Tunable Octdong and Spindle-Torus Fermi Surfaces in Kramers Nodal Line Metals

Domain: physics
Published: 2025-12-12
Editorial outcome: accept

Abstract
Kramers nodal lines are doubly degenerate band crossings in achiral non-centrosymmetric crystals, arising from spin-orbit coupling and connecting time-reversal invariant momenta. When intersecting the Fermi level, they generate exotic three-dimensional Fermi surfaces, in some cases described by two-dimensional massless Dirac fermions, enabling enhanced graphene-like physics such as quantized optical conductivity and large anomalous Hall effects. However, no experimental realization of such materials has been reported. Here, we identify Kramers nodal line metals beyond the case of Fermi surfaces enclosing a single time-reversal invariant momentum. Using angle-resolved photoemission spectroscopy and first-principles calculations, we show that 3R-TaS2 and 3R-NbS2 host open Octdong and Spindle-torus Fermi surfaces, respectively. We observe a filling-controlled transition between these configurations and evidence of size quantization in 3R-TaS2 inclusions within 2H-TaS2. We further predict a strain- or pressure-driven transition to a conventional metal. Our results establish 3R transition-metal dichalcogenides as a tunable platform for Kramers nodal line physics.

Introduction
The discovery of graphene revolutionized condensed matter physics by unveiling exotic phenomena such as quantized optical conductivity and light-induced anomalous Hall effects, which arise from the unique two-dimensional Dirac fermions on its Fermi surface. A key limitation of graphene, however, is the presence of only two Dirac fermions at the same energy, which significantly restrict the versatility and magnitude of the aforementioned phenomena. A major challenge in this field has thus been to extend these unique properties in materials with multiple Dirac fermions. Although the discovery of three-dimensional topological semimetals exhibiting multiple Weyl, Dirac or multi-fold fermions represented a significant milestone in that direction, these materials typically do not exhibit graphene-like physics, such as quantized optical conductivity due to the three-dimensional nature of their relativistic fermions. Additionally, they pose additional challenges for practical applications since the Dirac and Weyl cones can be located far from the Fermi level.

Recently, it has been predicted that all non-centrosymmetric achiral materials host so-called Kramers nodal lines (KNLs), which are doubly degenerate band crossings connecting time-reversal invariant momenta (TRIMs) in the Brillouin zone and arise due to the combination of time-reversal symmetry and achiral little group symmetries. When Kramers nodal lines are crossing the Fermi-level they form a Kramers nodal line metal (KNLM), characterized by exotic Octdong (figure-eight) and Spindle-torus Fermi-surfaces, where all electrons are described by a family of two-dimensional Dirac or Rashba Hamiltonians, respectively. Crucially, KNLM arise due to the presence of spin-orbit coupling (SOC) and have been predicted to lead to fascinating new phenomena: When the materials are confined, size quantization can lead to a quantized optical conductivity with multiple quantization steps, arising from the presence of multiple Dirac cones at different energies, in contrast to the single quantized value in graphene. Furthermore, one of the Dirac cones is pinned at the Fermi level, meaning that the onset frequency for the quantization is guaranteed to be zero. This is unlike the case of generic Dirac and Weyl points in other materials, and hence lead to a finite onset frequency. Moreover, under light illumination, giant anomalous Hall effects are expected to occur due to the large number of two-dimensional massless Dirac fermions forming the Fermi-surface. Finally, since KNLs are transformed into Kramers-Weyl fermions upon breaking of the mirror or roto-inversion symmetries, KNLM are the parent state for structurally chiral Kramers-Weyl semimetals, which have so far remained elusive in experiments. As a result, KNLM could realize strain induced spin-hedgehogs, orbital angular momentum monopoles, unconventional spin-orbit torques that could be exploited for novel spintronic devices, and a quantized circular photogalvanic effect without multiband corrections. However, due to the lack of material candidates, these theoretical predictions have so far remained untested experimentally.

It has previously been shown that if the Fermi surface encloses only one of the TRIMs connected by the KNL, the material is guaranteed to form a KNLM and the touching Fermi surfaces can be ascribed to either an octdong or spindle-torus. In the former (latter), two Fermi surface pockets that enclose different TRIMs (the same TRIMs) are touching and all electrons on the Fermi surface can be described by a family of Dirac (Rashba) Hamiltonians parametrized by the momentum along the nodal line.

Although several materials have been suggested to host KNLs away from the Fermi level, such as SmAlSi or transition metal ruthenium silicides, to the best of our knowledge, a KNLM where the KNLs are crossing the Fermi-level has not been conclusively identified in experiments yet. Whilst there have been investigations of KNLMs in YAuGe and the charge density wave state of RTe3, the crossing and splitting of the nodal line at the Fermi level is not resolved in the experiments. Moreover, although there have been predictions for the realization of a spindle-torus Fermi surface is real materials, a candidate material hosting an exotic octdong Fermi surface that could give rise to quantized optical conductivity has not yet been identified theoretically. In this work, by combining density functional theory (DFT) and angle-resolved photoelectrons spectroscopy (ARPES), we find the non-centrosymmetric rhombohedral phases (3R) of metallic MS2 (M=Nb,Ta) as candidates for highly tunable KNLMs, realizing both octdong and spindle-torus Fermi surfaces.

