and one would expect that 0.33 should behave the same way as 0.34 and 0.36"}, "author_response": "Response to Reviewer’s Comments:\nWe thank all reviewers for the careful reviewing of our paper and his/her constructive comments\nand suggestions to improve the manuscript. We also thank the reviewer for nicely capturing the\nimportance and significance of our work.\nResponse to Reviewer #1:\nThis manuscript reports the angle resolved photoemission spectroscopy (ARPES) study of Cox-\nTaS2 providing the evidence of the electronic structure to support the sharp magnetic transition\nfrom a non-coplanar 3Q order to a helical order upon electron doping at low temperature. Cox-\nTaS2 forms an intercalated Co triangular lattice between TaS2 layers and the doping of Co ions\ncan change the electronic structure dramatically near x=0.33, possibly accompanied by the mag-\nnetic transition. The main conclusion of this manuscript is that the electron doping effect mainly\nchanges Ta band minimum as evidenced from the ARPES measurements of both Co dopings and\nK depositions, while the changes of Co-derived bands are evident from only the Co doping effect\nwhich is attributed to the spin-exchange interaction change of itinerant bands with localized spins.\nThe manuscript is clearly written and address an important and interesting topic in these Van-\nder-Waals quantum magnets. While these findings can be of great interests to other similar in-\ntercalated transition metal dichalcogenides, I am not convinced that a rather simple tight-binding\nmodel that they used can explain the essential physics of Co-derived bands near the Fermi energy\nand the parameters they used are not justified.\n(1) Reviewer’s comment: To understand the ARPES band changes upon electron dopings, the\nauthors used a rather simple tight-binding model including the nearest-neighbor hopping pa-\nrameter t and the Kondo coupling J between itinerant electrons and the classical spins. However,\nthere is no justification of the parameters they used. In my opinion, it would be beneficial to use\nsome ab-initio calculations to verify the quantitative features of band structure changes near the\nFermi energy. For example, the authors should at least check if the itinerant Co d band (J=0) is\nconsistent with the Co projected DFT band structure of Co1/3TaS2. Moreover, these itinerant Co\ndz2 bands are covalently bonded with Ta dz2 bands. They should justify why these Ta dz2 orbitals\ncan be neglected in this simple tight-binding model. They should also check how sensitively the\nband structure near the M point changes due to the change of t and J parameters.\nOur response: We thank the referee for this insightful comment. Indeed, we employ a simple\ntight-binding model that couples to local magnetic moments to describe the flat Co-derived\nbands near the Fermi level. The model parameters, t and J, are not arbitrary; they are fitted to\nexperimental data. In our view, the ability of such a minimal model to capture the salient near-E\nF\nfeatures, such as the inverse Mexican-hat dispersion observed in ARPES, is quite remarkable and\nworth reporting. Furthermore, the simplicity of the model highlights the essential interplay be-\ntween the itinerant electrons and the local moments – an aspect that is often difficult to isolate in\nmore complex approaches. As we discuss below, DFT sometimes encounters difficulties in dealing\nwith particularly flat-band features, primarily due to strong electron-electron correlations arising\nfrom the flatness and the presence of local magnetic moments. Under these circumstances, we\nbelieve it is reasonable to employ a phenomenological model to account for the flat-band physics\nthat DFT misses.\nTo address the referee’s concern, we performed orbital-projected DFT calculations for Co TaS\n1/3 2\nwithout and with the 3Q magnetic order and compared the results with our ARPES data. Figures\nR1(a, b) and R1(d, e) show the bulk band structures for the non-magnetic and 3Q-ordered states,\nrespectively. The highly-dispersive 𝛼 and 𝛽 bands and the near-Ef flat-bands (𝛾 and 𝛿 bands)\nare qualitatively reproduced. Moreover, by comparing the band structures projected onto the Co\n3d (Fig. R1(a, d)) and Ta 5d (Fig. R1(b, e)) orbitals, we find that the near-Ef flat bands are primarily\nderived from Co 3d orbitals, while the 𝛼 and 𝛽 bands mainly originate from Ta 5d orbitals,\nwhich justifies neglecting Ta 5d orbitals in our tight-binding model.\nNevertheless, there are several reasons for not relying solely on DFT to describe our results. First,\nintroducing the 3Q magnetic order (Fig. R1(d, e)) produces results that deviate notably from the\nnon-magnetic calculation (Fig. R1(a, b)). Such sensitivity of the band structure to the presence/ab-\nsence of 3Q magnetic order is not observed experimentally. Second, DFT struggles to accurately\nreproduce the detailed dispersions of the experimental 𝛾 and 𝛿 bands, likely due to strong cor-\nrelation effects associated with their flatness and the influence of local magnetic moments. Third,\nDFT results are quite sensitive to the computational setup. For instance, previous reports [1, 2]\nshow qualitative agreement with ours but differ in quantitative details, suggesting that even rel-\natively small changes in parameter settings may easily obscure shallow 𝛾 and 𝛿 features ob-\nserved experimentally. Finally, the band-unfolding process introduces additional modifications to\nthe calculated band structures, which further complicate the identification of individual bands (see\nFig. R1f). Given these limitations of DFT and the distinct orbital characters of the highly-dispersive\nand near-Ef flat bands, we consider it is more appropriate to employ a phenomenological tight-\nbinding model on a Co triangular lattice. This model effectively captures the essential flat-band\nphysics and its coupling to local moments that DFT cannot describe with sufficient accuracy.\nIn the revised manuscript, we have added Fig. R1 as Supplementary Fig. S5, and the correspond-\ning DFT Calculations section in Methods. In addition, we included a discussion on the orbital\ncharacters (page 6, line 162): “We further performed DFT calculations for Co TaS and identified\n1/3 2\nhighly-dispersive bands mainly originating from Ta orbitals together with near-E flat bands\nF\nprimarily deriving from Co orbitals (see Supplementary Fig. S5).” Moreover, we have discussed\nthe feasibility of using a tight-binding model on a Co triangular lattice (page 8, line 220): “Since\nDFT struggles to accurately describe the dispersion of the near-Ef flat bands, primarily due to\nstrong electron-electron correlations and the presence of local magnetic moments, it cannot\nprovide a reliable account of our system. At the same time, the near-Ef flat bands are dominated\nby Co 3d orbitals, whereas the dispersive 𝛼 and 𝛽 bands mainly originate from TaS-derived\n2\nstates (see Supplementary Fig. S5). Given both the limitations of DFT and the distinct orbital char-\nacters of these states, we adopt a phenomenological tight-binding model on a Co triangular\nlattice to account for the flat bands and their coupling to local moments.”\nFig. R1. DFT band structures of Co TaS without and with 3Q magnetic order. (a, b) Calculated\n1/3 2\nbulk band structures of Co TaS without 3Q magnetic order, projected onto the Co 3d (a) and\n1/3 2\nTa 5d (b) orbitals. (c) Brillouin zones of 2H-TaS (solid hexagon), Co TaS without 3Q order (black\n2 1/3 2\ndashed hexagon), and Co TaS with 3Q order (red dashed hexagon). (d, e) Calculated bulk band\n1/3 2\nstructures of Co TaS with 3Q magnetic order, projected onto the Co 3d (d) and Ta 5d (e) orbitals.\n1/3 2\n(f) DFT band structure of Co TaS with 3Q magnetic order unfolded to the Brillouin zone of the\n1/3 2\nnon-magnetic Co TaS.