Abstract
Antiferromagnetic states with a spin-split electronic structure give rise to spintronic, magnonic and electronic phenomena despite (near-)zero net magnetization1,2,3,4,5,6,7. The simplest odd-parity spin splitting—p wave—was originally proposed to emerge from a collective instability in interacting electron systems8,9,10,11,12. Recent theory has identified a distinct route to realize p-wave spin-split electronic bands without strong correlations13,14, termed p-wave magnetism. Here we demonstrate an experimental realization of a metallic p-wave magnet. The odd-parity spin splitting of delocalized conduction electrons arises from their coupling to an antiferromagnetic texture of localized magnetic moments: a coplanar spin helix whose magnetic period is an even multiple of the chemical unit cell, as revealed by X-ray scattering experiments. This texture breaks space-inversion symmetry but approximately preserves time-reversal symmetry up to a half-unit-cell translation—thereby fulfilling the symmetry conditions for p-wave magnetism. Consistent with theoretical predictions, our p-wave magnet shows a characteristic anisotropy in the electronic conductivity13,14,15. Relativistic spin–orbit coupling and a tiny spontaneous net magnetization further break time-reversal symmetry, resulting in a giant anomalous Hall effect (Hall conductivity >600 S cm−1, Hall angle >3%), for an antiferromagnet. Our model calculations show that the spin-nodal planes found in the electronic structure of p-wave magnets are readily gapped by a small perturbation to induce the anomalous Hall effect. We establish metallic p-wave magnets as an ideal platform to explore the functionality of spin-split electronic states in magnets, superconductors, and in spintronic devices.
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Data availability
All experimental data to reproduce the figures are available on Zenodo at https://doi.org/10.5281/zenodo.17035626 (ref. 46).
Code availability
The source code used to perform the calculations described in this paper is available from the corresponding authors upon request.
References
Ahn, K.-H., Hariki, A., Lee, K.-W. & Kuneš, J. Antiferromagnetism in RuO2 as d-wave Pomeranchuk instability. Phys. Rev. B 99, 184432 (2019).
Naka, M. et al. Spin current generation in organic antiferromagnets. Nat. Commun. 10, 4305 (2019).
Šmejkal, L., González-Hernández, R., Jungwirth, T. & Sinova, J. Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets. Sci. Adv. 6, eaaz8809 (2020).
Å mejkal, L., Sinova, J. & Jungwirth, T. Emerging research landscape of altermagnetism. Phys. Rev. X 12, 040501 (2022).
Å mejkal, L., Sinova, J. & Jungwirth, T. Beyond conventional ferromagnetism and antiferromagnetism: a phase with nonrelativistic spin and crystal rotation symmetry. Phys. Rev. X 12, 031042 (2022).
Ezawa, M. Third-order and fifth-order nonlinear spin-current generation in g-wave and i-wave altermagnets and perfectly nonreciprocal spin current in f-wave magnets. Phys. Rev. B 111, 125420 (2025).
Yu, Y. et al. Odd-parity magnetism driven by antiferromagnetic exchange. Phys. Rev. Lett. 135, 046701 (2025).
Hirsch, J. E. Spin-split states in metals. Phys. Rev. B 41, 6820–6827 (1990).
Wu, C., Sun, K., Fradkin, E. & Zhang, S.-C. Fermi liquid instabilities in the spin channel. Phys. Rev. B 75, 115103 (2007).
Jung, J., Polini, M. & MacDonald, A. H. Persistent current states in bilayer graphene. Phys. Rev. B 91, 155423 (2015).
Kiselev, E. I., Scheurer, M. S., Wölfle, P. & Schmalian, J. Limits on dynamically generated spin-orbit coupling: absence of l = 1 Pomeranchuk instabilities in metals. Phys. Rev. B 95, 125122 (2017).
Wu, Y.-M., Klein, A. & Chubukov, A. V. Conditions for l = 1 Pomeranchuk instability in a Fermi liquid. Phys. Rev. B 97, 165101 (2018).
Hellenes, A. B. et al. P-wave magnets. Preprint at https://arxiv.org/abs/2309.01607 (2024).
Jungwirth, T. et al. From superfluid 3He to altermagnets. Preprint at https://arxiv.org/abs/2411.00717 (2024).