For the non-centrosymmetric space group R3m (no. 160) of the 3R transition metal dichalcogenides (TMDCs), each mirror plane supports two KNLs, protected by both time-reversal symmetry and the mirror symmetry, connecting Γ to T as well as L to F (see also Section A of Supplementary Material). While the former is pinned along the high symmetry path connecting the two TRIMs, the latter is only pinned at L and F but is otherwise free to disperse on the mirror plane, hence the name almost movable KNL (AMKNL). The typical Fermi surface of a metallic 3R-TMDC, exemplified by 3R-TaS2, together with an illustration of the nodal lines are shown in Fig. 1b, c, while the representation of the mirror plane torus, obtained by identifying the points k~k+G, is shown in Fig. 1d. Although in TMDCs the Fermi surfaces typically enclose more than one single TRIM, depending on the dispersion of the AMKNL, the realization of a KNLM is still possible. The touching Fermi surfaces are then realizing an open version of the octdong (Fig. 1e) or open spindle-torous (Fig. 1f) that are described by the same type of Hamiltonians as their closed counterparts, and therefore lead to equivalent physical responses.

Interestingly, as we predict theoretically and demonstrate experimentally in this work, the Fermi surfaces of TMDCs can undergo a Lifshitz transition from octdong and spindle-torus induced by a change of the band filling, which can be also be achieved by doping or gating. In particular, our DFT calculations predict that due to the lower band filling, 3R-TaS2 is hosting open octdong Fermi-surfaces. On the other hand, due to a higher band filling, NbS2 realizes open spindle torus Fermi surfaces where the two touching Fermi-surfaces enclose the same TRIMs (Fig. 1c), Γ and T. We confirm these predictions experimentally by directly visualizing these Fermi surfaces by ARPES.

While the 2H-phase of MS2 can be synthesized in stoichiometric single crystals, the 3R-phase has been found to require doping and intercalation to be stable at ambient conditions. This presents a formidable challenge for experimental confirmation of KNLs, because doping and intercalation can lead to broadening of experimental ARPES spectra. However, it is well known that as grown single crystals of hexagonal transition metal dichalcogenides, in particular transition metal disulfides, are susceptible to stacking faults, which can lead to locally distinct stacking configurations and the realization of a 3R-stacking configuration without the need for additional doping. Some of the present authors have recently used microfocused ARPES to obtain relatively sharp spectra of few layer 3R-stacking configurations that were embedded within a single crystal of the 2H-polytype. Here, we make use of these new capabilities to access small domains of 3R-TaS2 with very sharp bands suggesting a low intercalation level for ARPES measurements, which enables us to confirm the presence of octdong Fermi-surfaces in this compound. We furthermore present ARPES measurements of Nb self-intercalated 3R-NbS2, which shows that the measured Fermi-surface is consistent with a spindle-torus Fermi surface. Our DFT calculations show that one can tune between those two Fermi-surface types by changing the band filling, which can in principle be achieved by chemical doping or gating. Based on a tight-binding model, we finally demonstrate that due to the fact that the Fermi surfaces enclose multiple TRIMs, applying pressure or strain can transform these materials from Kramers-nodal line metals into ordinary metals, where the KNL does not pierce the Fermi surface, by changing the topology of the AMKNL.

Results
KNLM with octdong Fermi surface in 3R-TaS2
Because both SOC and non-centrosymmetric space groups are required for the realization of a KNLM, TMDCs involving transition metals, like Nb or Ta, are ideal candidates. During our micro-ARPES measurements of commercially available cleaved TaS2 single crystals nominally in the 2H phase, besides the typical Fermi surface of the 2H phase, we often observed variably sized domains of another phase exhibiting one single band crossing the Fermi level along the Γ−M direction, which we identify as the 3R phase. The crystal structures of the two phases, as well as the characteristic sizes of their respective domains, are shown in Fig. 2a, b. Figure 2c, d show the corresponding x-ray diffraction (XRD) patterns recorded in the 2θ−θ geometry. In the range 2θ=35°−45°, several reflections are observed that are uniquely attributable to either the 2H (blue) or 3R (green) TaS2 polytypes. In particular, two characteristic 3R reflections-(015) and (018)-were recorded via transverse (φ) scans, revealing an intensity ratio of approximately 3.4:1.0. The positions and relative intensities of these peaks are consistent with reported values for the 3R polytype in Pearson’s Crystal Data, which lists a ratio of approximately 3.75:1.00. Notice that the distinction between 3R, 2H, and any other possible stacking can also be made based on fundamental band theory arguments as presented in the Section B of the Supplementary Material.

The measured Fermi surface of a 2H-TaS2 domain is shown in Fig. 2e, together with the corresponding density functional theory (DFT) calculation in Fig. 2f. The experimental Fermi surface shows a central hole pocket surrounded by three ’dog-bone’ electron pockets along the Γ−M direction (along the mirror plane). Importantly, the two pockets are not connected, as can also be seen from the experimental band dispersion along Γ−M displayed in Fig. 2g, which is clearly showing two bands crossing the Fermi level. These two bands are spin-degenerate because of the presence of time-reversal- and inversion-symmetry and are mostly split by interlayer coupling, which is also confirmed by our DFT calculations shown in Fig. 2h.

