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

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... triangular lattice Heisenberg model is a textbook example that can host diverse quantum states with small variations of short-range exchange interactions. The generic ground state for nearest-neighbour (NN) antiferromagnetic interactions is the three-sublattice 120° structure shown in Fig. 1a. This spiral structure is characterized by an ordering wave vector located at ± K-points of the hexagonal Brillouin zone (Fig. 1d). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1b, whose ordering wave vector is one of the three ...

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... host diverse quantum states with small variations of short-range exchange interactions. The generic ground state for nearest-neighbour (NN) antiferromagnetic interactions is the three-sublattice 120° structure shown in Fig. 1a. This spiral structure is characterized by an ordering wave vector located at ± K-points of the hexagonal Brillouin zone (Fig. 1d). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1b, whose ordering wave vector is one of the three M-points of the 3 Brillouin zone (see Fig. 1e). Remarkably, for S = ½, these two phases seem to be separated by a quantum spin liquid ...

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... interactions is the three-sublattice 120° structure shown in Fig. 1a. This spiral structure is characterized by an ordering wave vector located at ± K-points of the hexagonal Brillouin zone (Fig. 1d). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1b, whose ordering wave vector is one of the three M-points of the 3 Brillouin zone (see Fig. 1e). Remarkably, for S = ½, these two phases seem to be separated by a quantum spin liquid state 5-11 whose nature is not yet fully ...

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... is characterized by an ordering wave vector located at ± K-points of the hexagonal Brillouin zone (Fig. 1d). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1b, whose ordering wave vector is one of the three M-points of the 3 Brillouin zone (see Fig. 1e). Remarkably, for S = ½, these two phases seem to be separated by a quantum spin liquid state 5-11 whose nature is not yet fully ...

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... small effective four-spin interactions can induce a fundamentally different chiral antiferromagnetic order in triangular lattice antiferromagnets. This state is the "triple-Q" version of the stripe order, where the three different M-ordering wave vectors (see Fig. 1f) coexist in the same phase giving rise to a noncoplanar four-sublattice magnetic ordering (see Fig. 1c). The spins of each sublattice point along the all-in or all-out principal directions of a regular tetrahedron. Theoretical studies suggest that this state can appear naturally in metallic TLAFs, where effective four-spin interactions ...

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... four-spin interactions can induce a fundamentally different chiral antiferromagnetic order in triangular lattice antiferromagnets. This state is the "triple-Q" version of the stripe order, where the three different M-ordering wave vectors (see Fig. 1f) coexist in the same phase giving rise to a noncoplanar four-sublattice magnetic ordering (see Fig. 1c). The spins of each sublattice point along the all-in or all-out principal directions of a regular tetrahedron. Theoretical studies suggest that this state can appear naturally in metallic TLAFs, where effective four-spin interactions arise from the exchange interaction between conduction electrons and localized spin degrees of freedom ...

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... ordering is noteworthy for its topological nature, as it can be viewed as the short-wavelength limit of a magnetic skyrmion crystal 17 . The three spins of each triangular plaquette span one-quarter of the solid angle of a sphere, implying that each skyrmion (one flux quantum) is confined to four triangular plaquettes. As illustrated in Fig. 1c, the two-dimensional (2D) magnetic unit cell of the tetrahedral ordering consists of eight triangular plaquettes, meaning there are two skyrmions per magnetic unit cell. In other words, the tetrahedral triple-Q ordering creates a very strong effective magnetic field of one flux quantum divided by the area of 4 triangular plaquettes in ...

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... Moreover, the calculated low-energy magnon spectra of the tetrahedral ordering agree with the spectra measured by inelastic neutron scattering. Finally, we discuss the robustness of the tetrahedral ordering against an applied magnetic field in Co1/3TaS2. Co1/3TaS2 is a Co-intercalated metal comprising triangular layers of magnetic Co 2+ ions (Fig. 1g). Previous studies on Co1/3TaS2 in the 1980s reported the bulk properties of a metallic antiferromagnet with S = 3/2 (a high-spin d 7 configuration of Co 2+ ), 23-25 including a neutron diffraction study that reported an ordering wave vector qm = (1/3, 1/3, 0) characteristic of a 120° ordering 25 . More recently, an experimental study ...

