SUPPLEMENTARY INFORMATION. Large anomalous Hall effect in a half-heusler antiferromagnet

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1 Large anomalous Hall effect in a half-heusler antiferromagnet T. Suzuki, 1 R. Chisnell, 2 A. Devarakonda, 1 Y.-T. Liu, 1 W. Feng, 3 D. Xiao, 4 J. W. Lynn, 2 and J. G. Checkelsky 1 1 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 3 School of Physics, Beijing Institute of Technology, Beijing , China 4 Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA NATURE PHYSICS 1

2 S1. METHOD Single crystals of GdPtBi were grown by a flux method out of Bi flux [S1]. A mixture of Gd pieces (with a purity of 99.9 %), Pt wire (99.99 %), and Bi needle ( %) in the molar ratio of 1 : 1 : 20 was put in alumina crucible and sealed in an evacuated quartz tube with MPa Ar gas. The sample was heated at 1100 C for 2 hours and then cooled slowly between 900 and 600 C at a rate of 2 C/h, followed by centrifuging to dissociate the GdPtBi crystals from the Bi flux. For neutron scattering measurements, 160 Gd isotopeenriched single crystals were grown in the same way except for starting from the mixture of 97.7 % isotope-enriched 160 Gd (with a purity of 99.9 %), Pt (99.99 %), and Bi ( %) powders. The obtained crystals were confirmed to be of a single phase by powder X-ray diffraction method. The crystals were aligned by X-ray back Laue reflection and/or single crystal X-ray diffraction techniques. Transport measurements were performed with a conventional 4-probe method. Au wires were attached to the sample with Ag paste (DuPont 4922N). The electrical current of typically 1 ma was supplied by a Keithley K6221 current source. The voltage was amplified by a low noise preamplifier (DL Instrument DL1201 and/or Stanford Research SR560) and measured with a Stanford Research SR830 lock in amplifier. To correct for contact misalignment, the measured longitudinal and transverse voltages were field symmetrized and antisymmetrized, respectively. Magnetization measurements were performed using a commercial SQUID magnetometer. Torque magnetometry was performed using a BeCu cantilever with a dimension of mm 3. The gap distance between the BeCu foil and Au bottom electrode was approximately 0.25 mm. The capacitance between the gap was measured by a capacitance bridge. Neutron scattering measurements were carried out on the triple-axis spectrometer BT-7 at NIST Center for Neutron Research [S2]. The incident and outgoing beams were monochromated at λ =2.359 Å using the (002) reflection of the pyrolytic graphite (PG) monochromator and analyzer, respectively. The collimator configuration was open and PG filters were used before and after the sample to suppress higher-order wavelength. Temperature was controlled down to 2 K by a closed cycle He refrigerator and magnetic field was applied up to 9.9 T perpendicular to the scattering vector of the neutron. The sample mounted on an aluminum plate was aligned in such a way that the magnetic field was parallel to the [1 10] direction of the crystal and scattering 2 NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION vector of the neutron lay in the (HHL) plane. The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology. Electronic structure calculations were preformed using the projector augmented wave method, implemented in the Vienna ab initio simulation package [S3]. The fully relativistic projector augmented potentials were adopted in order to include the spin-orbit coupling. The generalized gradient approximation of Perdew-Burke-Ernzerhof parametrization was used for the exchange-correlation potential [S4]. The valence configurations of Gd, Pt, and Bi atoms taken into account in the calculations were 4f 7 5d 1 6s 2,5d 9 6s 1, and 6s 2 6p 3, respectively. Plane-wave energy cutoff of 400 ev and the Monkhorst-Pack k-point mesh of were used for static electronic structure calculations. S2. RESISTIVITY AND MAGNETIC SUSCEPTIBILITY Figure S1a shows the temperature (T ) dependence of resistivity (ρ) up to 300 K. ρ monotonically decreases with increasing T above the onset of the kink concomitant with the antiferromagnetic phase transition. The temperature dependence of the susceptibility (M/H) in 0.01 T is shown in Fig. S1b. M/H in the high T -region is fit to M/H = χ 0 + C/(T Θ D ), where the first term is the sum of the T -independent contribution and the second is the Curie-Weiss type contribution from the localized Gd magnetic moment with Curie constant C and Curie-Weiss temperature Θ D, as shown by the solid line in Fig. S1b. From the fit, we obtain C =7.56 cm 3 / mol K and Θ D = 38 K. This yields µ eff = 3k B C/N =7.78µ B, where k B, N, and µ B are Boltzmann constant, Avogadro s number, and Bohr magneton, respectively. (7.94 µ B ) from Gd 3+ ion (S =7/2 with Landè g-factor g = 2). This is comparable with the expected value S3. TORQUE MAGNETOMETRY Figure S2a plots the magnetic torque (τ) in magnetic field (B) upto30t.b is applied along the direction at different angles (θ) to the [110] direction. τ shows a θ-independent distinct kink at B C 25 T, which is more clearly seen as a peak of d 2 τ/db 2 in Fig. S2b. The peak position of d 2 τ/db 2 versus θ is shown in the inset of Fig. 1c in the main text. NATURE PHYSICS 3