In contrast, the experimental Fermi-surface of the 3R phase (Fig. 2i) shows that the two pockets are connected into a degenerate point, forming a Fermi surface shaped like an hourglass. Only a single band is resolvable in the band dispersion along Γ−M in Fig. 2k, while two bands are still visible from the experimental Fermi-surface away from the mirror plane. Our DFT calculations of 3R-TaS2, shown in Fig. 2j, l, reproduce the same features observed by ARPES, strongly supporting the identification of the 2H and 3R polytypes.

This implies that the hourglass touching point along the mirror plane, if pierced by the AMKNL, can be ascribed to the open octdong formed by joining together the ’dog-bone’ electron pocket to the central hole pocket. By comparing four band dispersions along a direction perpendicular to the mirror plane (Fig. 2), the experimental data clearly reveals the Kramers nodal line crossing the Fermi level in agreement with the DFT calculations. In particular, our DFT calculations also confirm that the observed splitting when moving away from the mirror plane is due to spin-orbit coupling (solid and dashed lines in the bottom panels of Fig. 3c) and also predict a nodal line that is fully winding twice around the mirror plane torus (Fig. 3b), implying that it must pierce the Fermi surface. The Dirac-likedispersion that is shifting through the Fermi level when going from touching hole to electron pockets (Fig. 3a, c) is the predicted hallmark of the octdong Fermi surface of KNLMs and has never been observed in experiments (c.f. Fig. 3e in ref. 6). Note that the gapless nodal line only occurs at specific out-of-plane momenta kz within the mirror plane, which would technically correspond to a specific photon energy in an ARPES experiment. However, due to the limited probing depth of our experiment that results in out-of-plane momentum broadening, we observe a gapless nodal line spectrum for all photon energies within our experimental resolution. Furthermore, here we only show the first Brillouin zone, but as shown in the Section C of the Supplementary Material, the touching point between the two pockets is reproduced also in neighboring Brillouin zones.

Naturally occurring quantum confinement in 3R-TaS2 samples
Identifying a naturally occuring quantum confinement of the 3R-TaS2 within the 2H-TaS2 sample would be particularly appealing since the octdong Fermi surface can be described by a two-dimensional Dirac Hamiltonian, which leads to a quantization of the optical conductivity when the momentum along the nodal line is quantized. This was originally proposed by considering size quantization in the thin film limit. Intriguingly, our ARPES measurements of the 3R-TaS2 domains show signatures of quantum well states in the valence band (Fig. 4a–c), similar to the 3R-NbS2 and 3R-MoS2 thin film inclusions that were recently reported in commercial single crystals nominally of the 2H polytype. This observation suggests that the 3R phase on the surface of the commercially available 2H-TaS2 could consist of only a few layers, which would also lead to a quantization of the out-of-plane momentum, implying that it could directly exhibit a quantization of the optical conductivity. Based on the number of quantized bands, we estimate that the 3R phase should be approximately 5-monolayers thick. This value is also supported by the analysis of the width of the fitted XRD peaks (see the Section D of the Supplementary Material). To get a reliable estimate of the bulk 2H- vs. 3R-phase volume ratio in the crystal, we performed additional quantitative single crystal X ray diffraction experiments which allowed us to estimate a bulk volume ratio of approximately 8:1 in favor of the 2H phase (see the Section D of the Supplementary Material). This partially deviates from the ARPES data, which show nearly equal surface contributions of 2H and 3R. We attribute the enhanced presence of the 3R phase at the surface either to preferential cleaving at 2H and 3R stacking faults or to cleaving induced stacking faults.

An alternative explanation of our data could be surface band bending, which has also been invoked to explain quantum well states in the valence band of TaSe2, due to a confining surface potential. Such band-bending induced surface confinement would also lead to a quantization of the conduction band and thus quantized optical conductivity similar to the thin film case, which could be investigated in future optical experiments. A third explanation could be that the observed valence band features are actually genuine surface states or surface resonances rather than quantum well states, in which case the conduction band would not have to be quantized. Finally, beyond the well-known quantum well confinement at III-V semiconductor phase boundaries—exploited in optoelectronic devices—similar confinement potentials are also known to arise in metallic quantum wells because of relative and hybridization gaps.

Tuning the octdong into a spindle-torus by increased band filling in 3R-TMDCs
Another interesting feature of the 3R-TMDCs is that, because of their characteristic band structure along the Γ−K path, increasing the filling factor can lead to a Lifshitz transition from the open octdong to the open spindle-torus configuration, as shown by the DFT calculation comparing isoenergy surfaces at different binding energy of 3R-TaS2 in Fig. 4d. Since only the octdong Fermi surface is expected to show quantized optical conductivity, doping or gating these compounds could be used to significantly tune the optoelectronic properties of these materials. To demonstrate this tunability with band filling, we also investigated the commercially available 3R-NbS2 polytype, which is known to have an increased band-filling due to self-doping by Nb interstitials during growth. This material is predicted by DFT to exhibit two touching concentric Fermi surfaces centered at Γ (Fig. 5a), surrounded by six disconnected Fermi surface pockets that are split due to spin-orbit coupling. The predicted Fermi surface is also confirmed by ARPES, showing a central hole pocket surrounded by six disconnected pockets (Fig. 5b). Since the spin-orbit coupling in this compound is relatively small and the spectral features are broad, likely due to the presence of interstitial Nb atoms, we were not able to resolve the predicted spin-orbit splitting of the bands in our ARPES measurement. However, our DFT calculations predict an almost movable nodal line that fully winds twice around the mirror plane torus, so that it is guaranteed to cross the Fermi surface at four points. This would imply that the two hole pockets enclosing Γ and T realize an open spindle-torus Fermi surface, which is consistent with our ARPES measurements.