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... 1/3, 0) characteristic of a 120° ordering 25 . More recently, an experimental study on single-crystal Co1/3TaS2 observed a significant anomalous Hall effect (AHE) comparable to those in ferromagnets below 26.5 K, which is the second transition temperature (TN2) of the two antiferromagnetic phase transitions at TN1 = 38 K and TN2 = 26.5 K (see Fig. 1h) 26 . Based on the 120° ordering reported in Ref. 25 and a symmetry argument, the authors of Ref. 26 suggested that the observed AHE, σ xy (í µí°‡í µí°‡ = 0) ≠ 0, can be attributed to a ferroic order of cluster toroidal dipole moments. However, our latest neutron scattering data reported in this work reveals an entirely different ...

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... most symmetric triple-Q ordering that produces the same neutron diffraction pattern as that of Fig. 2h is illustrated in Fig. 2i. This is precisely the four-sublattice tetrahedral ordering shown in Fig. 1c, except that Co1/3TaS2 has an additional 3D structure with an AB stacking pattern. Such a triple-Q counterpart can be obtained through a linear combination of three symmetrically-equivalent single-Q states ( í µí°Œí µí°Œ í µí±–í µí±– í µí¼ˆí µí¼ˆ = í µíº«í µíº« í µí¼ˆí µí¼ˆ cos(í µí°ªí µí°ª í µí±ší µí±š í µí¼ˆí ...

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... observation suggests that the Co 3d bands would retain their localized character, while itinerant electrons mainly arise from the Ta 5d bands. Our density functional theory (DFT) calculations confirm this picture and reveal that the density of states near the Fermi energy has a dominant Ta 5d orbital character ( Supplementary Fig. 11). In this situation, a magnetic Co 2+ ion can interact with another Co 2+ ion only via the conduction electrons in the Ta 5d bands. ...

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... complete comparison between our data and the two calculations is shown in Supplementary Fig. 10. Additionally, the linear magnon modes of Co1/3TaS2 are also slightly gapped (~ 0.5 meV, see Fig. 3i). ...

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... measured the powder X-ray diffraction (XRD) pattern of Co1/3TaS2 using a high-resolution (Smartlab, Rigaku Japan) diffractometer, which confirmed the desired crystal structure without any noticeable disorder ( Supplementary Fig. 1). In particular, the intercalation profile of Co atoms was carefully checked by the (10L) superlattice peak pattern in the low-2θ region. ...

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... functional theory (DFT) calculations. We performed first-principles calculations using 'Vienna ab initio simulation package (VASP)' 43-45 based on projector augmented wave (PAW) potential 46 and within Perdew-Burke-Ernzerhof (PBE) type of GGA functional 47 (see Supplementary Fig. 11). DFT+U method 48,49 was adopted to take into account localized Co-3d orbitals properly, where U = 4.1 eV and JHund = 0.8 eV were used as obtained by the constrained RPA for CoO and Co 50 . ...

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... triangular lattice Heisenberg model is a textbook example that can host a diverse spectrum of quantum states with small variations of short-range exchange interactions. For any spin value, the generic ground state for nearest-neighbour (NN) antiferromagnetic interactions is the three-sublattice 120° structure shown in Fig. 1a. This spiral structure is characterized by an ordering wave vector located at ± K-points of the hexagonal Brillouin zone (Fig. ...

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... states with small variations of short-range exchange interactions. For any spin value, the generic ground state for nearest-neighbour (NN) antiferromagnetic interactions is the three-sublattice 120° structure shown in Fig. 1a. This spiral structure is characterized by an ordering wave vector located at ± K-points of the hexagonal Brillouin zone (Fig. ...

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... a relatively small second NN antiferromagnetic interaction gives rise to the twosublattice collinear stripe spin configuration shown in Fig. 1b, whose ordering wave vector is one of the three M-points of the Brillouin zone (see Fig. 1e). Remarkably, for S = 1/2, these two phases seem to be separated by a quantum spin liquid state 3-9 whose nature is not yet fully ...