\n1/3 2\nIn Fig. R2, we study the sensitivity of the band dispersions near the M point to variations of the\nparameters t and J within our tight-binding model. Specifically, we calculated the band structures\nfor different values of J at fixed t, and for different values of t at fixed J. We find that a finite\nexchange coupling J can induce the emergence of an inverse Mexican hat-like dispersion around\nthe M point. Meanwhile, changing the magnitudes of J and t primarily affects the bandwidth and\nthe energy separation between the M-point minimum and the neighboring maxima. Therefore,\nthe inverse Mexican hat-like feature is qualitatively preserved as long as the 3Q magnetic order\nis present. In the main text, we adopt the parameter set that best reproduces the experimental\nobservations for comparison.\nIn the revised manuscript, we have included Fig. R2 as Supplementary Fig. S12 to demonstrate\nthe preservation of the inverse Mexican hat-like band in the presence of 3Q order. In addition,\nwe added the following sentence on page 10, line 283: “We further tested the robustness of this\nresult by varying hopping amplitude t and coupling constant J, and found that the inverse Mexi-\ncan hat-like dispersion persists as long as the 3Q order is present (See Supplementary Fig. S12).”\nFig. R2. Sensitivity of the band structures near the M point to variations of the hopping amplitude\nt and exchange coupling J. (a) Band structures calculated for different values of t at a fixed J =\n0.04. (b) Band structures calculated for different values of J at a fixed t =-0.062. All calculations\nwere performed with the chemical potential 𝜇=2.545t.\n(2) Reviewer’s comment: The authors argue that Fig.3a Fermi surface is obtained from the ab-\nsence of the 3Q magnetic order- “namely, when the magnetic moments are randomly oriented”.\nHowever, it looks like the Fermi surface can be obtained using the tight binding model with itin-\nerant bands (J=0) at the particular ¾ filling. Does this mean that the Fermi surface is obtained\nfrom the “non-magnetic” band structure? “Non-magnetic” and “Randomly oriented” are different.\nThey should clarify this.\nOur response: We thank the reviewer for the correction. We agree that describing the Co mo-\nments as “randomly oriented” in the paramagnetic state is misleading. An accurate term should\nbe “non-magnetic”, which reflects the absence of long-range magnetic order, rather than imply-\ning the presence of fluctuating local moments with random orientations. To avoid confusion, we\nhave revised the wording in the manuscript accordingly. Specifically, the sentence “namely, when\nthe magnetic moments are randomly oriented” has been changed to “that is, in the non-magnetic\nstate (exchange coupling constant J=0)” on page 8, line 228.\n(3) Reviewer’s comment: The occupation of Co d bands at the ¾ filling seems to be a special\npoint to realize the Van Hove singularity and the triple-Q nesting vectors. However, it is not clear\nhow this particular occupation will be realized in the Co intercalated TaS2 structure. Is there any\nsimple argument for this?\nOur response: We thank the referee for this insightful comment. According to a simple electron-\ncounting scheme, the Co-intercalated compounds can be regarded as alternating layers of\n[Co ]+ and [TaS ]− (Fig. R3(a)). As shown in Fig. R3(b), the coordination environment of Co3+\n1/3 2\nin Co TaS is a trigonally distorted pseudo-octahedron with D3d crystal field symmetry, leading\n1/3 2\nto a splitting of the Co 3d orbitals into e2 (dxy, dx2−y2), a1 (dz2), and e1 (dxz, dyz) (Fig. R1(c)) [3,\n4].\nPolarization-dependent ARPES measurements (Figure 1 of our manuscript) suggest that the shal-\nlow 𝛾 and 𝛿 bands near the Fermi level are most likely dominated by Co dz2 character. We\ntherefore focus on the filling of the dz2 orbital. For a Co3+ ion with six d electrons within a D3d\ncrystal field, if the crystal field from the TaS layers were very weak, the electrons would adopt a\n2\nhigh-spin configuration as sketched in the left panel of Fig. R3(c), with n = 4 unpaired electrons.\nIn this case, the dz2 orbital would be only half-filled (1/2-filling). However, such a configuration\nwould yield an effective magnetic moment of μ (μ )= √n(n+2) ≈ 4.9, which is noticeably\neff B\nlarger than the experimental value of μ (μ )= 3.7 obtained from magnetic susceptibility\neff B\nmeasurements [5]. This suggests that the crystal field may be not as weak as in the pure high-\nspin limit, and that an intermediate spin configuration is stabilized by the crystal field and cova-\nlency. In this situation, more electrons are transferred from the dxz/dyz orbitals into the dz2 and\ndxy/dx2−y2 states, effectively increasing the dz2 occupancy from the 1/2-filling expected in the\nhigh-spin limit toward ~3/4 filling, as illustrated in the right panel of Fig. R3(c).\nWhile this picture is simplified, it provides a sound rationale for the 3/4-filling condition of the Co\nlattice in Co TaS, consistent with the resemblance between the ARPES-observed 𝛾 Fermi sur-\n1/3 2\nface and the tight-binding prediction for a 3/4-filled Co band (see Figure 3 in our manuscript).\nFig. R3. (a) Crystal structure of Co TaS, illustrating the formal charges of the TaS and Co layers\n1/3 2 2\non a simple electron-counting scheme [4]. (b) Local coordination of a Co center with surrounding\nS atoms in Co TaS, forming a D3d ligand field [4]. (c) Schematic d-orbital splitting diagrams for\n1/3 2\nCo3+ (3d6) in a D3d ligand field. Left panel: high-spin configuration; Right panel: relatively lower-\nspin configuration.\n(4) Reviewer’s comment: The authors show the comparison of the Fermi surfaces with/without\nthe 3Q order in Fig.3a and Fig.3b. What about the 1Q magnetic order Fermi surface using the\nsame Hamiltonian? The experimental phase diagram shows the 3Q magnetic order changes to\nthe 1Q order as temperature is elevated. Do the authors expect the Fermi surface or the band\nstructure also changes qualitatively at higher temperatures? Do they have the experimental evi-\ndence of ARPES measurements?\nOur response: We thank the referee for this insightful comment. To address this point, we have\nsimulated the band structure of the 1Q ordered state using a tight-binding model. Figure R4(a)\nshows the 1Q ordered structure with two adjacent layers, while Fig. R4(b) illustrates the Brillouin\nzones of the unit cell (a×a) without magnetic order and of the supercell (2a×a) with 1Q order.\nThe band structures are modeled by the following Hamiltonian defined on the 2a×a lattice:\n𝐻 = −∑ 𝑡 𝑐† 𝑐 −∑ 𝑡 𝑐† 𝑐 −∑ 𝑡 𝑐† 𝑐 −𝜇∑ 𝑐† 𝑐 −𝐽∑ 𝐒 ∙\n<𝑖𝑗>1,𝛼 1 𝑖𝛼 𝑗𝛼 <𝑖𝑗>2,𝛼 2 𝑖𝛼 𝑗𝛼 <𝑖𝑗>3,𝛼 3 𝑖𝛼 𝑗𝛼 𝑖,𝛼 𝑖𝛼 𝑖𝛼 𝑖 𝑖\n𝑐† 𝝈 𝑐 .\n𝑖𝛼 𝛼𝛽 𝑖𝛽\nWe tested a wide range of parameter sets for t1, t2, and t3 to reproduce the overall hole-like\nband along the K-M-K’ direction, and then examined the effect of varying the exchange coupling\nconstant J. In Fig. R4(c), we show a few representative cases. Our calculations consistently reveal\nthat, regardless of whether J=0 (without 1Q magnetic order) or J≠0 (with 1Q magnetic order),\nthe band dispersion along K-M-K’ remains hole-like. This indicates that the presence or absence\nof the 1Q magnetic order does not qualitatively alter the dispersion along this path.\nFig. R4. Band structure of the 1Q ordered state. (a) Schematic illustration of the 1Q ordered structure\nwith two adjacent layers. Large (small) dots denote sites in the upper (lower) layer. Blue/red colors\nindicate opposite out-of-plane magnetic-moment orientations. The hopping parameters t1, t2, and\nt3 are assigned according to the bond lengths, and the supercell (2a×a) is shown in comparison with\nthe unit cell (a×a). (b) Brillouin zones of the unit cell (a×a) without 1Q order and of the supercell\n(2a×a) with 1Q order, showing the high-symmetry K-M-K’ path. (c) Band dispersions calculated from\nthe tight-binding Hamiltonian on the 2a×a lattice for different values of J with fixed t1=0.08, t2=-0.04,\nt3=-0.005, and 𝜇=-2.85t1.\nTo address the referee’s question on temperature dependence, we performed temperature-de-\npendent ARPES measurements on Co TaS along K-M-K (Fig. R5). As shown in Fig. R5(a), at 6 K\n0.31 2\nwithin the 3Q ordered state, the near-Ef band exhibits an inverse Mexican hat-like dispersion.