Nodal line topology and its tunability by strain
As already mentioned in the previous sections, while the existence of the nodal lines is symmetry-enforced, the dispersion of the AMKNL is material-dependent so that one can make a distinction between those that are guaranteed to cross the Fermi surface and those that are not. To do so, one can start by realizing that the quotient space of the mirror plane with the prescription k~k+G, where G is a reciprocal lattice vector, is a torus T2. On the other hand, both the mirror-invariant points of the Fermi surface and the AMKNL connecting L and F can be ascribed to one-dimensional loops S1⊂T2. The different mappings f of loops onto T2 are elements of the fundamental homotopy group π1(T2)=Z×Z where each class is identified by a set of two winding numbers, respectively along the meridian and longitude, f∈[wm,wl]∈π1(T2). If we take the high-symmetry path Γ−T to lie along the torus’ meridian, then we have that the Fermi surface loops and the pinned KNL connecting Γ and T both belong to the classes [±1,0]. Because for symmorphic mirror symmetries there can only be an odd number of lines originating from each TRIM due to TRS exchanging the mirror symmetry eigenvalues, the AMKNL is restrained to the classes [2Z+1,2Z]. Then, if f∈[1,wl≠0], each Fermi surface loop must be pierced by the AMKNL at least wl times. On the other hand, if f∈[wm,0] the crossing is not guaranteed and will depend on the relative position of the nodal line and the Fermi surface. This mean that there can be two types of KNLM, those with wl≠0 where the Fermi surface crossing is guaranteed, and those with wl=0, which are accidental (Fig. 6). Because each wl identifies topologically distinct nodal lines, a transition from one class to the other must happen through a Lifshitz transition of the AMKNL. For both NbS2 and TaS2 the DFT calculations predict wl=2, however we expect it to be possible to enforce a transition to wl=0 by applying strain to the system to affect the out-of-plane dispersion of the nodal line.

To provide an intuitive understanding of this, we construct a tight-binding model with a single orbital and a spin-1/2 degree of freedom. The model accounts for in-plane terms up to fifth nearest-neighbor and out-of-plane terms up to third nearest-neighbor resulting in a Hamiltonian consisting of a sum of 17 matrix components and corresponding hopping terms. In order to model the effects of uniaxial strain, all hopping parameters involving any out-of-plane displacement are scaled, thereby emulating the modification of orbital overlaps. Importantly, the spin-orbit splitting within the mirror planes, and thus the shape of the nodal lines, is fully determined by the relative sizes of out-of-plane and in-plane hopping terms. A dominant in-plane hopping results in nodal lines that close the spin-orbit gap when traversing the Brillouin zone in-plane, as seen in Fig. 6a. Conversely, when out-of-plane spin-orbit terms dominate, the spin-orbit gap closes along the vertical direction, and the nodal line crosses the Γ−T path (Fig. 6b). Strain-induced tuning of these terms can thus drive a transition in the nodal line connectivity. Details of the full tight-binding Hamiltonian are provided in Methods, and comparison with the DFT band structure is shown in the Section E of the Supplementary Material.

Methods
ARPES
Single crystals of 3R-NbS2 and 2H-TaS2 were obtained commercially from HQ Graphene, Groningen. All crystals were used as received from HQ Graphene and were not subjected to any chemical, thermal, or mechanical treatment beyond standard handling for XRD and ARPES measurements. The presence of Ta interstitials revealed by our XRD refinements (see Section D of the Supplementary Material) provides a plausible growth-related mechanism for stabilizing the 3R-TaS2 domains, consistent with previous literature. ARPES measurements of TaS2 were performed at the I05 beamline of Diamond Light Source, with a Scienta DA-30 analyser measured at a pressure of approx. 1-2×10−10 mbar, and using photon energies in the range 30-150 eV, at sample temperatures between 80 and 100 K. The combined energy resolution was approximately 30-40 meV with an approximate beam spot diameter (FWHM) of 4 μm. On the other hand, the measurements of NbS2 were performed at the URANOS beamline at the National Synchrotron Radiation Centre SOLARIS in Krakow (Poland) using a SCIENTA OMICRON DA30L photoelectron spectrometer, at the sample temperature of 15 K. The photon energy used ranged from 20-100 eV.

DFT
We performed density functional theory (DFT) calculations implemented in the Vienna Ab Initio Simulation Package (VASP) utilizing projector-augmented wave method with kinetic energy cutoff for the plane-wave basis 520 eV. Both 3R phases are described by symmetries of SG 160 (R3m) with atoms located at 3a Wyckoff position. Optimized positions of atoms for 3R-NbS2 were taken from Materials Project with lattice parameters a=b=c= 7.087 Å, α=β=γ= 27.403°, the compound has the ID mp-966 (ICSD 4209935). For the calculation of 2H-TaS2 (SG 194) electronic structure, we relaxed a hexagonal cell with lattice parameters a=b= 3.42 Å, c= 13.76 Å, α=β= 90°, γ= 120°. The compound has the IDs mp-1984 and ICSD 68488. Relaxed positions of atoms for 3R-TaS2 were taken for the compound with the code mp-10014 (ICSD 43410), the lattice parameters were a=b=c= 7.089 Å, α=β=γ= 27.281°. To compute the Fermi surface we performed the calculations on a k-grid 41×41×41. For the slab calculations of 3R-TaS2, we constructed the supercell with 5 monolayers and relaxed it until the total energy change between the two steps was less than 1e−4.