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... a relatively small second NN antiferromagnetic interaction gives rise to the twosublattice collinear stripe spin configuration shown in Fig. 1b, whose ordering wave vector is one of the three M-points of the Brillouin zone (see Fig. 1e). Remarkably, for S = 1/2, these two phases seem to be separated by a quantum spin liquid state 3-9 whose nature is not yet fully ...

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... small effective four-spin interactions can induce a fundamentally different chiral antiferromagnetic order in triangular lattice antiferromagnets. This state is the "triple-Q" version of the stripe order, where the three different M-ordering wave vectors (see Fig. 1f) coexist in the same phase giving rise to a noncoplanar four-sublattice magnetic ordering (see Fig. 1c). The spins of each sublattice point along the all-in or all-out principal directions of a regular tetrahedron. Theoretical studies suggest that this state can appear naturally in metallic TLAFs, where effective four-spin interactions ...

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... the calculated magnon spectra of the tetrahedral ordering agree with the low-energy spectra measured by inelastic neutron scattering. Finally, we discuss the robustness of the tetrahedral ordering against an applied magnetic field in Co1/3TaS2. Co1/3TaS2 is a Co-intercalated metal comprising triangular layers of magnetic Co 2+ ions (Fig. 1g). While most studies from the 1980s focused on the typical bulk properties of a metallic antiferromagnet 19,20 , a previous neutron diffraction study reported the ordering wave vector qm = (1/3, 1/3, 0) characteristic of 120° ordering 21 . In contrast, the latest experimental study of the single-crystal Co1/3TaS2 reported two ...

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... the typical bulk properties of a metallic antiferromagnet 19,20 , a previous neutron diffraction study reported the ordering wave vector qm = (1/3, 1/3, 0) characteristic of 120° ordering 21 . In contrast, the latest experimental study of the single-crystal Co1/3TaS2 reported two antiferromagnetic phase transitions at TN1 = 38 K and TN2 = 26.5 K (Fig. 1h), as well as a significant anomalous Hall effect (AHE) comparable to that of ferromagnets below TN2 22 . The authors of Ref. 22 interpreted the observed AHE as being based on the ferroic order of cluster toroidal dipole moments, which is the only scenario that gives rise to finite σ xy (í µí°‡í µí°‡ = 0) for the single-Q 120° ordering ...

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... observation suggests that the Co 3d bands would retain their localized character, while itinerant electrons mainly arise from the Ta 5d bands. Our density functional theory (DFT) calculations confirm this picture and reveal that the density of states near the Fermi energy has a dominant Ta 5d orbital character ( Supplementary Fig. 12). In this situation, a magnetic Co 2+ ion can interact with another Co 2+ ion only via the conduction electrons in the Ta 5d bands. ...

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... intensity of the quadratic magnon mode is much stronger than that of the triple-Q spectra, in apparent disagreement with our INS data. A complete comparison between our data and the two calculations is shown in Supplementary Fig. 10. Additionally, the linear magnon modes of Co1/3TaS2 are slightly gapped (~ 0.5 meV, see Fig. 3i). ...

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... the linear magnon modes of Co1/3TaS2 are slightly gapped (~ 0.5 meV, see Fig. 3i). This feature can be explained for tetrahedral ordering by including the higher-order single-ion anisotropy í µí±‚í µí±‚ 6 6 (see Supplementary Fig. 11). Moreover, í µí±‚í µí±‚ 6 6 can describe the tetrahedral spin configuration obtained from the Rietveld refinement (Fig. 2i), where one of the four spins is parallel to the c-axis and the in-plane components of the other three spins are parallel to the a*-axis (see Supplementary Fig. 11). ...

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... feature can be explained for tetrahedral ordering by including the higher-order single-ion anisotropy í µí±‚í µí±‚ 6 6 (see Supplementary Fig. 11). Moreover, í µí±‚í µí±‚ 6 6 can describe the tetrahedral spin configuration obtained from the Rietveld refinement (Fig. 2i), where one of the four spins is parallel to the c-axis and the in-plane components of the other three spins are parallel to the a*-axis (see Supplementary Fig. 11). ...