\nThe energy distribution curves (EDCs) obtained at M (k1) and k2 [Fig. R5(d)] show that the peak\nposition of EDC(k1) is located at a slightly yet clearly higher binding energy than that of EDC(k2).\nThis behavior is consistent with a 3Q order-induced reconstruction that produces the inverse\nMexican hat-like dispersion described in our manuscript.\nBy contrast, at 31 K (1Q order) and 82 K (paramagnetic state), the dispersions in Fig. R5(b, c) no\nlonger exhibit the inverse Mexican hat-like structure. Instead, the EDC peak positions at k1 and\nk2 [Fig. R5(e, f)] are nearly identical, indicating a hole-like band dispersion along K-M-K.\nTaken together, these results demonstrate that the inverse Mexican hat-like dispersion emerges\nexclusively in the 3Q phase, while the 1Q and paramagnetic phases show only a hole-like band.\nThis provides direct spectroscopic evidence that the 3Q magnetic order strongly modifies the\nlow-energy electronic structure.\nIn the revised manuscript, we have added Fig. R5 as Supplementary Fig. S14, Fig. R4 as Supple-\nmentary Fig. S15, and the corresponding Theory section in Methods. In addition, we have ex-\npanded the discussion on page 11, line 304 to clarify this point: “Temperature-dependent meas-\nurements provide further evidence for the role of 3Q magnetic order. As shown in Supplementary\nFig. S14, the inverse Mexican hat-like dispersion is observed only in the 3Q ordered phase at low\ntemperature. At higher temperatures, corresponding to the 1Q ordered and paramagnetic states,\nthe dispersion evolves into a simple hole-like band along the K-M-K direction. These observa-\ntions are in close agreement with tight-binding calculations, which shows that only the 3Q or-\ndered state gives rise to the inverse Mexican hat-like dispersion (Fig. 4 and Supplementary Fig.\nS15). Together, these results demonstrate that the reconstruction of the near-Ef bands is a dis-\ntinctive fingerprint of the 3Q order.”\nFig. R5. Temperature-dependent ARPES spectra of Co TaS along the K-M-K direction. (a-c) Band\n0.31 2\nstructures measured at 6K, 31K and 82K, respectively. Arrows mark the momenta k1(M point) and k2.\n(d-f) EDCs extracted at k1 and k2 from panels (a-c).\n(5) Reviewer’s comment: The authors argue that the Fermi surface nesting vectors at the ¾ filling\nare (pi,0)-G in Fig.3a. Are these vectors consistent with three magnetic ordering Q vectors in the\nnon-coplanar magnetic structure measured experimentally when x<0.33?\nOur response: We thank the referee for this helpful question. We clarify at the outset that the\nFermi-surface nesting vectors (π, 0)-G at 3/4 filling correspond directly to the three magnetic\nordering Q vectors. As shown in Supplementary Fig. S9, which we reproduced here as Fig. R6, the\n3Q magnetic order in samples with x<0.33 leads to a 2×2 magnetic unit cell in real space (Fig.\nR6(b)), relative to the original Co triangular lattice (Fig. R6(a)). The original lattice has two basis\nvectors with equal length a separated by 60°, and the 2×2 magnetic superstructure thus doubles\nthe periodicity along both directions.\nAccordingly, the periodicity in reciprocal space is reduced (Fig. R6(c)). In the original lattice, the\nreal-space periodicity a corresponds to a reciprocal lattice periodicity of 2π/a, which defines the\nextent of the original Brillouin zone. The 2×2 enlargement in real space leads to a folding of the\nBrillouin zone, and new magnetic ordering wave vectors Q of magnitude 2π/2a (= π/a) emerge\nalong three high-symmetry directions separated by 120°. These Q vectors are Q= (π, 0), Q= (-\n1 2\nπ/2, √3π/2), and Q = (-π/2, -√3π/2), expressed in units of 1/a.\nhexagonal Fermi surface at 3/4 filling, is equivalent to Q up to a reciprocal lattice vector. This\n1\nsupports our interpretation that the observed Fermi surface reconstruction originates from the\n3Q magnetic order, consistent with our discussion in the manuscript.\nFig. R6. The 3Q magnetic order in real space and the reconstructed Brillouin zone. (a) The trian-\ngular lattice of intercalated Co atoms (indicated by gray dashed lines) and its unit cell (outlined\nby black lines). (b) Configuration of the 3Q magnetic order with four sublattices and the resultant\n2×2 magnetic unit cell (black lines). (c) The Brillouin zone of the Co triangular lattice, shown both\nwithout (gray dashed lines) and with (black solid lines) the 3Q magnetic order. The arrow indicates\nthe folding wave vector (π,0). (d) Nesting wave vector ((π, 0)-G) that connects the edge centers\nof the hexagonal Fermi surface at 3/4-filling. The nesting vector ((π,0)-G) is equivalent to (π, 0)\nup to a reciprocal lattice vector G.\n(6) Reviewer’s comment: The experimental ARPES data shows that the Co band minimum with\nthe Van Hove singularity shifts toward the Fermi energy upon electron doping. Can this be ex-\nplained using the tight-binding model they used? Is this change originated from the reduction of\nthe spin-exchange J or the hopping parameter t, or the chemical potential?\nOur response: We thank the referee for the careful observation. It is indeed true that the Co band\nminimum shifts toward the Fermi level as the Co composition increases from 0.34 to 0.36. How-\never, this behavior cannot be fully captured within our tight-binding model. As shown in Fig. R2(a),\nreducing J would broaden the bandwidth, which is opposite to the experimentally observed up-\nward shift of the Co band minimum. Moreover, although a reduction in t (Fig. R2(b)) could in\nprinciple narrow the bandwidth and shift the band minimum upward, we do not find any physical\nreason for t to decrease with increasing Co content. Lastly, a rigid-band picture governed by the\nchemical potential also fails to explain the observation, since 𝜇 remains experimentally pinned\nnear the band top and cannot account for the upward shift. To reasonably explain the data, we\npropose that the helical antiferromagnetic order emerging for x>0.33 plays a dominant role,\nstrongly modifying the electronic structure and pushing the Co band bottom upward. However,\nthe complexity of this order along the b*-c plane lies beyond the scope of our tight-binding\nmodel.\n(7) Reviewer’s comment: It would be worthwhile to check if the change of Ta band minimums\ndue to the doping effect makes sense quantitatively (Fig.2e). I would suggest performing DFT\nband calculation of TaS2 and plot the Fermi energy vs the doping level (total number of electrons)\nwith the rigid-band shift approximation. One could check if the slope of the plot is consistent with\nthe Ta band minimum slope in Fig.2e.\nOur response: We thank the referee for this constructive suggestion. Following the comment, we\nperformed DFT calculations for 2H-TaS along the M -Γ -K -M direction and obtained the\n2 0 0 0 0\ncorresponding density of states (DOS) integrated over the entire momentum space, as shown in\nFig. R7a. From this DOS, we determined the relation between the Fermi energy and the doping\nlevel, expressed as the number of additional electrons per TaS unit cell (Fig. R7b).\n2\nnominally dopes approximately 3x electrons to the TaS layers, giving a total electron count of 3x\n2\nper unit cell. We then plotted the evolution of the 𝛼-band minimum along the Γ -M direction\n0 0\nas a function of this nominal electron doping (black curve in Fig. R7c, reproduced from Fig. 2e of\nour manuscript). When comparing this experimental trend with the calculated dependence of\nFermi-level shift on electron doping (blue curve in Fig. R7c), we find excellent agreement not only\nin the absolute electron count but also in the slope of the relation between the number of elec-\ntrons and the Fermi-level shift.