Tight binding model
The complete Hamiltonian is expressed as
H(k)= ∑n=1 17 Hn(k), (1)
where, using the primitive lattice vectors a1, a2, a3 and a spin-1/2 basis, the matrices Hn(k) comprise hopping terms in-plane up to 5th NN and out-of-plane up to 3rd order hopping terms.
H1(k)=2< (t1) I2×2 (2)
(H2(k))12=2i t2 +t*2 eiπ 3 ei2 3 π sin(k·a1)+sin(k·a2)+e−i2 3 π sin(k·a3) (3)
(H3(k))12=2i t3 +t*3 ei2 3 π ei2 3 π sin(k·(a1−a2))+sin(k·(a2−a3)) +e−i2 3 π sin(k·(a3−a1)) (4)
(H4(k))11=4< (t4) cos(k·a1)+cos(k·a2)+cos(k·a3) (5)
(H5(k))11=2< (t5) cos(k·(a1−a2))+cos(k·(a2−a3))+cos(k·(a3−a1)) +2=(t5) sin(k·(a1−a2))+sin(k·(a2−a3))+sin(k·(a3−a1)) (6)
(H6(k))12=4=(t6) e−iπ 3 sin(k·(a1−a2+a3))−sin(k·(a1+a2−a3)) +eiπ 3 sin(k·(−a1+a2+a3)) (7)
(H7(k))11=4< (t7) cos(k·(−a1+a2+a3))+cos(k·(a1−a2+a3)) +cos(k·(a1+a2−a3)) (8)
(H8(k))12=2i t8 +t*8 eiπ 3 ei2 3 π sin(k·(2a1−a2−a3))+sin(k·(−a1+2a2−a3)) +e−i2 3 π sin(k·(−a1−a2+2a3)) (9)
(H9(k))11=4< (t9) cos(k·(2a1−a2−a3))+cos(k·(−a1+2a2−a3)) +cos(k·(−a1−a2+2a3)) (10)
(H10(k))12=4=(t10) −sin(k·(a1+a2))+eiπ 3 sin(k·(a2+a3)) +e−iπ 3 sin(k·(a3+a1)) (11)
(H11(k))12=2it11 e−i2 3 π sin(k·(2a1−a2))+sin(k·(2a2−a3))+e−i2 3 π sin(k·(2a3−a1)) +2it*11 eiπ 3 sin(k·(2a2−a1))−sin(k·(2a1−a3))+e−iπ 3 sin(k·(2a3−a2)) (12)
(H12(k))11=4< (t12) (cos(k·(a1+a2))+cos(k·(a2+a3))+cos(k·(a3+a1))) (13)
(H13(k))11=2< (t13) cos(k·(a1−2a2))+cos(k·(a2−2a3))+cos(k·(a3−2a1)) +cos(k·(2a1−a2))+cos(k·(2a2−a3))+cos(k·(2a3−a1)) +2=(t13) sin(k·(a1−2a2))+sin(k·(a2−2a3))+sin(k·(a3−2a1)) +sin(k·(2a1−a2))+sin(k·(2a2−a3))+sin(k·(2a3−a1)) (14)
(H14(k))12=2i t14 +t*14 ei2 3 π ei2 3 π sin(k·2(a1−a2))+sin(k·2(a2−a3)) +e−i2 3 π sin(k·2(a3−a1)) (15)
(H15(k))11=2< (t15) cos(k·2(a1−a2))+cos(k·2(a2−a3))+cos(k·2(a3−a1)) +2=(t15) sin(k·2(a1−a2))+sin(k·2(a2−a3))+sin(k·2(a3−a1)) (16)
(H16(k))12=2it16 eiπ 3 sin(k·(2a1+a2−3a3))+e−iπ 3 sin(k·(−3a1+2a2+a3)) −sin(k·(a1−3a2+2a3)) +2it*16 ei2 3 π sin(k·(a1+2a2−3a3))+sin(k·(−3a1+a2+2a3)) +e−i2 3 π sin(k·(2a1−3a2+a3)) (17)
(H17(k))11=2< (t17) cos(k·(3a1−2a2−a3))+cos(k·(−3a1+a2+2a3)) +cos(k·(−a1+3a2−2a3))+cos(k·(2a1−3a2+a3)) +cos(k·(−2a1−a2+3a3))+cos(k·(a1+2a2−3a3)) +2=(t17) sin(k·(3a1−2a2−a3))+sin(k·(−3a1+a2+2a3)) +sin(k·(−a1+3a2−2a3))+sin(k·(2a1−3a2+a3)) +sin(k·(−2a1−a2+3a3))+sin(k·(a1+2a2−3a3)) (18)
The Hamiltonian is Hermitian, H(k)=H(k)†, and the off-diagonal entries in the matrices Hn(k) are zero. The parameters t1 to t17 are adjusted to match the DFT band structure of 3R-TaS2 on 20 cuts parallel to the path S-Γ-L that are evenly spaced in kz between Γ and T, resulting in Table 1. We approximate the effect of c-axis pressure by multiplying all out-of-plane hopping terms, t2, t4, t6, t7, t10, t11, t12, t13, by a common factor. The length of the out-of-plane component of all relevant hopping vectors are the same, hence we include only one factor for all out-of-plane terms.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_2": "The revised version of the manuscript has improved considerably through the detailed characterization by x-ray crystallography of the 2H-TaS2 crystal from HQ Graphene studied by the authors. They report 11% 3R phase in the bulk 2H-TaS2 crystal from their new XRD analysis. So, it is evident that a mixed-phase crystal has been studied. The authors could not satisfactorily answer my previous comment, given below in italics: “It is likely that the presence of the 3R phase in a 2H crystal, on which their whole study is based, depends on the growth conditions. So, this incomplete information indicates the possibility that, as I mentioned in my earlier report, the amount of the 3R phase in a different batch of crystals from different growers/suppliers might vary significantly. Thus, future researchers, who might want to work further on the present interesting findings reported here, will be put in a disadvantageous position. So, the authors should provide the details of the crystal growth. Alternately, the authors should show the existence of a similar amount of 3R phase at the surface (Fig. 2b) as obtained here on at least two crystals from other sources. This will show that the presence of the 3R phase is independent of any specific condition(s) of crystal growth.” The reasons for my above comment, which is extremely relevant, are explained below: The XRD study by the authors shows that the bulk 2H-TaS2 crystal is multiphase with 11% 3R phase. The impression in the earlier version of the manuscript is that the 50% 3R phase occurs at the surface. But now it is clear that even the bulk crystal is multiphase. Importantly, this is in contradiction with the XRD, EDAX, and Raman studies on the 2H-TaS2 crystals by HQ Graphene, where the 2H-TaS2 crystals are shown to be >99.995% pure (see attachment below, reproduced from https://www.hqgraphene.com/TaS2.php). Moreover, the papers listed on this website, where a 2H-TaS crystal from HQ Graphene was used, do not report 2 evidence of the 3R phase. The author's justification that “HQ Graphene does not disclose specific growth conditions— an unfortunate but common limitation when working with commercially sourced samples, which are now widely used for experimental studies in the condensed matter community” could possibly be acceptable if the crystals were pure single phase. However, in the case of complicated crystals and multiphase crystals, and in particular when the work fully depends on the impurity phase of the multiphase crystal, the details of crystal growth must be provided. In such