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... measured the powder X-ray diffraction (XRD) pattern of Co1/3TaS2 using a high-resolution (Smartlab, Rigaku Japan) diffractometer, which confirmed the desired crystal structure without any noticeable disorder ( Supplementary Fig. 1). In particular, the intercalation profile of Co atoms was carefully checked by the (10L) superlattice peak pattern in the low-2θ region. ...

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... the INS cross-section calculation with including higher-order single-ion anisotropies ( Supplementary Fig. 11 & Supplementary Table 5), we used the calculation package Su(n)ny which is based on Landau-Lifshitz dynamics (LLD) 41,42 . We used a 100×100×10 supercell and the Langevin time step of dt = 0.1/(J1S 2 ) for this calculation. ...

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... functional theory (DFT) calculations. We performed first-principles calculations using 'Vienna ab initio simulation package (VASP)' 43-45 based on projector augmented wave (PAW) potential 46 and within Perdew-Burke-Ernzerhof (PBE) type of GGA functional 47 (see Supplementary Fig. 12). DFT+U method 48,49 was adopted to take into account localized Co-3d orbitals properly, where U = 4.1 eV and JHund = 0.8 eV were used as obtained by the constrained RPA for CoO and Co 50 . ...

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... triangular lattice Heisenberg model is a textbook example of a model that can host a diverse spectrum of quantum states with small variations of short-range exchange interactions. For any spin value, the generic ground state for nearest-neighbor (NN) antiferromagnetic interactions is the three-sublattice 120° structure shown in Fig. 1(A). This spiral structure is characterized by an ordering wave vector located at K-points of the hexagonal Brillouin zone ( Fig. 1(D)). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1(B), whose ordering wave vector is one of the three ...

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... states with small variations of short-range exchange interactions. For any spin value, the generic ground state for nearest-neighbor (NN) antiferromagnetic interactions is the three-sublattice 120° structure shown in Fig. 1(A). This spiral structure is characterized by an ordering wave vector located at K-points of the hexagonal Brillouin zone ( Fig. 1(D)). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1(B), whose ordering wave vector is one of the three M-points of the Brillouin zone (see Fig. 1(E)). Remarkably, for S = 1/2, these two phases seem to be separated by a quantum spin ...

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... interactions is the three-sublattice 120° structure shown in Fig. 1(A). This spiral structure is characterized by an ordering wave vector located at K-points of the hexagonal Brillouin zone ( Fig. 1(D)). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1(B), whose ordering wave vector is one of the three M-points of the Brillouin zone (see Fig. 1(E)). Remarkably, for S = 1/2, these two phases seem to be separated by a quantum spin liquid state (1-7) whose nature is not yet fully ...

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... is characterized by an ordering wave vector located at K-points of the hexagonal Brillouin zone ( Fig. 1(D)). Adding a relatively small second NN antiferromagnetic interaction gives rise to the two-sublattice collinear stripe spin configuration shown in Fig. 1(B), whose ordering wave vector is one of the three M-points of the Brillouin zone (see Fig. 1(E)). Remarkably, for S = 1/2, these two phases seem to be separated by a quantum spin liquid state (1-7) whose nature is not yet fully ...

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... a fundamentally different chiral antiferromagnetic order can emerge in a triangular lattice if small effective four-spin interactions are present. This state is the "triple-Q" version of the stripe order, where the three different M-ordering wave vectors (see Fig. 1(F)) coexist in the same phase. This state is a non-coplanar four-sublattice magnetic ordering (see Fig. 1(C)) in which spins point along the all-in or all-out principal directions of a regular tetrahedron. Theoretical studies suggest that this state can appear naturally in metallic TLAFs, where effective four-spin interactions can ...

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... chiral antiferromagnetic order can emerge in a triangular lattice if small effective four-spin interactions are present. This state is the "triple-Q" version of the stripe order, where the three different M-ordering wave vectors (see Fig. 1(F)) coexist in the same phase. This state is a non-coplanar four-sublattice magnetic ordering (see Fig. 1(C)) in which spins point along the all-in or all-out principal directions of a regular tetrahedron. Theoretical studies suggest that this state can appear naturally in metallic TLAFs, where effective four-spin interactions can arise from the magnetic exchange interactions between conduction electrons and localized spin degrees of freedom ...