4 S4. SHUBNIKOV-DE HAAS OSCILLATION Shubnikov-de Haas (SdH) oscillations are observed in the high B region. Figure S3 plots d 2 ρ xx /db 2 of GdPtBi sample #4 as a function of 1/B up to B = 31 T. Dip-type structure in ρ xx and enhancement of ρ yx arising from the spin texture is observed at B 2.5 T for this specimen. The periodicity and intensity of SdH oscillation observed up to 1/B =0.13 T 1 (corresponds to B = 7.7 T) does not match to the structure observed around 1/B =0.4 T 1 (B = 2.5 T). This indicates that the pronounced structure in ρ xx and ρ yx observed in the low B region does not originate from Landau quantization. From the oscillation period, k F is estimated to be Å 1. S5. HALL EFFECT ARISING FROM SPIN TEXTURE The total Hall resistivity can be expressed as ρ yx = ρ N yx + ρ A yx + ρ T yx, where ρ N yx is the ordinary Hall effect, ρ A yx is the anomalous Hall effect coupled to the magnetization, and ρ T yx is the Hall effect arising from spin texture (see main text). First, we estimate ρ N yx by using ρ yx at 50 K ( T N ). Figure S4 shows ρ yx and ρ yx ρ yx (T ) ρ yx (50K) ρ yx ρ N yx at 2.5 K. The distinct feature of ρ yx observed at B = 4 T becomes less pronounced in the high B region. ρ A yx can be expressed as the combination of the skew scattering, side-jump, and intrinsic contributions [S5], ρ A yx = αmρ xx0 + βmρ 2 xx0 + bmρ 2 xx, (S1) where ρ xx0 and M are residual resistivity and magnetization, respectively. Using the observed M and assuming ρ xx = ρ xx0 at 2.5 K, we fit ρ yx above 6 T to Eq. (S1). The resulting ρ T yx = ρ yx ρ N yx ρ A yx is shown in the inset. The maximum value of ρ T yx is 0.18 mω cm at B = 3.9 T. S6. ELASTIC NEUTRON SCATTERING MEASUREMENT IN MAGNETIC FIELD Figure S5a and S5b show the profiles of the (001) and ( ) magnetic Bragg peaks at2k(<t N ) in various magnetic field, respectively. The former peak is allowed in the presence of the 3-q type-i antiferromagnetic ordering of Gd, but no intensity around this 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION peak position was observed within the experimental error. On the other hand, for the latter peak, which is allowed in the presence of type-ii ordering, the intensity is enhanced up to B = 6 T and shows saturation above that field. This tendency is seen more clearly in the integrated Gaussian area (G) obtained by fitting the profile to a single Gaussian function (shown in solid lines) as a function of B, as shown in Fig. 2d of the main text. It should be noted that the result for B = 0 T is consistent with the previous studies [S6] and we found that the type-ii ordering survives in field. Note that for these experiments B is applied along the [1 10] direction to allow access to scattering from the possible type-i and type-ii structures; the transport response is qualitatively similar with the field along both of these directions. We also performed neutron scattering measurements at all half-integer (HHL) locations with Q < 3.7 Å 1 with ( ) having the highest Q as listed in Table S1. Peaks at larger Q were inaccessible given the experimental setup, and expected to be weak due to the decrease in magnetic form factor at high Q. Magnetic Bragg peaks were observed at all measured locations, and all show the same trends as the ( ) peak shown in Fig. 2d in the main text and Fig. S5 in the supplement: an increase intensity with increasing B that saturates above B of 6 T. The profiles of the (111) peak are shown in Fig. S5c. The peak originates from the nuclear Bragg scattering as well as the magnetic one from the ferromagnetic component of the spin ordering, whose intensities are additive. The magnetic intensity increases with B and the increase of G scales with the square of the magnetization (M 2 ) as shown in Fig. 2d of the main text. This indicates the presence of the ferromagnetic canting of type-ii ordering away from the zero-field collinear state in finite B. In contrast to the behavior of the ( ) peak, the (111) peak intensity is monotonically enhanced up to 9.9 T, which is consistent with the distinct kink observed in the high field torque magnetometry. S7. SCALING RELATION Figure S6 shows σ xy as a function of σ xx at several temperatures (T ). At fixed T, σ xy obtained from the differerent pieces with different carrier concentrations can be fit to σ xy σxx α with temperature dependent exponent α, as shown in Fig. 3b in the main text and Fig. S6. α at 2 K is 1.2 and decreases with increasing T (Fig. 3c in the main text). NATURE PHYSICS 5