cases, the norm in the scientific community is to include the crystal grower(s) as coauthor(s) of the manuscript. In this work, it is even more necessary because the multiphase nature of the 2H-TaS2 crystal is in contradiction with the data shown by crystal grower(s) (HQ Graphene). I suggested in my previous report that the authors should show the existence of a similar amount of 3R phase (as obtained here) on at least two crystals from other sources, in case they are unable to provide the details of crystal growth. In the absence of any details about crystal growth, this would have shown that their results are not dependent on a particular batch of crystals from HQ Graphene. The authors have failed to do that. The above discussion shows that the foundation of the present work is rather weak, since it is done on the “impurity” phase (3R) of a multiphase crystal declared to be pure by the crystal grower (HQ Graphene). An important implication of this disagreement is that the results of the earlier work in the literature (see the attached website where these works are listed) performed with HQ Graphene crystals on bulk 2H-TaS crystals, which do not find 2 evidence of the 3R phase, will be shown as a corollary of this work to be rather doubtful. This will create an undesirable situation for the research community working with commercial crystals. So, the authors must carefully scrutinize the history of the treatment done to the crystal(s) they have studied after these arrived in their laboratory. This will help them to pinpoint what led to the appearance of such a large 3R phase. At present, the authors are unable to provide any clue about how their crystals have 11% 3H phase in the bulk of a >99.995% pure 2H-TaS crystal that they obtained from HQ Graphene. 2 This is essential for the scrutiny and reproducibility of the data presented here by the scientific community, which is the cornerstone of scientific research. However, since the authors were unable to address this important issue, manuscript cannot be recommended for acceptance in Nature Communications."}, "author_response": "Response to the comments by the reviewers, round 3 Reviewer’s comment The revised version of the manuscript has improved considerably through the detailed characterization by x-ray crystallography of the 2H-TaS2 crystal from HQ Graphene studied by the authors. They report 11% 3R phase in the bulk 2H-TaS2 crystal from their new XRD analysis. So, it is evident that a mixed-phase crystal has been studied. Author’s reply We thank the referee for their detailed report and for acknowledging the improvements in the revised manuscript, in particular our detailed additional XRD analysis in our revised manuscript that revealed the presence of a 3R phase that we had earlier already detected with our ARPES measurements. Reviewer’s comment The authors could not satisfactorily answer my previous comment, given below in italics: “It is likely that the presence of the 3R phase in a 2H crystal, on which their whole study is based, depends on the growth conditions. So, this incomplete information indicates the possibility that, as I mentioned in my earlier report, the amount of the 3R phase in a different batch of crystals from different growers/suppliers might vary significantly. Thus, future researchers, who might want to work further on the present interesting findings reported here, will be put in a disadvantageous position. So, the authors should provide the details of the crystal growth. Alternately, the authors should show the existence of a similar amount of 3R phase at the surface (Fig. 2b) as obtained here on at least two crystals from other sources. This will show that the presence of the 3R phase is independent of any specific condition(s) of crystal growth.” The reasons for my above comment, which is extremely relevant, are explained below: The XRD study by the authors shows that the bulk 2H-TaS2 crystal is multiphase with 11% 3R phase. The impression in the earlier version of the manuscript is that the 50% 3R phase occurs at the surface. But now it is clear that even the bulk crystal is multiphase. Importantly, this is in contradiction with the XRD, EDAX, and Raman studies on the 2H-TaS2 crystals by HQ Graphene, where the 2H-TaS2 crystals are shown to be >99.995% pure (see attachment below, reproduced from https://www.hqgraphene.com/TaS2.php). Moreover, the papers listed on this website, where a 2H-TaS crystal from HQ Graphene was used, do not report 2 evidence of the 3R phase. Author’s reply We thank the reviewer for highlighting this point and for directing us to the HQ Graphene characterization. To clarify, the supplier’s specification (>99.995 %) refers to elemental/chemical purity, most likely measured by EDX, rather than to structural phase purity. Standard XRD or Raman measurements as reported by HQ Graphene cannot determine polytype fractions with that level of precision, so the quoted figure must describe elemental composition. High elemental purity is fully compatible with small fractions of different stacking sequences (i.e., 2H/3R intergrowth), which are well documented to occur during preparation of layered dichalcogenides samples (e.g., TaS2: Chem. Eur. J. 23, 8082–8091 (2017); MoS2: Cryst. Growth Des. 19, 5762–5767 (2019); NbS2: Dalton Trans. 