Ezawa, M. Purely electrical detection of the spin-splitting vector in p-wave magnets based on linear and nonlinear conductivities. Phys. Rev. B 112, 125412 (2025).
Brekke, B., Sukhachov, P., Giil, H. G., Brataas, A. & Linder, J. Minimal models and transport properties of unconventional p-wave magnets. Phys. Rev. Lett. 133, 236703 (2024).
Ezawa, M. Topological insulators and superconductors based on p-wave magnets: electrical control and detection of a domain wall. Phys. Rev. B 110, 165429 (2024).
Gladyshevskii, R. E., Strusievicz, O. R., Cenzual, K. & Parthé, E. Structure of Gd3Ru4Al12, a new member of the EuMg5.2 structure family with minority-atom clusters. Acta Crystallogr. B 49, 474–478 (1993).
Niermann, J. & Jeitschko, W. Ternary rare earth (R) transition metal aluminides R3T4Al12 (T = Ru and Os) with Gd3Ru4Al12 type structure. Z. Anorg. Allg. Chem. 628, 2549–2556 (2002).
Nakamura, S. et al. Spin trimer formation in the metallic compound Gd3Ru4Al12 with a distorted kagome lattice structure. Phys. Rev. B 98, 054410 (2018).
Matsumura, T., Ozono, Y., Nakamura, S., Kabeya, N. & Ochiai, A. Helical ordering of spin trimers in a distorted kagomé lattice of Gd3Ru4Al12 studied by resonant X-ray diffraction. J. Phys. Soc. Jpn 88, 023704 (2019).
Lovesey, S. W. & Collins, S. P. X-ray Scattering and Absorption by Magnetic Materials Oxford Series on Synchrotron Radiation No. 1 (Clarendon Press, Oxford Univ. Press, 1996).
McGuire, T. & Potter, R. Anisotropic magnetoresistance in ferromagnetic 3d alloys. IEEE Trans. Magn. 11, 1018–1038 (1975).
Okumura, S., Kato, Y. & Motome, Y. Lock-in of a chiral soliton lattice by itinerant electrons. J. Phys. Soc. Jpn 87, 033708 (2018).
Hodt, E. W., Bentmann, H. & Linder, J. Fate of p-wave spin polarization in helimagnets with Rashba spin-orbit coupling. Phys. Rev. B 111, 205416 (2025)
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539 (2010).
Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212–215 (2015).
Nayak, A. K. et al. Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3Ge. Sci. Adv. 2, e1501870 (2016).
Ghimire, N. J. et al. Large anomalous Hall effect in the chiral-lattice antiferromagnet CoNb3S6. Nat. Commun. 9, 3280 (2018).
Takagi, H. et al. Spontaneous topological Hall effect induced by non-coplanar antiferromagnetic order in intercalated van der Waals materials. Nat. Phys. 19, 961–968 (2023).
Park, P. et al. Tetrahedral triple-Q magnetic ordering and large spontaneous Hall conductivity in the metallic triangular antiferromagnet Co1/3TaS2. Nat. Commun. 14, 8346 (2023).
Šmejkal, L., MacDonald, A. H., Sinova, J., Nakatsuji, S. & Jungwirth, T. Anomalous Hall antiferromagnets. Nat. Rev. Mater. 7, 482–496 (2022).
Hedayati, A. A. & Salehi, M. Transverse spin current at normal-metal /p-wave magnet junctions. Phys. Rev. B 111, 035404 (2025).
Ãlvarez Pari, N. A., Jaeschke-Ubiergo, R., Chakraborty, A., Å mejkal, L. & Sinova, J. Nonrelativistic linear Edelstein effect in helical EuIn2As2. Phys. Rev. B 112, 024404 (2025).
Choy, T. P., Edge, J. M., Akhmerov, A. R. & Beenakker, C. W. J. Majorana fermions emerging from magnetic nanoparticles on a superconductor without spin–orbit coupling. Phys. Rev. B 84, 195442 (2011).
Martin, I. & Morpurgo, A. F. Majorana fermions in superconducting helical magnets. Phys. Rev. B 85, 144505 (2012).