\nThis quantitative consistency strongly supports that the 𝛼 and 𝛽 bands are predominantly de-\nrived from TaS orbitals, and that their energy shifts upon electron doping are well captured by\n2\nthe rigid-band shift approximation. To substantiate this point, we have added Fig. R7 as Supple-\nmentary Fig. S4 and incorporated the following sentence into the manuscript (page 6, line 149):\n“Density Functional Theory (DFT) calculations for 2H-TaS reveal that the observed ~300meV\n2\nband shift can be reproduced by adding one electron per TaS unit cell, in good agreement with\n2\na simple electron-counting picture of Co TaS consisting of alternating [Co ]+ and [TaS]- layers\n1/3 2 1/3 2\n(see Supplementary Fig. S4).” In addition, we have added another sentence on page 7, line 189:\n“The observed slope of the shift of the 𝛼-band minimum shift as a function of Co doping is well\nreproduced by a rigid-band-shift approximation applied to the TaS band structure (see Supple-\n2\nmentary Fig. S4)”\nFig. R7. Calculated and experimental electron-doping dependence of TaS bands. (a) Calculated\n2\nband structure of 2H-TaS along the M -Γ -K -M direction. The right panel displays the cal-\n2 0 0 0 0\nculated density of states (DOS) integrated over the entire Brillouin zone as a function of energy.\n(b) Calculated relation between the number of additional electrons per TaS unit cell and the\n2\nupward shift of the Fermi level. (c) Comparison between experiment and calculation: the black\ncurve represents the evolution of the 𝛼-band minimum along the Γ -M direction as a function\n0 0\nof the nominal electrons per TaS unit cell (3x, taken from Fig. 2e in the manuscript), while the\n2\nblue curve shows the calculated relation between electron doping and Fermi-level shift.\n(8) Reviewer’s comment: I think that each Co ion obeys the trigonal crystal field symmetry (D3h)\ndue to the surrounding S ions and, as a result, dxz and dyz orbitals should be degenerate and have\nhigher energies as they are pointing toward S ion directions. At the same time, Co dz2 orbitals are\ncovalently bonded with Ta dz2 orbitals and they form the bonding/anti-bonding states with partially\nfilled bands near the Fermi energy. Based on the linearly polarized ARPES measurements, the\nauthors argue that the partially filled Co bands are mostly dz2 orbitals and dxy/dyz orbitals are\nlocated ~0.75eV below the Fermi energy. Is the dxy orbital typo? I think it should be dxz orbital.\nOverall, the authors should extend their arguments about the orbital characters of measured bands\nto be consistent with the local symmetry and the crystal field splitting of Co ions. Moreover, they\ncould compare the ARPES measured bands to the orbital-projected DFT band structure obtained\nfrom ab-initio calculations.\nOur response: We thank the referee for these insightful comments. As explained in the response\nto Comment #3, the coordination environment of Co3+ in Co TaS is a trigonally distorted\n1/3 2\npseudo-octahedron (D3d crystal field symmetry), leading to a qualitative d-orbital splitting dia-\ngram of e2(dxy, dx2-y2), a1 (dz2), and e1 (dxz, dyz). For simplicity, we treat Co3+ in the high-spin\nd6 configuration as shown in the left panel of Fig. R8(b). In contrast, Ta3+ has D3h symmetry, giving\nthe d-orbital splitting: a1 (dz2), e2(dxy, dx2-y2), and e1 (dxz, dyz) [4, 5], with a d2 configuration as\nshown in the right panel of Fig. R8(b).\nWe then consider what determines the orbital spectral weight in ARPES. The spectral weight is\nstrongly modulated by the dipole matrix element |Mk|2 ∝ |⟨ϕk|ε̂∙r|ϕk⟩|2, where ϕk (ϕk) de-\nf,i f i i f\nnotes the initial (final) state wavefunction of the photoelectron, and ε̂ is the unit vector along the\npolarization direction of the incident light (LH: even parity; LV: odd parity). Approximately, only\norbitals sharing the same parity as the incident polarization can be detected in ARPES [6]. When\nthe analyzer slit is aligned along the Γ -𝐾 direction (parallel to the x-axis as defined in Fig. R8(c)),\n0 0\nthe d orbital symmetries of Co and Ta (a1, e1, and e2 sets) with respect to the scattering plane\n(x-z plane) are illustrated in Fig. R8(d). The even dz2 orbitals are expected to be visible with LH\npolarization, whereas the e1 (dxz, dyz) and e2 (dxy, dx^2-y^2) sets are not symmetric collectively\nwith respect to the scattering plane and may appear under both LH and LV polarizations. Ac-\ncordingly, the observed bands can be classified into three categories: (1) the 𝛾 and 𝛾 bands,\n𝐾 𝑀\nvisible only under LH polarization and thus mainly derived from dz2 orbitals; (2) the 𝛼 and 𝛽\nbands, visible under both LH and LV polarizations but significantly stronger in LH, which can arise\nfrom all orbitals but are dominated by dz2 character; and (3) the 𝜀 band, which appears pre-\ndominantly under LV polarization is mainly contributed by dxz/dyz and dxy/dx2-y2 sets.\nTogether with the DFT-calculated band structures projected onto individual Co and Ta orbitals\n(see Fig. R9), the above polarization-dependent analysis can be further extended. The DFT results\nenable a clear separation of Co- and Ta-derived contributions, indicating that the near-Ef 𝛾\n𝐾\nand 𝛾 bands mainly arise from Co 3dz2 orbital; whereas the highly-dispersive 𝛼 and 𝛽 bands\n𝑀\nare dominated by Ta 5dz2 and Ta 5dxy/dx2-y2 states.\nTo strength our argument, we have added the phrase “a pronounced reduction of spectral weight\nin the highly-dispersive 𝛼 and 𝛽 bands” on page 6, line 170, and included “and the DFT-calcu-\nlated orbital projected band structures (see Supplementary Note and Supplementary Fig. S6-S7)”\non page 6, line 172. In addition, we have revised the sentence on page 7, line 174, to read: “the\nnear-Ef 𝛾 and 𝛾 bands mainly originate from Co 3dz2 orbital, the highly-dispersive 𝛼 and\n𝐾 𝑀\n𝛽 bands are dominated by Ta 5dz2 Ta 5dxy/dx2-y2 orbitals, and the 𝜀 band is primarily\ncontributed by dxz/dyz and dxy/dx2-y2 states.” We have also added Fig. R8 as Supplementary Fig.\nS6, Fig. R9 as Supplementary Fig. S7, and included the corresponding discussion as a Supple-\nmentary Note in the Supplementary Materials.\nFig. R8. Orbital Symmetries of Co TaS. (a) Local coordination environments of Co and Ta centers\n1/3 2\nand their surrounding S atoms in Co TaS. (b) Qualitative diagrams of d-orbital splitting for iso-\n1/3 2\nlated Co and Ta centers under their respective local ligand field. (c) Real-space projections on the\nx-y plane showing the local coordination geometries of Co and Ta. (d) Symmetries of the d or-\nbitals for Co and Ta with respect to the scattering plane, defined as the x-z plane aligned along\nthe Γ -𝐾 direction [4].\n0 0\nFig. R9. DFT-calculated orbital-projected band structures of bulk Co TaS with 3Q magnetic or-\n1/3 2\nder. (a-c) Calculated band structures of Co TaS with 3Q order projected onto Co dz2, Co\n1/3 2\ndxz/dyz, Co dxy/dx2-y2 orbitals, respectively. (d-f) Corresponding orbital projections onto Ta dz2,\nTa dxz/dyz, Ta dxy/dx2-y2 orbitals, respectively. The color scale represents the orbital-weight\nintensity.\nAs a result, I would not recommend the current version of the manuscript to the publication in\nNature Communications.\nWe sincerely thank the Reviewer for their careful evaluation and constructive feedback. We hope\nthat the revised version has adequately addressed their concerns and that they may now find it\nsuitable for publication in Nature Communications.\nResponse to Reviewer #2:\nThe manuscript by Luo et al. reports electronic signature of the peculiar 3Q magnetic order in a\nVan der Waals quantum magnet CoTaS2 (x~0.33). The authors performed high-resolution ARPES\nmeasurements on samples with different dopings and revealed an “inverse Mexican hat” disper-\nsion along the K-M-K direction, which was attributed to the van Hove singularity from the 3Q\nmagnetic order. The key evidence for the 3Q magnetic order is the doping dependence of the\nvan Hove singularity, where a change in the intensity ratio between the tip and M points and a\nvariation of the band dispersion near the M point can be identified. In general, the experiments\nwere carried out systematically on samples with different dopings and the experimental data were\nof good quality. The results are reasonably captured by model calculations. The paper also deals\nwith an interesting topic of a novel magnetic order on a 2D triangular lattice, which can be of\nbroad interest for the community. As far as I know, the paper also presents for the first time the\nelectronic signature of the 3Q magnetic order, which is very timely and well suited for publication\nin Nature Communications.