50, 3216–3223 (2021)). Likewise, Raman spectroscopy does not reliably distinguish the 2H and 3R phases because the high-frequency intralayer phonon modes—those reported by HQ Graphene—are essentially identical for both stacking sequences (see, for example, ACS Nano 10, 1948– 1953 (2016)). In addition, the single-crystal XRD plots provided by HQ Graphene display only the (00l) reflections, where the 2H and 3R peaks occur at nearly the same 2θ values; such data alone cannot rule out a minority 3R component. On the other hand, our XRD measurements, using a much broader set of reflections along multiple directions and explicit structural refinement, resolve this ambiguity and clearly reveal a small 3R minority phase (~11%). We therefore see no contradiction with the HQ Graphene data. Rather, our analysis complements it by providing higher sensitivity to stacking order. We also note that several peaks in the cited single-crystal XRD pattern from HQ graphene show a systematic skewness toward lower 2θ, which would be expected from a minor 3R admixture, as the database entries show that the 3R phase is at slightly smaller angles than the 2H phase. Reference 2H-TaS2 XRD (Pearson’s database) Reference 3R-TaS2 XRD (Pearson’s database) HQ graphene data Reviewer’s comment The author's justification that “HQ Graphene does not disclose specific growth conditions— an unfortunate but common limitation when working with commercially sourced samples, which are now widely used for experimental studies in the condensed matter community” could possibly be acceptable if the crystals were pure single phase. However, in the case of complicated crystals and multiphase crystals, and in particular when the work fully depends on the impurity phase of the multiphase crystal, the details of crystal growth must be provided. In such cases, the norm in the scientific community is to include the crystal grower(s) as coauthor(s) of the manuscript. In this work, it is even more necessary because the multiphase nature of the 2H-TaS2 crystal is in contradiction with the data shown by crystal grower(s) (HQ Graphene). Author’s reply We appreciate the reviewer’s concern about transparency in sample sourcing and growth conditions. We agree that such details are valuable whenever they are available. In this case, however, our findings do not contradict the characterization provided by HQ Graphene. Their data and specifications address elemental purity, whereas our measurements reveal a small minority 3R stacking variant, something not excluded by the supplier’s reports and well known to occur in layered dichalcogenides (references given above). Our own XRD and ARPES analyses provide direct structural evidence for this multiphase nature, giving us confidence in the result even without specific growth details. We therefore believe the conclusions of our work remain well-supported and reproducible despite the absence of proprietary growth information from the commercial supplier. Reviewer’s comment I suggested in my previous report that the authors should show the existence of a similar amount of 3R phase (as obtained here) on at least two crystals from other sources, in case they are unable to provide the details of crystal growth. In the absence of any details about crystal growth, this would have shown that their results are not dependent on a particular batch of crystals from HQ Graphene. The authors have failed to do that. The above discussion shows that the foundation of the present work is rather weak, since it is done on the “impurity” phase (3R) of a multiphase crystal declared to be pure by the crystal grower (HQ Graphene). An important implication of this disagreement is that the results of the earlier work in the literature (see the attached website where these works are listed) performed with HQ Graphene crystals on bulk 2H-TaS crystals, which do not find 2 evidence of the 3R phase, will be shown as a corollary of this work to be rather doubtful. This will create an undesirable situation for the research community working with commercial crystals. So, the authors must carefully scrutinize the history of the treatment done to the crystal(s) they have studied after these arrived in their laboratory. This will help them to pinpoint what led to the appearance of such a large 3R phase. At present, the authors are unable to provide any clue about how their crystals have 11% 3H phase in the bulk of a >99.995% pure 2H-TaS crystal that they obtained from HQ Graphene. 2 This is essential for the scrutiny and reproducibility of the data presented here by the scientific community, which is the cornerstone of scientific research. However, since the authors were unable to address this important issue, manuscript cannot be recommended for acceptance in Nature Communications. Author’s reply We are grateful for the reviewer’s suggestions and share the goal of ensuring that our results are robust and reproducible. However, we respectfully disagree with the view that our conclusions rest on a weak foundation simply because the 3R phase is a minority component of a commercially sourced crystal. Our central finding, the observation of a Kramers nodal-line metal with an Octdong and spindle-torus Fermi surface, is an intrinsic property of any 3R polytype itself. It does not depend on how that polytype is obtained or on its fraction within a given crystal. We share the reviewer’s commitment to reproducibility. Importantly, however, reproducibility does not depend on access to HQ Graphene’s undisclosed growth conditions. What matters is the ability to identify the 3R phase, which can be done straightforwardly and conclusively by careful X-ray diffraction experiments, as we have demonstrated. Indeed, most of the HQ Graphene samples we examined contained measurable 3R fractions (see Supplementary Fig. 11). The opportunity for other groups to reproduce our results is therefore not in question. Researchers may either obtain crystals directly from HQ Graphene and verify the presence of 3R inclusions by XRD, or intentionally grow phase-pure 3R-TaS2 single crystals following established procedures in the literature (e.g., Gotoh et al., J. Alloys Compd. 