6 S8. COMPARISON OF ANOMALOUS HALL EFFECT TO OTHER ANTIFERRO- MAGNETS It is instructive to compare the anomalous Hall effect of GdPtBi with other antiferromagnets. Table S2 lists the Hall resistivity arising from spin textures (ρ T xy) and anomalous Hall angle (Θ AH ) along with the mechanism for selected antiferromagnets. AuFe alloy with 8 atomic % of Fe is a canonical spin glass, where non-vanishing net total spin chirality χ ijk = S i S j S k gives rise to finite anomalous Hall effect [S7, S8]. MnSi and MnGe are known to have a large-scale spin texture called a Skyrmion lattice [S9, S10]. In this phase, the charge carriers acquire a Berry phase in real space, where the fictitious B is determined by the size of the Skyrmion (spin-orbit coupling strength is irrelevant) [S11]. Nd 2 Mo 2 O 7, Mn 5 Si 3, and UCu 5 have non-coplanar spin structures with finite spin chirality χ ijk [S12, S13, S14]. The charge carriers acquire a Berry phase in real space by traversing the spin texture with finite χ ijk, which results in an anomalous Hall effect. Mn 3 Sn and Mn 3 Ge have a non-collinear 120 spin ordering of the magnetic Mn on the kagome sublattice [S15, S16]. Because of their sufficiently low symmetry and the presence of the (avoided) band crossing in the electronic band structure, the anomalous Hall effect originating from the Berry curvature emerges as in the case of ferromagnets [S17]. Note that the anomalous Hall effect of GdPtBi is also explained by this mechanism, where the required symmetry breaking is realized by the magnetic structure in B. Θ AH tends to show a larger value when the Berry curvature in the momentum space is responsible for the anomalous Hall effect. GdPtBi has the largest ρ T yx and Θ AH, which are more than one order of magnitude larger than other antiferromagnets. S9. ANOMALOUS HALL EFFECT ABOVE MAGNETIC TRANSITION TEMPER- ATURE As shown in Fig. 3a in the main text, the anomalous Hall effect of GdPtBi is maintained above the antiferromagnetic transition temperature T N = 9.2 K, unlike the case of Mn 3 Sn and Mn 3 Ge where the anomalous Hall effect is not shown evident above the long-range ordered phase. The finite anomalous Hall effect of GdPtBi above T N can be attributed to the symmetry breaking by the short-range spin correlations (see main text). We note that 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION Mn 3 Sn and Mn 3 Ge are also magnetically frustrated systems (T N = 365 K and Curie-Weiss temperature θ CW = 853 K for Mn 3 Ge [S18]). Those compounds exhibit 120 spin ordering on a kagome sublattice which has clockwise and counterclockwise triangular configurations which are degenerate when only the symmetric exchange interaction is considered [S19, S20]. The antisymmetric exchange interaction determines the ordered state and is uniquely determined on the kagome sublattice. It is known that the regular and inverse 120 ordering states have opposite contributions to the anomalous Hall effect, and thus fluctuations have canceling contributions [S17]. On the other hand, the type-ii antiferromagnetic phase for GdPtBi contains no such canceling degenerate spin correlations. A more detailed investigation of this effect will be subject of future work. S10. MAGNETORESISTANCE GdPtBi shows negative longitudinal magnetoresistance. Figure S7a plots the longitudinal resistivity (ρ xx ) as a function of magnetic field (B) with current (I) parallel to B. In this configuration, ρ xx decreases with increasing B, where its B dependence is superlinear in the low B region (almost quadratic up to 2 T) but becomes sublinear in the high B region. On the other hand, positive magnetoresistance is observed for B I. When B is tilted away from the I direction at angle θ (the geometry is shown in Fig. S7b), ρ xx in B fixed at 9 T exhibits the θ dependence shown in Fig. S7c, where ρ xx (θ) can be fit to the following formula: ρ xx (θ) =ρ 0 + ρ 1 cos 2 θ + ρ 2 cos 4 θ. (S2) We note that the qualitatively similar magnetoresistance is obtained with a 2-probe measurement, indicating the negative longitudinal magnetoresistance does not originate purely from the current jetting effect [S21]. A possible origin of ρ xx (θ) of the form in Eq. (S2) is anisotropic magnetoresistance [S22]. In this mechanism, spin-orbit coupling creates an anisotropy in the scattering rate of the charge carriers between ẑ I and ẑ I, where ẑ is the spin-axis. By rotating B in the (001) plane, the in-plane component of the spin-axis is rotated, which gives rise to the θ dependence of ρ xx. An alternative explanation is a chiral magnetic effect induced by the chiral anomaly, if the system has Weyl points, which also gives a negative magnetoresistance for B I [S23, S24]; a similar report has recently appeared [S25, S26]. Both mechanisms NATURE PHYSICS 7