\n(1) Reviewer’s comment: While the doping dependence provides electronic signature of the 3Q\nmagnetic order, another key (probably more convincing) evidence should come from the tem-\nperature-dependent measurements, which unfortunately are not discussed at all in the current\npaper. This is my major concern for this paper. Considering that the transition temperatures (~40\nK from PM to single Q, ~20 K from single Q to 3Q) are easily accessible for state-of-art ARPES\nmeasurements, results from temperature-dependent ARPES measurements are highly desirable\nand can be very important to confirm the origin. In particular, one would expect that the change\nof the electronic structure from the 3Q magnetic order can be verified by varying the temperature\nacross the magnetic transition, similar to the change caused by doping. On the other hand, if no\nobvious temperature-driven change can be observed, the origin of the van Hove singularity might\nhave to be reconsidered. Therefore, I would strongly recommend that the authors present the\ntemperature-dependent ARPES data for samples with 3Q magnetic order and discuss the impli-\ncations of the results.\nOur response: We thank the reviewer for raising this important point, which we also consider\ncrucial for establishing the role of the 3Q magnetic order. To address the reviewer’s concern, we\nperformed temperature-dependent ARPES measurements on Co TaS along the K-M-K direc-\n0.31 2\ntion (Fig. R5). These results clearly demonstrate the temperature-driven evolution of the band\nstructure, consistent with the change from the 3Q to 1Q magnetic order. A detailed description\nand discussion of these results are provided in our response to Comment#4 of Referee#1, where\nwe compare the temperature-dependent data with tight-binding calculations and discuss dis-\ntinctive role of the 3Q order.\n(2) Reviewer’s comment: The authors use two different approaches to understand the observed\nelectronic states in CoTaS2. Namely, the dispersive bands are explained by electron doping of the\npristine TaS2 bands (Fig. 1), while the flat bands near EF (from Co 3d) are compared with model\ncalculations of a 2D triangular lattice (Fig. 4). Strictly speaking, the electronic states should be\nexplained within the same framework, e.g., using DFT calculations for CoTaS2. I assume that DFT\ncalculations for bulk CoTaS2 cannot account well for the observed bands. Some explanation on\nthe rational for such separate treatment of electronic states might be helpful for the readers.\nOur response: We thank the referee for raising this important point. We agree that, ideally, both\nthe dispersive and flat bands should be treated within a unified theoretical framework. However,\nit is well known that density functional theory (DFT) struggles to accurately describe flat bands\nand local magnetic moments, primarily due to the strong electron-electron correlations involved\n(see Fig. R1). To further justify this separation, we note that the near-Ef flat bands are dominated\nby Co 3d orbitals, whereas the dispersive 𝛼 and 𝛽 bands are primarily derived from Ta 5d states\n(see Fig. R1). Since the orbital characters of these states are quite distinct, in the absence of a fully\nfirst-principles description of all bands, we adopt a hybrid approach: a phenomenological tight-\nbinding model on a Co triangular lattice is employed to describe the flat bands and their coupling\nto local moments, while a rigid-band-shifted DFT of pristine TaS is used to describe the more\n2\ndispersive 𝛼 and 𝛽 bands (see Fig. R7). This combined framework yields a good quantitative\nagreement with the ARPES data.\nTo help readers better understand the rationale behind our separate treatments of the dispersive\nand flat-band states, we have clarified this point in the revised manuscript (page 8, line 219):\n“Since DFT struggles to accurately describe the dispersion of the near-Ef flat bands, primarily due\nto strong electron-electron correlations and the presence of local moments, it cannot provide a\nreliable account of our system. At the same time, the near-Ef flat bands are dominated by Co 3d\norbitals, whereas the dispersive 𝛼 and 𝛽 bands are primarily derived from Ta 5d states (see\nthese states, we adopt a phenomenological tight-binding model on a Co triangular lattice to\naccount for the flat bands and their coupling to local magnetic moments.”\n(3) Reviewer’s comment: Regarding the hole-like delta pocket at the M point: it spears to me\nthat this band might simply be the tail of the gamma pocket centered at K. Why the authors\nspecifically assign it as a separate delta band? Some explanation might be helpful.\nOur response: We agree with the reviewer that the δ band at the M point and the γ pocket\ncentered at K originate from the same band, exhibiting anisotropic dispersions along different\nhigh-symmetry directions, similar to how the 𝛼 and 𝛽 bands behave along both Γ-M and Γ-\nK directions.\nHowever, since we present more detailed studies of these two branches in later sections (e.g.,\ndoping-dependent experiments), we find it clearer to distinguish them by direction for subse-\nquent discussion. Therefore, we have renamed the current γ band as γ band (along Γ-K) and\nK\nthe previous δ band as γ band (along Γ-M), to indicate that they are directional branches of\nM\nthe same band. Consequently, the previous labels for the ε and ζ bands are shifted forward by\none Greek letter and are now labeled δ and ε, respectively. This relabeling is purely for clarity\nand does not affect our conclusions.\n(4) Reviewer’s comment: Page 8, third line from the bottom: “we attribute the observed enlarge-\nment of the gamma pocket to the weakening of the 3Q magnetic order above xc”. This is con-\nfusing: the enlargement of the gamma pocket should be simply caused by increased Co doping.\nWhy is it related to the 3Q magnetic order directly?\nOur response: We thank the reviewer for this thoughtful comment. While increased Co doping\ncould, in principle, lead to band filling, which would simultaneously cause a downward shift of the\nband bottom and a corresponding enlargement of the 𝛾 pocket, our data do not support this\nexpectation. Specifically, as shown in Fig. 2(e), the bottom of the γ band does not shift down-\nK\nward in energy with increasing doping. This behavior contradicts the expectation from a simple\nband-filling scenario.\nTherefore, the observed enlargement of the 𝛾 pocket cannot be solely attributed to charge dop-\ning. A plausible explanation is the weakening of the 3Q magnetic order above x ≈0.33, which\nC\nalters the Fermi surface topology and could lead to an enlarged 𝛾 Fermi pocket size, as seen\nwhen comparing Fig. 3a to Fig. 3b in our manuscript. Thus, our interpretation goes beyond a\nsimple doping picture and highlights the essential role of the 3Q magnetic order in shaping the\nFermi surface.\nTo clarify this point, we have added the following sentence on page 9, line 255: “rather than to\nincreased Co doping. This assignment is supported by the observation that the 𝛾 band bottom\n𝐾\ndoes not shift downward with doping (see Fig. 2), which is inconsistent with a simple band-filling\npicture. Instead, the weakening of the 3Q order alters the Fermi surface topology, naturally lead-\ning to the enlarged 𝛾 Fermi pockets.”\n(5) Reviewer’s comment: What is the difference between Fig. 3(h) and Fig. 4(b)? They look very\nsimilar and appear to be redundant.\nOur response: We thank the reviewer for raising this point. Although Fig. 3(h) and Fig. 4(b) display\nsimilar overall trends, they provide complementary perspectives. Specifically, Fig. 3(h) illustrates\nthe redistribution of spectral weight at the Fermi surface between the tip of the 𝛾 pocket and\nthe M point, while Fig. 4(b) presents the corresponding redistribution in the band dispersion\nthrough the ratio of EDC peak intensities at the same momenta. Taken together, these two figures\noffer a more complete picture of how the 3Q order-induced band reconstruction affects both the\nFermi surface topology and the underlying band dispersions.