270, 115 (1998)). Both routes provide a clear and reliable pathway to the 3R polytype whose intrinsic properties underpin our findings. We appreciate the referee’s suggestion to study samples from other suppliers. However, such measurements would not necessarily resolve the role of growth conditions, since uncontrolled differences between suppliers could either suppress or enhance 3R fractions. We therefore believe that the most informative approach for our readers is to provide a prescription to thoroughly characterize the actual structure of the crystals used in our study, as we have done. Reviewer’s comment An important implication of this disagreement is that the results of the earlier work in the literature (see the attached website where these works are listed) performed with HQ Graphene crystals on bulk 2H-TaS 2 crystals, which do not find evidence of the 3R phase, will be shown as a corollary of this work to be rather doubtful. This will create an undesirable situation for the research community working with commercial crystals. Author’s reply We would like to reiterate that there is no disagreement between our findings and the published characterization of the supplier. Moreover, we believe our findings provide helpful context for the community by highlighting that commercially supplied crystals, even those advertised as highly pure, may contain small fractions of different structural phases. Recognizing this possibility should assist researchers in interpreting their own data and, in our view, ultimately strengthen rather than undermine future work with commercially sourced crystals. Reviewer’s comment So, the authors must carefully scrutinize the history of the treatment done to the crystal(s) they have studied after these arrived in their laboratory. This will help them to pinpoint what led to the appearance of such a large 3R phase. Author’s reply We thank the reviewer for this helpful suggestion. We have carefully examined the full handling history of our crystals and find no evidence of any post-growth treatment that could have induced the 3R domains. Our XRD refinements already reveal Ta interstitials within the 3R regions, providing a microscopic mechanism for their stabilization that is consistent with published reports. These observations indicate that the 3R phase formed during growth rather than in our laboratory. To reflect this discussion, we have added a statement to our methods section: “All crystals were used as received from HQ Graphene and were not subjected to any chemical, thermal, or mechanical treatment beyond standard handling for XRD and ARPES measurements. The presence of Ta interstitials revealed by our XRD refinements (see Supplementary Fig. 12) provides a plausible growth-related mechanism for stabilizing the 3R domains, consistent with previous literature [16].” Reviewer’s comment At present, the authors are unable to provide any clue about how their crystals have 11% 3H phase in the bulk of a >99.995% pure 2H-TaS crystal that they obtained from HQ Graphene. 2 This is essential for the scrutiny and reproducibility of the data presented here by the scientific community, which is the cornerstone of scientific research. However, since the authors were unable to address this important issue, manuscript cannot be recommended for acceptance in Nature Communications. Author’s reply We would like to reiterate that the >99.995 % figure provided by HQ Graphene refers to elemental purity as determined by EDX, not to structural polytypism. Because the 2H and 3R phases share the same elemental composition, neither EDX nor standard Raman measurements can accurately distinguish them. Consequently, the supplier’s specification is fully compatible with the minority 3R stacking sequence that we detect in our XRD and ARPES measurements. We fully share the reviewer’s concern for reproducibility, and we emphasize that the opportunity to reproduce our results is not in question. Our XRD refinements reveal Ta interstitials that provide a well-documented mechanism for stabilizing the 3R phase, consistent with prior literature. Moreover, established recipes for the intentional growth of phase-pure 3R-TaS₂ are available (e.g., Gotoh et al., J. Alloys Compd. 270, 115 (1998)), so other groups can either (i) obtain crystals from HQ Graphene and verify 3R inclusions by XRD, or (ii) grow phase-pure 3R crystals directly. Most importantly, the central advance of our work, the observation of a Kramers nodal-line metal with an Octdong and spindle-torus Fermi surface, is an intrinsic property of the 3R polytype itself and does not depend on how the 3R phase is obtained. We therefore believe the scientific foundation of our study remains strong and that the results will be of broad value to the condensed-matter community."}]

Decision letter