Nadj-Perge, S., Drozdov, I. K., Bernevig, B. A. & Yazdani, A. Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor. Phys. Rev. B 88, 020407 (2013).
Klinovaja, J., Stano, P., Yazdani, A. & Loss, D. Topological superconductivity and Majorana fermions in RKKY systems. Phys. Rev. Lett. 111, 186805 (2013).
Maeda, K., Lu, B., Yada, K. & Tanaka, Y. Theory of tunneling spectroscopy in unconventional p-wave magnet-superconductor hybrid structures. J. Phys. Soc. Jpn 93, 114703 (2024).
Song, Q. et al. Electrical switching of a p-wave magnet. Nature 642, 64–70 (2025).
Aharoni, A. Demagnetizing factors for rectangular ferromagnetic prisms. J. Appl. Phys. 83, 3432–3434 (1998).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Harmon, B., Antropov, V., Liechtenstein, A., Solovyev, I. & Anisimov, V. Calculation of magneto-optical properties for 4f systems: LSDA + Hubbard U results. J. Phys. Chem. Solids 56, 1521–1524 (1995).
Shick, A. B., Liechtenstein, A. I. & Pickett, W. E. Implementation of the LDA+U method using the full-potential linearized augmented plane-wave basis. Phys. Rev. B 60, 10763–10769 (1999).
Yamada, R. Dataset for: A metallic p-wave magnet with commensurate spin helix. Zenodo https://doi.org/10.5281/zenodo.17035626 (2025).
Ikhlas, M. et al. Large anomalous Nernst effect at room temperature in a chiral antiferromagnet. Nat. Phys. 13, 1085–1090 (2017).
Chen, T. et al. Anomalous transport due to Weyl fermions in the chiral antiferromagnets Mn3X, X = Sn, Ge. Nat. Commun. 12, 572 (2021).
Liu, Z. H. et al. Transition from anomalous Hall effect to topological Hall effect in hexagonal non-collinear magnet Mn3Ga. Sci. Rep. 7, 515 (2017).
Hayashi, H. et al. Large anomalous Hall effect observed in the cubic-lattice antiferromagnet Mn3Sb with kagome lattice. Phys. Rev. B 108, 075140 (2023).
Zuniga-Cespedes, B. E. et al. Observation of an anomalous Hall effect in single-crystal Mn3Pt. New J. Phys. 25, 023029 (2023).
Sürgers, C. et al. Anomalous Nernst effect in the noncollinear antiferromagnet Mn5Si3. Commun. Mater. 5, 176 (2024).
Kotegawa, H. et al. Large anomalous Hall effect and unusual domain switching in an orthorhombic antiferromagnetic material NbMnP. npj Quantum Mater. 8, 56 (2023).
Kotegawa, H. et al. Large spontaneous Hall effect with flexible domain control in the antiferromagnetic material TaMnP. Phys. Rev. B 110, 214417 (2024).
Kotegawa, H. et al. Large anomalous Hall conductivity derived from an f-electron collinear antiferromagnetic structure. Phys. Rev. Lett. 133, 106301 (2024).
Acknowledgements
We acknowledge A. Kikkawa for help with crystal growth. We acknowledge support from the Japan Society for the Promotion of Science (JSPS) under grant numbers JP22H04463, JP23H05431, JP22F22742, JP22K20348, JP23K13057, JP23H00171, JP24H01607, JP24H01604, JP25K17336, JP22K13998, and JP23K25816, and from the Murata Science Foundation, Yamada Science Foundation, Hattori Hokokai Foundation, Mazda Foundation, Casio Science Promotion Foundation, Inamori Foundation, Kenjiro Takayanagi Foundation, Foundation for Promotion of Material Science and Technology of Japan, Yashima Environment Technology Foundation, Yazaki Memorial Foundation for Science and Technology, and ENEOS Tonengeneral Research/Academic Foundation. This work was partially supported by the Japan Science and Technology Agency via JST CREST grant numbers JPMJCR1874, JPMJCR20T1 and JPMJCR20T2 (Japan), JST FOREST (JPMJFR2238), and JST PRESTO (JPMJPR259A, JPMJPR2595). It was also supported by Japan Science and Technology Agency (JST) as part of Adopting Sustainable Partnerships for Innovative Research Ecosystem (ASPIRE), grant number JPMJAP2426. M.H. is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Transregio TRR 360 – 492547816. P.R.B. acknowledges Swiss National Science Foundation (SNSF) Postdoc.Mobility grant P500PT_217697 for financial assistance. M.M.H. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number 518238332. J.M. acknowledges funding from the DFG under the project number 547968854. Resonant X-ray scattering at Photon Factory (KEK) was carried out under proposal numbers 2022G551 and 2023G611. The neutron experiments at the Materials and Life Science Experimental Facility of J-PARC were performed under a user programme (proposal number 2020B0347, TAIKAN, and number 2020A0282, SENJU).