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... ordering is noteworthy for its topological nature, as it can be viewed as the short-wavelength limit of a magnetic skyrmion crystal (10). The three spins of each triangular plaquette span one-quarter of the solid angle of a sphere, implying that each skyrmion (one flux quantum) is confined to four triangular plaquettes. As illustrated in Fig. 1(C), the two-dimensional (2D) magnetic unit cell of the tetrahedral ordering consists of eight triangular plaquettes, meaning there are two skyrmions per magnetic unit cell. In other words, the tetrahedral triple-Q ordering creates a very strong effective magnetic field of one flux quantum divided by the area of 4 triangular plaquettes in ...

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... is a Co-intercalated metal in which magnetic Co ions (Co 2+ ) form a layered triangular lattice ( Fig. 1(G)). While most studies from the 1980s focused on the typical bulk properties of a metallic antiferromagnet (14,15), a previous neutron diffraction study reported the ordering wave vector qm = (1/3, 1/3, 0) characteristic of 120° ordering (16). In contrast, the latest experimental study of the single-crystal Co1/3TaS2 reported two ...

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... typical bulk properties of a metallic antiferromagnet (14,15), a previous neutron diffraction study reported the ordering wave vector qm = (1/3, 1/3, 0) characteristic of 120° ordering (16). In contrast, the latest experimental study of the single-crystal Co1/3TaS2 reported two antiferromagnetic phase transitions at TN1 = 38 K and TN2 = 26.5 K ( Fig. 1(H)), as well as a significant anomalous Hall effect (AHE) comparable to that of ferromagnets below TN2 (17). The authors of Ref. (17) interpreted the observed AHE as being based on the ferroic order of cluster toroidal dipole moments, which is the only scenario that gives rise to finite σ í µí°‡ í µ 0 for the single-Q 120° ordering ...

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... triple-Q ordering producing the same neutron diffraction pattern as that of Fig. 2(H) is illustrated in Fig. 2(I). This is precisely the four-sublattice tetrahedral ordering shown in Fig. 1(C), except that Co1/3TaS2 has an additional 3D structure with an AB stacking pattern. Such a triple-Q counterpart can be obtained through a linear combination of three symmetrically-equivalent single-Q states ( í µí°Œ í µíº« cosí ...

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... 4(G)) successfully describe the measured INS spectra. For comparison, the magnon spectra of the stripe ordering are also presented in Fig. 4(H). The intensity of the quadratic magnon mode is much stronger than that of the triple-Q spectra, which disagrees with our INS data. A full comparison between our data and the two calculations is shown in Fig. S10. Additionally, the linear magnon modes of Co1/3TaS2 are slightly gapped (~ 0.5 meV, see Fig. 4(I)). This feature, along with the tetrahedral ground state obtained experimentally (Fig. 2(I)), can be explained by including the higher-order single-ion anisotropy í µí±‚ (see Fig. ...

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... comparison between our data and the two calculations is shown in Fig. S10. Additionally, the linear magnon modes of Co1/3TaS2 are slightly gapped (~ 0.5 meV, see Fig. 4(I)). This feature, along with the tetrahedral ground state obtained experimentally (Fig. 2(I)), can be explained by including the higher-order single-ion anisotropy í µí±‚ (see Fig. ...

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... section. Therefore, the sign change at í µí°» represents the transition between tetrahedral orderings with positive and negative í µí¼’ values (see Figs. 4(A)-(B)). Additionally, we explicitly examined the intrinsic í µí¼Ž í µí°‡ í µ 0 of Co1/3TaS2 with the magnetic structure shown in Fig. 4(A) using density functional theory calculations (See Fig. S12). The calculated band structure gives a finite |í µí¼Ž í µí°‡ í µ 0)| value of ~ 125 Ω cm , a similar order of magnitude to the experimental ...

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... effect of an in-plane magnetic field was investigated using single-crystal neutron diffraction. Fig. 4(E) shows field-dependent (H // í µí°ª = (1/2, 0, 0)) intensities of the three magnetic Bragg peaks, each originating from three different í µí°ª of the tetrahedral ordering. Interestingly, the equal intensity of the three peaks remains almost unchanged by a magnetic field up to 10 T, implying the robustness of the tetrahedral ordering against an in-plane magnetic field. ...