8 possess qualitatively the same θ dependence. Detailed studies of the Fermi level dependence of magnetoresistance and magnetic anisotropy may help to distinguish between these two mechanisms and provide further insight into the large anomalous Hall effect observed here. S11. PARTIAL DENSITY OF STATES Figure S8 shows the partial density of states of GdPtBi. The characteristics of the bands are in qualitative agreement with the previous reports [S27]. The bands near the Fermi level (E F ) mainly originate from Pt and Bi. The f orbitals from Gd are located 8 ev below and 3 ev above E F and do not contribute directly to the states near E F. It should be noted, however, that they influence the detailed structure near E F through magnetism as shown in Fig. 4 in the main text. [S1] Canfield, P. C. & Fisk, Z. Growth of single crystals from metallic fluxes. Phil. Mag. B 65, (1992) [S2] Lynn, J. W., Chen, Y., Chang, S., Zhao, Y., Chi, S., Ratcliff, W., Ueland, B. G., & Erwin, R. W. Double-focusing thermal triple-axis spectrometer at NCNR. J. Res. Natl. Inst. Stand. Technol. 117, (2012) [S3] Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, (1996) [S4] Perdew, J. P., Burke, K., & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, (1996) [S5] Li, Y., Kanazawa, N., Yu, X. Z., Tsukazaki, A., Kawasaki, M., Ichikawa, M., Jin, X. F., Kagawa, F., & Tokura, Y. Robust Formation of Skyrmion and Topological Hall Effect Anomaly in Epitaxial Thin Films of MnSi. Phys. Rev. Lett. 110, (2013) [S6] Kreyssig, A., Kim, M. G., Kim, J. W., Pratt, D. K., Sauerbrei, S. M., March, S. D., Tesdall, G. R., Bud ko, S. L., Canfield, P. C., McQueeney, R. J., & Goldman, A. I. Magnetic order in GdBiPt studied by x-ray resonant magnetic scattering. Phys. Rev. B 84, (R) (2011) [S7] Taniguchi, T., Yamanaka, K., Sumioka, H., Yamazaki, T., Tabata, Y., & Kawarazaki, S. Direct Observation of Chiral Susceptibility in the Canonical Spin Glass AuFe. Phys. Rev. Lett. 8 NATURE PHYSICS