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M.H., Y.M., T.-h.A. and Y.T. conceived of the project. R.Y. and M.H. grew and characterized the single crystals. R.Y., R.N. and M.H. performed the magnetization and transport measurements on bulk samples. M.T.B. prepared the focused-ion-beam devices and performed all the device measurements. P.R.B., R.N., S.G., H.S., H.N. and M.H. performed the resonant X-ray scattering experiments and analysed the data with guidance from T.-h.A. T. Nakajima, K.O., K.K.K., T.O., R.K. and M.H. performed the neutron scattering experiments, and the data were analysed by P.R.B. and Y.I. S.O., M.M.H., M.E., J.M. and Y.M. performed the symmetry analysis and model calculations. T. Nomoto performed the ab initio calculations. R.Y., M.T.B., P.R.B. and M.H. wrote the paper with help from I.B. All authors discussed the results and commented on the paper.
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Extended data figures and tables
Extended Data Fig. 1 Fermi surfaces of a p-wave band.
a, Band dispersion with p-wave splitting, spin-orbit coupling (SOC), and time-reversal breaking mz = 0.01. Here, we set p = 1.2 and λ = 0.4, where p-wave exchange splitting is larger than the size of SOC and the net magnetization is small but non-zero. b,c, Fermi surfaces of a p-wave split band with (mz = 0, λ = 0) and without (mz = 0.01, λ = 0.4) time-reversal symmetry, respectively. The two bands are degenerate on the kx = 0 plane for mz = 0. However, the degeneracy is lifted due to band hybridization when mz is finite in Eq. (1). The equal energy surface is calculated at E = 10 in panels (b, c). Contrary to the low-energy model for a p-wave Pomeranchuk instability, \(| \langle {S}_{x}\rangle | < \hslash /2\) for the p-wave magnet of conduction electrons coupled to a spin helix (Methods, Supplementary Fig. 15 and Note 15).
Extended Data Fig. 2 Resonant elastic X-ray scattering of magnetic satellites in Phase-I.
a, The magnetic reflection intensity at \({{\boldsymbol{Q}}}_{{\rm{M}}}(\,\equiv \,{{\boldsymbol{G}}}_{440}-{{\boldsymbol{k}}}_{{\rm{m}}{\rm{a}}{\rm{g}}}^{(1)})\) and total fluorescence signal obtained at T = 4 K strongly depend on the energy E of the incoming synchrotron X-ray beam. The maximum intensity due to magnetic scattering appears at E = 7.935 keV, in resonance with the Gd-L2 absorption edge. Inset: schematic of momentum space and magnetic satellites around the (440) fundamental (structural) reflection. We indicate the center of momentum space, Γ, and lattice vectors of momentum space. b, Measurement geometry for detecting the QM reflection: ki and kf are the wavevectors of the incoming and outgoing X-ray beams, respectively. The angles ω, 2θ defined here are referred to in Eq. (S5) and following in Supplementary Note 7. c, Polarisation analysis of magnetic satellites close to (440), (250), and (700) fundamental reflections using Eq. (S6) in Supplementary Note 7 reveals good agreement with a distorted spin helix model. Dashed green (yellow) lines represent expected behaviour for a harmonic cycloid (harmonic helix). The data are consistent with a distorted helix, elliptically squeezed into the hexagonal basal plane. Even when the commensurate helix shows a distorted spin rotation plane, the symmetries [C2x∥t1/2] and \([{\mathcal{T}}\,\parallel {{\bf{t}}}_{1/2}]\) are preserved.