\nResponse to Reviewer #3:\nIn the article “Discovery of Van Hove Singularities: Electronic Fingerprints of 3Q Magnetic Order\nin a Van der Waals Quantum Magnet” H.-L. Luo et al. report an indirect evidence of a 3Q magnetic\norder formation in Co-doped 2H-TaS2.\nSingle crystals of pure 2H-TaS2 and Co-doped TaS2, with doping levels ranging from 0.29 to\n0.36, were investigated using angle-resolved photoemission spectroscopy (ARPES), with pure\nTaS₂ serving as the reference sample. A critical doping level of 0.33 was identified, above which\nthe magnetic order in Co-doped TaS2 transitions from a 3Q antiferromagnetic (AFM) state (at\nlow temperatures) to a coplanar helical AFM order. ARPES measurements reveal indirect signa-\ntures of the 3Q magnetic order, through an appearance of new bands whose dispersion cannot\nbe explained in the absence of a 3Q order.\nWhile I find the study interesting and appreciate the careful and detailed experimental work, I\nhave a few questions and concerns that I would like to see addressed before recommending this\nmanuscript for publication.\n(1) Reviewer’s comment: Page 2, lines 37-28: The authors write that “Magnetically intercalated TMD systems mark the emergence of a new frontier: van der Waals magnets”. Does this refer to\nan emergence of vdW magnets from a historic perspective, as today we have quite a few truly\nmagnetic vdW materials that do not need any intercalation?\nOur response: We thank the reviewer for this insightful comment. We agree that our original\nstatement was imprecise. Our intention was to highlight the historical significance of magnetically\nintercalated TMDs in the early exploration of van der Waals magnetism, rather than to imply that\nthey represent the first or only class of vdW magnets. To avoid confusion, we have revised the\nsentence on page 2, line 40, to read: “Magnetically intercalated TMDs, as an important class of\nvan der Waals magnets, open a new paradigm in TMD research and provide a versatile platform\nto investigate the interplay between magnetic order and electronic properties.”\n(2) Reviewer’s comment: What is the reason for majority of ARPES measurements to be done at\ndoping of 0.32? This is very close to a critical doping of 0.325 where 3Q order is replaced by\nhelical order.\nOur response: We thank the reviewer for raising this important point. We would like to clarify\nthat, among all the ARPES data presented in this manuscript, only Fig. 1 is based exclusively on\nthe sample with doping level x=0.32. In all other figures where data from x=0.32 are shown, they\nare always presented together with results from other doping levels as part of a broader system-\natic study, and are not used in isolation.\nIn Fig. 1, we chose the x = 0.32 sample because it provides particularly clear spectra for illustrating\nthe key differences between Co TaS and pristine TaS, namely the downward shift of TaS -de-\n1/3 2 2 2\nrived bands and the emergence of Co-derived bands. The use of the x = 0.32 sample does not\naffect these conclusions, as these features remain robust across the entire doping range studied.\nTherefore, the conclusions drawn from the x=0.32 sample in Fig. 1 are valid and representative of\nthe general behavior.\n(3) Reviewer’s comment: Page 5, lines 128-129: The authors write that the direct comparison of\nTaS2 and Co0.32TaS2 reveals several striking differences, but only 2 are listed (significant elec-\ntron-doping of α and β bands and appearance of a δ band). While two is technically several, it\nwould be more correct to explicitly write “two striking differences”, except if this refers to also\neffects (i)-(iV) on page 6.\nOur response: We thank the reviewer for the helpful suggestion. To improve clarity, we have\nmodified the wording on page 5, line 136, replacing “several” with “two.”\n(4) Reviewer’s comment: Page 10, growth methods: Are 0.36 samples grown the same way as\n0.29, 0.32 and 0.34? 0.36 samples are not listed in the growth methods.\nOur response: We thank the reviewer for pointing this out. The omission of x=0.36 was uninten-\ntional. The x=0.36 samples were grown using the same procedure as the other doping levels. We\nhave now corrected the text and explicitly included x=0.36 on page 12, line 331 in the Methods\nsection.\n(5) Reviewer’s comment: On page 10 it says that ARPES measurement temperature was 10 K,\nhowever, experimental methods list 7 K. What is the correct measurement temperature?\nOur response: We thank the reviewer for this careful observation. The correct ARPES measure-\nment temperature is 7 K. The mention of 10 K on page 4 was a typographical error, which we have\nnow corrected it in the revised manuscript (page 4, line101).\n(6) Reviewer’s comment: Page 11, ARPES details: What is the reason for performing TaS2 and\nCo-doped TaS2 measurements at different photon energies? Is it related to their matrix elements,\nor to a different size of the 3D unit cell?\nOur response: We thank the reviewer for this question. Different photon energies were employed\nfor TaS and Co-doped TaS because of matrix element effects. In our experiments, we performed\n2 2\nphoton energy-dependent measurements and selected the photon energy that provided the\nsharpest and most well-defined band structure for each sample. This approach ensures optimal\ncontrast and visibility of the relevant electronic features in ARPES spectra.\n(7) Reviewer’s comment: In the article authors write that there are 2 different terminations one\ncan end up with after cleaving, TaS2 one and Co one. Am I correct to assume that the data shown\nin the main article is from the TaS2 plane? This should be made more clear in the article.\nOur response: We thank the reviewer for this suggestion. All the band structure data presented\nin the main figures were indeed measured on the TaS-terminated surface. To clarify this point,\n2\nwe have revised the sentence on page 5, line 115, from “Overall, the data of the TaS-termination\n2\npresented in Fig. 1f show much sharper band structure.” to “All band structures presented here\nare from the TaS-terminated surface, as it provides much sharper band structures.”\n2\n(8) Reviewer’s comment: Related to my previous question, do authors observe the same type of\nbehaviour (appearance of the same bands, as well as the same trends with doping) on the Co\nplane as well?\nOur response: We thank the reviewer for raising this question. We show in Supplementary Fig. S1\nthe band structure measured on a Co-terminated surface. During cleaving, the Co atoms located\nbetween TaS layers are partially retained on the exposed surface and partially removed with the\n2\nopposite side, leading to a disrupted atomic arrangement. As a result, the residual Co atoms do\nnot form a well-ordered lattice, preventing the emergence of well-defined Co-derived bands. In\naddition, due to the limited photoelectron escape depth, Co-derived states beneath the first TaS\n2\nlayer are not detectable. Since the Co-derived bands are central to our study but are not clearly\nobserved on the Co-terminated surface, we did not pursue doping-dependent measurements\nfor this termination.\n(9) Reviewer’s comment: Page 9 in the SI, Figure S8: The authors observe a Mexican hat-like\ndispersion for samples with 0.29, 0.32 and 0.33 doping, while at 0.34 and 0.36 this turns into a\nsimple hole-like band without a dip at the M point. Can authors comment as to why they see the\ndip also at 0.33 doping since this is above the critical doping value (0.325 listed in the main text)\nand one would expect that 0.33 should behave the same way as 0.34 and 0.36\nOur response: We thank the reviewer for this insightful question. Although the doping level\nx = 0.33 is slightly above the critical point (x ≈ 0.325) where the 3Q order is expected to be re-\nplaced by the helical phase, we still observe a dip at the M point in the ARPES band structure,\nsimilar to lower dopings. We attribute this to the crossover nature of the magnetic transition: the\nchange from 3Q to helical order does not occur abruptly at a single doping value, but rather\nspans a finite doping range.