9 SUPPLEMENTARY INFORMATION 93, (2004) [S8] Kawamura, H. Anomalous Hall Effect as a Probe of the Chiral Order in Spin Glasses. Phys. Rev. Lett. 90, (2003) [S9] Lee, M., Kang, W., Onose, Y., Tokura, Y., & Ong, N. P. Unusual Hall Effect Anomaly in MnSi under Pressure. Phys. Rev. Lett. 102, (2009) [S10] Kanazawa, N., Onose, Y., Arima, T., Okuyama, D., Ohoyama, K., Wakimoto, S., Kakurai, K., Ishiwata, S., & Tokura, Y. Large Topological Hall Effect in a Short-Period Helimagnet MnGe. Phys. Rev. Lett. 106, (2011) [S11] Ritz, R., Halder, M., Franz, C., Bauer, A., Wagner, M., Bamler, R., Rosch, A., & Pfleiderer, C. Giant generic topological Hall resistivity of MnSi under pressure. Phys. Rev. B 87, (2013) [S12] Taguchi, Y., Oohara, Y., Yoshizawa, H., Nagaosa, N., & Tokura, Y. Spin Chirality, Berry Phase, and Anomalous Hall Effect in a Frustrated Ferromagnet. Science 291, (2001) [S13] Sürgers, C., Fischer, G., Winkel, P., & Löhneysen, H. V. Large topological Hall effect in the non-collinear phase of an antiferromagnet. Nature Commun. 5, 4300 (2014) [S14] Ueland, B. G., Miclea, C. F., Kato, Y., Ayala-Valenzuela, O., McDonald, R. D., Okazaki, R., Tobash, P. H., Torrez, M. A., Ronning, F., Movshovich, R., Fisk, Z., Bauer, E. D., Martin, I., & Thompson, J. D. Controllable chirality-induced geometrical Hall effect in a frustrated highly correlated metal. Nature Commun. 3, 1067 (2012) [S15] Nakatsuji, S., Kiyohara, N., & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212 (2015) [S16] Nayak, A. K., Fischer, J. E., Sun, Y., Yan, B., Karel, J., Kormarek, A. C., Shekhar, N., Schelle, W., Parkin, S. S. P., & Felser, C. Non-vanishing Berry curvature driven large anomalous Hall effect in non-collinear antiferromagnet Mn 3 Ge. Sci. Adv. 2, e (2016) [S17] Chen, H., Niu, Q., & MacDonald, A. H. Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism. Phys. Rev. Lett. 112, (2014) [S18] Yamada, N., Sakai, H., Mori, H., & Ohoyama, T. Magnetic properties of ε-mn 3 Ge. Physica B 149, (1988) [S19] Brown, P. J., Nunez, V., Tasset, F., Forsyth, J. B., & Radhakrishna, P. Determination of the magnetic structure of Mn 3 Sn using generalized neutron polarization analysis. J. Phys: Condens. Matter 2, (1990) NATURE PHYSICS 9

10 [S20] Zimmer, G. J. & Krén, E. Magnetic Structure of DO 19 Type Compounds. AIP Conf. Proc. 10, (1973) [S21] Pippard, A. B. Magnetoresistance in Metals (Cambridge University Press, New York, 2009) [S22] van Gorkom, R. P., Caro, J., Klapwijk, T. M., & Radelaar, S. Temperature and angular dependence of the anisotropic magnetoresistance in epitaxial Fe films. Phys. Rev. Lett. 63, (2001) [S23] Burkov, A. A. Chiral anomaly and transport in Weyl metals. J. Phys.: Condens. Matter 27, (2015) [S24] Xiong, J., Kushwaha, S. K., Liang, T., Krizan, J. W., Hirschberger, M., Wang, W., Cava, R. J., & Ong, N. P. Evidence for the chiral anomaly in Dirac semimetal Na 3 Bi. Science 350, (2015) [S25] Hirschberger, M., Kushwaha, S., Wang, Z., Gibson Q., Belvin, C. A., Bernevig, B. A., Cava, R. J., & Ong, N. P. The chiral anomaly and thermopower of Weyl fermions in the half-heusler GdPtBi. arxiv: (2016) [S26] Ruan, J., Jian, S.-K., Yao, H., Zhang, H., Zhang, S.-C., & Xing, D. Symmetry-protected ideal Weyl semimetal in HgTe-class materials. Nat. Commun. 7, (2016) [S27] Xiao, D., Yao, Y., Feng, W., Wen, J., Zhu, W., Chen, X.-Q., Stocks, G. M., & Zhang, Z. Half-Heusler Compounds as a New Class of Three-Dimensional Topological Insulators. Phys. Rev. Lett. 105, (2010) 10 NATURE PHYSICS

11 SUPPLEMENTARY INFORMATION Figures 6 a ρ xx (mωcm) b M/H (cm 3 /mol) M [001] ZFC 0.01 T T (K) FIG. S1: Temperature dependence of a resistivity (ρ) and b susceptibility (M/H) up to 300 K. M is measured along the [001] direction at 0.01 T in the zero field cooling condition. In panel b the solid circles denote the experimental data and the solid line is the result of the Curie-Weiss fit. NATURE PHYSICS 11