Extended Data Fig. 3 Magnetic order in Phase-I, probed by elastic neutron scattering.
a, Experimental geometry. A rectangular cuboid shaped crystal with Gd in natural abundance was installed inside a 4 T superconducting magnet. The magnet with sample assembly was rotated around the vertical axis by 29° in order to access reflections on the high-angle detector. The beamline is equipped with five detector banks; only the high-angle detector (shown in blue) was used to collect scattered neutrons. b, Reciprocal space map at L = 0, obtained at T = 2 K and B = 0 T. Data measured on an empty can at 15 K are used for background correction. Nuclear reflections are labelled. The red and green dashed lines represent the coordinate axes in the hexagonal system. Inset: six-fold symmetric magnetic satellite reflections in the region around G−200, marked by a white dashed box in the main panel. kmag-vectors are labelled according to Main Text Fig. 2e. c,d, Temperature and magnetic-field evolution of \({{\boldsymbol{Q}}}_{{\rm{M}}}(\equiv {{\boldsymbol{G}}}_{-200}+{{\boldsymbol{k}}}_{{\rm{mag}}}^{(1)})\) are shown in panels c and d, respectively. Solid lines represent back-to-back exponential fits as defined in Eq. (S1) in Supplementary Note 6. The curves are shifted by a constant value for better visibility. e, Line scan through the G−600 nuclear reflection as a function of total momentum transfer \(| {\boldsymbol{Q}}| \), tracked across the two T-induced transitions. f, Integrated intensities of G−600 and QM obtained from fitting a back-to-back exponential function. For the fits shown in panels c, d, and e, width and center position were kept constant while background and amplitude of the peaks were treated as free parameters. Experiments were performed at beamline BL15 (TAIKAN) of MLF, J-PARC (Tokai, Japan).
Extended Data Fig. 4 Step-like anisotropic magnetoresistance due to switching of p-wave domains, corresponding to switching the spin polarisation vector α.
a-c, Three types of α-domain configurations that can be controlled by rotating the magnetic field in the ab-plane. The relative angle between the a-axis and B is ϕ. The gray shaded wedges indicate the range of angles where a given domain is stable: ϕ = 0° ~ 60° (180° ~ 240°), 60° ~ 120° (240° ~ 300°), and 120° ~ 180° (300° ~ 360°), respectively, for Configuration 1, 2, and 3. d, Polar plot of AMR with illustrations of α-domain configurations. Step-like AMR observed in the p-wave phase originating from the switching of α-domains (see Supplementary Fig. 9). For current Ia∥a, a higher resistance \({R}_{xx}^{a}\) appears in Configuration 2 (red region), where I is nearly parallel to α. We also show the measurement configuration of anisotropic magnetoresistance (AMR). The current Ia is applied along the a-axis, while the magnetic field B is rotated within the ab-plane. e,f, AMR of a fabricated device with strain relief at T = 2 K and 30 K. The second y-axis on the right-hand side shows the relative magnitude of the AMR. g,h, AMR of a bulk sample at T = 2 K and 30 K. Bulk and device curves are in good qualitative agreement, although the AMR’s relative magnitude is three times larger in the suspended device. We note that the AMR measurement on devices allows us to measure the resistivity along several current directions simultaneously (see Fig. 3, Main Text). i,j, Separation of AMR into (i) step-type anomaly related to resistivity anisotropy in the p-wave magnet (green line, right axis) and (ii) six-fold patterns with the periodicity of the underlying crystal lattice (blue symbols, left axis). As the relative magnitude of the p-wave AMR is larger in the device, the six-fold patterns are harder to distinguish. Arrows in e,g indicate the sweep direction of the magnetic-field angle ϕ.
Extended Data Fig. 5 Electrical resistivity and Hall resistivity data for a bulk single crystal of Gd3(Ru0.95Rh0.05)4Al12.
a-c, Longitudinal resistivity Ïxx, Hall resistivity Ïyx, and Hall conductivity σxy at various temperatures. In support of Fig. 4 of the Main Text, here we show raw data of each quantity. Direction of magnetic field ramping is indicated by black arrows, with solid lines for ∂B/∂t < 0 and dashed lines for increasing magnetic field. The magnetic field is applied along the c-axis of the crystal. The offset between curves is set to be 2 μΩcm, 2 μΩcm, and 500 Ω−1cm−1 in panels a, b, and c, respectively.