\nFurthermore, even though the anomalous Hall effect is no longer observed at x = 0.33, the\nspectral signature at the M point may persist due to short-range magnetic order or spatial inho-\nmogeneity near the phase boundary. Such effects can produce a coexistence or competition of\nmagnetic orders within a narrow doping window. By contrast, for x = 0.34 and 0.36, the dip is\ncompletely absent, consistent with the system having fully entered the helical regime.\nTo clarify this point, we have added the following sentence to the caption of Supplementary\nFig. S8 (now Fig. S11): “For x = 0.33, although transport measurements no longer show an anom-\nalous Hall effect, a remnant dip at the M point remains visible in ARPES, which may originate from\nshort-range magnetic order or spatial inhomogeneity near the phase boundary.”\nSummary of Modifications:\n1. Following Comment #1 of Referee #1, we have added Fig. R1 as Supplementary Fig. S5, and the\ncorresponding DFT Calculations section in Methods. In addition, we have included a discussion\non page 6, line 162: “We further performed DFT calculations for Co TaS and identified highly-\n1/3 2\ndispersive bands mainly originating from Ta orbitals together with near-E flat bands primarily\nF\nderiving from Co orbitals (see Supplementary Fig. S5).”\n2. Following Comment #1 of Referee #1 and Comment #2 of Referee #2, we have added a discus-\nsion on page 8, line 219: “Since DFT struggles to accurately describe the dispersion of the near-\nEf flat bands, primarily due to strong electron-electron correlations and the presence of local\nmagnetic moments, it cannot provide a reliable account of our system. At the same time, the\nnear-Ef flat bands are dominated by Co 3d orbitals, whereas the dispersive 𝛼 and 𝛽 bands\nmainly originate from Ta 5d states (see Supplementary Fig. S5). Given both the limitations of DFT\nand the distinct orbital characters of these states, we adopt a phenomenological tight-binding\nmodel on a Co triangular lattice to account for the flat bands and their coupling to local magnetic\nmoments.”\n3. Following Comment #1 of Referee #1, we have included Fig. R2 as Supplementary Fig. S12 and\nadded the following sentence on page 10, line 283: “We further tested the robustness of this\nresult by varying hopping amplitude t and coupling constant J, and found that the inverse Mex-\nican hat-like dispersion persists as long as the 3Q order is present (See Supplementary Fig. S12).”\n4. Following Comment #2 of Referee #1, we have revised the sentence “namely, when the magnetic\nmoments are randomly oriented” to “that is, in the non-magnetic state (exchange coupling con-\nstant J=0)” on page 8, line 228.\n5. Following Comment #4 of Referee #1 and Comment #2 of Referee #2, we have added Fig. R5\nas Supplementary Fig. S14, Fig. R4 as Supplementary Fig. S15, and the corresponding Theory\nsection in Methods.\n6. Following Comment #4 of Referee #1, we have added the discussion on page 11, line 304: “Tem-\nperature-dependent measurements provide further evidence for the role of 3Q magnetic order.\nAs shown in Supplementary Fig. S14, the inverse Mexican hat-like dispersion is observed only in\nthe 3Q ordered phase at low temperature. At higher temperatures, corresponding to the 1Q\nordered and paramagnetic states, the dispersion evolves into a simple hole-like band along the\nK-M-K direction. These observations are in close agreement with tight-binding calculations,\nwhich shows that only the 3Q ordered state gives rise to the inverse Mexican hat-like dispersion\n(Fig. 4 and Supplementary Fig. S15). Together, these results demonstrate that the reconstruction\nof the near-Ef bands is a distinctive fingerprint of the 3Q order.”\n7. Following Comment #7 of Referee #1, we have added Fig. R7 as Supplementary Fig. S4\n8. Following Comment #7 of Referee #1, we have added a sentence on page 6, line 149: “Density\nFunctional Theory (DFT) calculations for 2H-TaS reveal that the observed ~300meV band shift\n2\ncan be reproduced by adding one electron per TaS unit cell, in good agreement with a simple\n2\nelectron-counting picture of Co TaS consisting of alternating [Co ]+ and [TaS]- layers (see\n1/3 2 1/3 2\nSupplementary Fig. S4).”\n9. Following Comment #7 of Referee #1, we have added a sentence on page 7, line 189: “The\nobserved slope of the shift of the 𝛼-band minimum shift as a function of Co doping is well\nreproduced by a rigid-band-shift approximation applied to the TaS band structure (see Sup-\n2\nplementary Fig. S4).”\n10. Following Comment #8 of Referee #1, we have added the phrase “a pronounced reduction of\nspectral weight in the highly-dispersive 𝛼 and 𝛽 bands” on page 6, line 170.\n11. Following Comment #8 of Referee #1, we have added “and the DFT-calculated orbital projected\nband structures (see Supplementary Note and Supplementary Fig. S6-S7)” on page 6, line 172.\n12. Following Comment #8 of Referee #1, we have revised the sentence on page 7, line 174, to read:\n“the near-Ef 𝛾 and 𝛾 bands mainly originate from Co 3dz2 orbital, the highly-dispersive 𝛼\n𝐾 𝑀\nand 𝛽 bands are dominated by Ta 5dz2 Ta 5dxy/dx2-y2 orbitals, and the 𝜀 band is primarily\ncontributed by dxz/dyz and dxy/dx2-y2 states.”\n13. Following Comment #8 of Referee #1, we have added Fig. R8 as Supplementary Fig. S6, Fig. R9\nas Supplementary Fig. S7, and included the corresponding discussion as a Supplementary Note\nin the Supplementary Materials.\n14. Following Comment #3 of Referee #2, we have renamed the current γ band as γ (along Γ-\nK\nK) and the previous δ band as γ (Γ -M). In addition, the previous labels for the ε and ζ\nM\nbands are now labeled δ and ε.\n15. Following Comment #4 of Referee #2, we have added the following sentences on page 9, line\n255: “rather than to increased Co doping. This assignment is supported by the observation that\nthe 𝛾 band bottom does not shift downward with doping (see Fig. 2), which is inconsistent with\n𝐾\na simple band-filling picture. Instead, the weakening of the 3Q order alters the Fermi surface\ntopology, naturally leading to the enlarged 𝛾 Fermi pockets.”\n16. Following Comment #1 of Referee #3, we have revised the sentence on page 2, line 40, to read:\n“Magnetically intercalated TMDs, as an important class of van der Waals magnets, open a new\nparadigm in TMD research and provide a versatile platform to investigate the interplay between\nmagnetic order and electronic properties.”\n17. Following Comment #3 of Referee #3, we have revised the wording on page 5, line 136, changing\n“several” to “two”.\n18. Following Comment #4 of Referee #3, we have included “x=0.36” on page 12, line 331 in the\nMethods section.\n19. Following Comment #5 of Referee #3, we have revised “10K” to “7K” on page 4, line101.\n20. Following Comment #7 of Referee #3, we have revised the sentence on page 5, line 115, to read:\n“All band structures presented here are from the TaS-terminated surface, as it provides much\n2\nsharper band structures.”\n21. Following Comment #9 of Referee #3, we have added the following sentence to the caption of\nSupplementary Fig. S11: “For x = 0.33, although transport measurements no longer show an\nanomalous Hall effect, a remnant dip at the M point remains visible in ARPES, which may origi-\nnate from short-range magnetic order or spatial inhomogeneity near the phase boundary.”\n22. We have moved Debasis Dutta and Guoqing Chang from the Acknowledgements to the list of\nco-authors, as they contributed to the DFT calculations. Their funding information has been\nadded to the Acknowledgments.\nReferences:\n1. P. Park, et al. Field-tunable toroidal moment and anomalous Hall effect in noncollinear antifer-\nromagnetic Weyl semimetal Co TaS. npj Quantum Materials 7, 42 (2022).\n1/3 2\n2. P. Park, et al. Tetrahedral triple-Q magnetic ordering and large spontaneous Hall conductivity in\nthe metallic triangular antiferromagnet Co TaS. Nature Communications 14, 8346 (2023)\n1/3 2\n3. L. Xie, et al. Structure and Magnetism of Iron- and Chromium-Intercalated Niobium and Tanta-\nlum Disulfides. Journal of the American Chemical Society 144, 9525-9542 (2022).