12 100 a 60 b τ (10 9 Nm) [110] θ B d 2 τ/db 2 (arb. units) θ = 15 T = 0.5 K B (T) θ = B (T) FIG. S2: Magnetic field (B) dependence of a magnetic torque (τ) and b d 2 τ/db 2. Experimental geometry is shown in the inset of panel a. 12 NATURE PHYSICS

13 SUPPLEMENTARY INFORMATION d 2 ρ xx /db 2 (arb. units) GdPtBi sample #4 T = 0.5 K /B (T 1 ) FIG. S3: d 2 ρ xx /db 2 as a function of 1/B up to B = 31 T. NATURE PHYSICS 13

14 ρ yx (mωcm) ρ yx T (mωcm) K B (T) T = 2.5 K ρ yx A ρ yx 0.5 ρ yx B (T) FIG. S4: Estimate of the Hall resistivity from spin texture ρ T yx. Curves labeled as ρ yx and ρ yx are the total Hall resistivity observed at 2.5 K and ρ yx (2.5K) ρ yx (50K), respectively. ρ A yx is the anomalous Hall resistivity estimated from Eq. (S1) (see text). Inset plots the estimated Hall resistivity arising from the spin texture ρ T yx ρ yx ρ N yx ρ A yx. 14 NATURE PHYSICS

15 SUPPLEMENTARY INFORMATION a b (0 0 1) ( ) c (1 1 1) Intensity (arb. units) B [1 1 0] T = 2 K B = 0 T 9 T 4 T Intensity (arb. units) T = 2 K B = 9 T 6 T 3 T 0 T Intensity (arb. units) T = 2 K B = 9 T 6 T 3 T 0 T θ ( ) 2θ ( ) 2θ ( ) FIG. S5: Neutron scattering peak profiles of a (0 0 1), b ( ), and c (1 1 1) Bragg peaks at 2 K in various magnetic fields. The solid lines denote the fitting results based on a single Gaussian function model. Error bars were obtained by taking the square root of the actual counts. S15 NATURE PHYSICS 15

16 σ xy (Ω 1 cm 1 ) a # 1 # 4 # 3 # 5 # 2 T = 4 K c 20 K b d σ xy (Ω 1 cm 1 ) K 30 K σ xx (Ω 1 cm 1 ) σ xx (Ω 1 cm 1 ) FIG. S6: σ xy versus σ xx at a T =4K,b 10 K, c 20 K, and d 30 K for samples #1 5. Solid lines are the fitting result at each T based on the scaling relation σ xy σ α xx. 16 NATURE PHYSICS

17 SUPPLEMENTARY INFORMATION 3 a b [010] B ρ xx (mωcm) 2 1 B I [100] T = 2 K ρ xx (mωcm) 4 2 [100] I [100] T = 2 K B = 9 T c B (T) θ ( ) θ FIG. S7: a Magnetic field (B) dependence of the longitudinal resistivity with B I [100]. b Geometry of the experiments for the angular (θ) dependence of magnetoresistance. c Longitudinal resistivity (ρ xx ) as a function of θ at T =2KandB = 9 T. Symbols denote the experimental data. The red solid line is the result of the fit to Eq. (S2). NATURE PHYSICS 17

18 12 6 x20 tot s p d f Gd DOS (states/ev/cell) Pt tot s p d f Bi tot s p d f Energy (ev) FIG. S8: Orbital decomposed partial density of states of GdPtBi. 18 NATURE PHYSICS

19 SUPPLEMENTARY INFORMATION Tables TABLE S1: Measured (HHL) locations in the neutron scattering experiment. H L NATURE PHYSICS 19

20 TABLE S2: Magnitude of the Hall resistivity arising from spin textures ( ρ T yx ) and anomalous Hall angle ( Θ AH ) of various antiferromagnets with different origins. compound ρ T yx (µω cm) Θ AH mechanism Ref. GdPtBi Berry curvature in momentum space This work AuFe (8 at. % Fe) spin chirality (chiral glass) S7 MnSi (high pressure) spin texture (Skyrmion) S9 MnGe spin texture (Skyrmion) S10 Nd 2 Mo 2 O spin chirality (noncoplanar spin structure) S12 Mn 5 Si spin chirality (noncoplanar spin structure) S13 UCu spin chirality (noncoplanar spin structure) S14 Mn 3 Sn Berry curvature in momentum space S15 Mn 3 Ge Berry curvature in momentum space S16 20 NATURE PHYSICS