Extended Data Fig. 6 Comparing transport properties of bulk single crystal and suspended device at T = 2 K.
The electric current and magnetic field are applied along the a-axis and c-axis, respectively. a Longitudinal resistivity Ïxx for a device fabricated by the focused ion beam technique. b, c Magnetic field dependence of Hall resistivity Ïyx and longitudinal conductivity σxx. d Hall conductivity σxy calculated for the device (red/blue) and for the bulk single crystal (gray line). The data are in good qualitative agreement. Inset: schematic measurement geometry. The hysteresis in Ïxx and σxx is attributed to the texture of p-wave domains, as discussed further in Supplementary Fig. 4. To calculate the conductivities and resistivities of the device, we approximate the width and length of the central circular patch as a rectangular bar, implying significant systematic errors due to the geometry of the device. Note: The Hall conductivity data presented in the Main Text was obtained from bulk single crystals to suppress systematic measurement errors for the absolute value of σxy.
Extended Data Fig. 7 Bulk magnetization data with B∥c.
a, Curie-Weiss fit of magnetic susceptibility χ = M/H, measured with 0.1 T external magnetic field applied along the c-axis. b, M(H) curves obtained at constant temperatures T as indicated in the panel. c, Magnetization isotherm and its corresponding field derivative at T = 2 K. The dashed vertical line, labelled as Bc1, represents the hysteretic boundary between Phase-I and Phase-II. Around this region, ramping up and ramping down direction of external magnetic field is indicated by solid and dashed arrows, respectively. d, Derivative of magnetization with respect to applied field at various T, showing peak-like features at Bcoerc (red arrow) and Bc1 (black arrows) defined in Fig. 4a. Consecutive curves are shifted by 0.2 μB f.u.−1 T−1 for better visibility. Similar to Panel-c, solid and dashed lines denote the direction of magnetic field ramping. e, Temperature dependent derivative dM/dT of magnetization data measured at constant magnetic field, showing two clear transitions, TN1 and TN2 as defined in Fig. 3. Each consecutive curve is shifted by 0.002 emu mol−1 K−1 for better visibility. Inset: Zoom-in of data at low magnetic field.
Extended Data Fig. 8 Anomalous Hall conductivity of a distorted p-wave magnet.
Anomalous Hall conductivity as a function of the uniform net magnetization \({S}_{{\boldsymbol{k}}=0}^{z}\). Here, S0 is the total length of the magnetic moment (saturation moment). In the low-energy model, only mz is varied while p = 0.1, kBT = 0.04 – as well as spin-orbit coupling strength λ and chemical potential μ, written in each panel – are fixed. We calculate \({S}_{{\boldsymbol{k}}=0}^{z}\) from mz/p and the parameters J, t of a tight-binding Hamiltonian, as described in Supplementary Note 12. A sharp anomaly emerges for small \({S}_{{\boldsymbol{k}}=0}^{z}\) when λ is weak, λ/p < 1. In the gray region, the spin helix collapses as discussed in Supplementary Note 13.
Extended Data Fig. 9 Solitonic spin states with a net magnetization from biquadratic magnetic interactions, following Supplementary Note 8.
To explain the observed distorted spin helix state with a weak net magnetization, we develop a simple 1D two-parameter model in Eq. (S8) in Supplementary Note 8. In this model, we introduce the exchange anisotropy Δ > 0 and \({\Delta }^{{\prime} } > 0\) to favour a spin helix with net magnetization along the z-axis; we also use the RKKY interaction J = 1, and the four spin interaction K. We optimize the energy to obtain the ground state of the model by simulated annealing. This figure shows the calculation results obtained for Δ = 0.1. Magnetic states with higher harmonics and spontaneous magnetization appear when the four-spin interaction is sufficiently large, K > 0.25.
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Yamada, R., Birch, M.T., Baral, P.R. et al. A metallic p-wave magnet with commensurate spin helix. Nature 646, 837–842 (2025). https://doi.org/10.1038/s41586-025-09633-4
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DOI: https://doi.org/10.1038/s41586-025-09633-4