\n4. L. Xie, et al. Comparative Electronic Structures of the Chiral Helimagnets Cr NbS and Cr TaS.\n1/3 2 1/3 2\nChemistry of Materials 35, 7239-7251 (2023).\n5. S. Parkin, et al. 3d Transition-Metal Intercalates of the Niobium and Tantalum Dichalcogenides.\nI. Magnetic Properties. Philosophical Magazine B 41, 65-93 (1980).\n6. A. Damascelli, et al. Angle-Resolved Photoemission Studies of the Cuprate Superconductors.\nReviews of Modern Physics 75, 473-541 (2003)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I went through the authors' response letter, and I think they addressed my questions/concerns properly. Therefore, I can recommend the publication of this manuscript.", "reviewer_2": "I thank the authors for the efforts to answer all the relevant questions. Although I find that most of issues are well addressed, I am still not convinced that the new data in Fig. S14 provides “additional” support for the electronic signature of the 3Q magnetic phase. Specifically, the temperature-dependent change in the EDC of Fig. S14(d-f) is very subtle and can be easily affected by the thermal broadening, i.e., the Mexican-hat bands might still be present in the non-magnetic phase, but it becomes invisible due to thermal broadening at elevated temperatures. In general, the temperature-induced change in the experimental band structure is very small compared to theoretical predictions. Some clarification on this point will certainly be important to a comprehensive understanding of the possible 3Q magnetic phase."}, "author_response": "Response to Reviewer’s Comments:\nResponse to Reviewer #1:\nI went through the authors' response letter, and I think they addressed my\nquestions/concerns properly. Therefore, I can recommend the publication of this\nmanuscript.\nWe thank the referee for carefully reading our revised manuscript and response letter\nand for recommending publication.\nResponse to Reviewer #2:\nI thank the authors for the efforts to answer all the relevant questions. Although I find\nthat most of issues are well addressed, I am still not convinced that the new data in Fig.\nS14 provides “additional” support for the electronic signature of the 3Q magnetic phase.\nSpecifically, the temperature-dependent change in the EDC of Fig. S14(d-f) is very subtle\nand can be easily affected by the thermal broadening, i.e., the Mexican-hat bands might\nstill be present in the non-magnetic phase, but it becomes invisible due to thermal\nbroadening at elevated temperatures. In general, the temperature-induced change in the\nexperimental band structure is very small compared to theoretical predictions. Some\nclarification on this point will certainly be important to a comprehensive understanding\nof the possible 3Q magnetic phase.\nWe thank the referee for this thoughtful comment. To clarify whether the Mexican hat-\nlike dispersion can be obscured by thermal broadening at 82K, we performed quantitative\nsimulations that incorporate the sources of thermal broadening: the Fermi-Dirac cutoff\nand the temperature evolution of the intrinsic scattering rate.\nTo model the ARPES spectra, we start from an intrinsic spectral peak represented by a\nLorentzian 𝐿(𝐸) centered at 𝐸−𝐸 = 𝐸 with a full width at half maximum (FWHM) Γ.\n𝐹 0\nThe temperature-dependent occupation is introduced by multiplying this spectrum by\nthe Fermi-Dirac distribution 𝑓(𝐸,𝑇) . The resulting curve is then convolved with the\nexperimental energy resolution, which is modeled as a Gaussian 𝐺(𝐸;𝛥𝐸 ) with FWHM\n𝑟𝑒𝑠\n𝛥𝐸 . The simulated energy distribution curve (EDC) is therefore\n𝑟𝑒𝑠\n𝐼 (𝐸,𝑇)= [𝐿(𝐸)𝑓(𝐸,𝑇)]⨂𝐺(𝐸),\n𝑠𝑖𝑚\nwhere\n1 Γ⁄2 1\n𝐿(𝐸) = , 𝑓(𝐸,𝑇)= ,\n𝜋(𝐸−𝐸0)2+(Γ⁄2)2 𝑒𝑥𝑝( 𝐸−𝐸𝐹)+1\n𝑘𝐵𝑇\nand\n𝐺(𝐸;𝛥𝐸 ) =\n1\n𝑒𝑥𝑝(-\n𝐸2\n), 𝜎 =\n𝛥𝐸𝑟𝑒𝑠\n, \n𝑟𝑒𝑠 𝜎√2𝜋 2𝜎2 2√2 𝑙𝑛2\nwith 𝑘 the Boltzmann constant and ⊗ denoting convolution [1].\n𝐵\nTo simulate the 6K spectra shown in Fig. R1g, we set T=6K, ∆𝐸 ≈\n𝑟𝑒𝑠\n√∆𝐸 2+∆𝐸 2=10.6meV, and Γ=24meV, and choose 𝐸 =-6meV and -2meV to\n𝑎𝑛𝑎𝑙𝑦𝑧𝑒𝑟 ℎ𝑣 0\nbest reproduce experimental EDC(k1) and EDC(k2) from Fig. R1d, respectively. The inverse\nMexican hat-like dispersion is reflected in the higher binding energy of the EDC(-6meV)\npeak relative to the EDC(-2meV) peak. To examine whether an increased temperature\ncould smear out the Mexican-hat feature, we then simulate the 31K and 82K spectra (Fig.\nR1h and R1i). In these simulations, the peak-center energies 𝐸 are kept fixed at -6meV\n0\nand -2meV for the orange and blue curves, respectively, and two physically motivated\nquantities are varied: (i) the temperature in the Fermi-Dirac distribution and (ii) the\nLorentzian FWHM (Γ), which is adjusted to best reproduce the experimental data. The\nlatter is allowed to increase modestly with temperature because the intrinsic linewidth\nreflects the imaginary part of the self-energy (i.e., the intrinsic scattering rate), which\ngenerally grows with temperature due to electron-phonon and electron-electron\ninteractions.\nImportantly, these simulations show that over the temperature range studied (6-82 K),\neven after including both sources of thermal broadening, an inverse Mexican hat-like\ndispersion that persists to high temperature would still produce a clearly deeper EDC\npeak for 𝐸 =-6meV than for 𝐸 =-2meV. Thermal broadening alone therefore cannot\n0 0\nhide such an inverse Mexican hat-like dispersion within our experimental temperature\nwindow. In contrast, the 82K data (Fig. R1f) exhibit the opposite behavior: the EDC(k1)\npeak becomes slightly shallower than the EDC(k2) peak. Thus, the comparison between\npanels (d-f) and (g-i) shows that the hole-like dispersion observed at 31K and 82K cannot\nbe generated by thermal broadening of an inverse Mexican hat-like band and is instead\nintrinsic. Notably, stronger thermal broadening could in principle smear out this\ncharacteristic inverse Mexican hat-like band. As illustrated by the example in the inset of\nFig. R1i, when the temperature is raised to 200 K and the FWHM is increased to Γ=100\nmeV, the two EDCs nearly coincide, such that the underlying inverse Mexican hat-like\nband becomes effectively indiscernible.\nIn the revised Supplementary Materials, we have incorporated Fig. R1g-R1i into\nSupplementary Fig. S14, added Supplementary Note 2 to explain the temperature-\ndependent data, and included the above discussion as Supplementary Note 3 to clarify\nthat the hole-like dispersion at 31K and 82K is intrinsic rather than a thermally masked\ninverse Mexican hat-like dispersion. In the revised main text (page 11), we explicitly refer\nto Supplementary Notes 2 and 3 when discussing the temperature-dependent\nmeasurements.\nFig. R1. Temperature-dependent ARPES spectra of Co TaS along the K-M-K direction. (a-\n0.31 2\nc) Band structures measured at 6K, 31K, and 82K, respectively. Arrows mark the momenta\nk1(M point) and k2. (d-f) EDCs extracted at k1 and k2 from panels (a-c). (g-i) Simulated EDCs\nat 6 K, 31 K, 82 K, and 200K (inset of panel i), assuming that the underlying band dispersion\nremains the same inverse Mexican hat-like dispersion as at 6 K, i.e., all orange and blue curves\ngenerated from a Lorentzian lineshape 𝐿(𝐸) centered at 𝐸 =-6meV and -2meV,\n0\nrespectively. The FWHM value Γ used for simulations at 6 K, 31 K, and 82 K are chosen to\nbest reproduce the experimental EDCs in panels (d-f).\nReference:\n1. A. Damascelli, et al. Angle-Resolved Photoemission Studies of the Cuprate\nSuperconductors. Reviews of Modern Physics 75, 473-541 (2003)."}, {"round": 3, "reviewer_comments": {"reviewer_2": "I thank the authors for addressing this question. The detailed simulation now provides a compelling evidence for the electronic transition associated with the triple-Q phase. Therefore, I recommend the publication of this paper in Nature Communications in the current form."}, "author_response": "We thank the referee for carefully reading our revised manuscript and response letter and for recommending publication."}]

Decision letter