Abstract
A fundamental question in condensed matter physics concerns how topological electronic states are influenced by many-body correlations, and magnetic Weyl semimetals represent an important material platform to address this problem. However, the magnetic structures realized in these materials are limited, and in particular, no clear example of an undistorted helimagnetic state has been definitively identified. Here, we report clear evidence of a harmonic helimagnetic cycloid with an incommensurate magnetic propagation vector Q in the Weyl semimetal GdAlSi via resonant elastic X-ray scattering, including rigorous polarization analysis. This cycloidal structure is consistent with the Dzyaloshinskii–Moriya interaction prescribed by the polar crystal structure of GdAlSi. Upon applying a magnetic field, the cycloid undergoes a transition to a novel multi-Q state. This field-induced, noncoplanar texture is consistent with our numerical spin model, which incorporates the Dzyaloshinskii–Moriya interaction and, crucially, anisotropic exchange interactions. The perfectly harmonic Weyl helimagnet GdAlSi serves as a prototypical platform to study electronic correlation effects in periodically modulated Weyl semimetals.
Data availability
All experimental data to reproduce the figures are available on Zenodo at https://doi.org/10.5281/zenodo.1822896745.
Code availability
The source code used to perform the calculations described in this paper is available from the corresponding authors upon request.
References
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Wang, Z. & Zhang, S.-C. Chiral anomaly, charge density waves, and axion strings from Weyl semimetals. Phys. Rev. B 87, 161107 (2013).
Roy, B. & Sau, J. D. Magnetic catalysis and axionic charge density wave in Weyl semimetals. Phys. Rev. B 92, 125141 (2015).
Roy, B., Goswami, P. & Juričić, V. Interacting Weyl fermions: phases, phase transitions, and global phase diagram. Phys. Rev. B 95, 201102 (2017).
Sehayek, D., Thakurathi, M. & Burkov, A. A. Charge density waves in Weyl semimetals. Phys. Rev. B 102, 115159 (2020).
Yi, J., Ying, X., Gioia, L. & Burkov, A. A. Topological order in interacting semimetals. Phys. Rev. B 107, 115147 (2023).
Yi, J. & Burkov, A. Disordered Weyl semimetal as an array of coupled Hubbard chains. Phys. Rev. B 110, 165114 (2025).
Chiu, W.-C. et al. Causal structure of interacting Weyl fermions in condensed matter systems. Nat. Commun. 14, 2228 (2023).
Chang, H.-R., Zhou, J., Wang, S.-X., Shan, W.-Y. & Xiao, D. RKKY interaction of magnetic impurities in Dirac and Weyl semimetals. Phys. Rev. B 92, 241103 (2015).
Hosseini, M. V. & Askari, M. Ruderman-Kittel-Kasuya-Yosida interaction in Weyl semimetals. Phys. Rev. B 92, 224435 (2015).
Araki, Y. & Nomura, K. Spin textures and spin-wave excitations in doped Dirac-Weyl semimetals. Phys. Rev. B 93, 094438 (2016).
Gaudet, J. et al. Weyl-mediated helical magnetism in NdAlSi. Nat. Mater. 20, 1650 (2021).
Yao, X. et al. Large topological Hall effect and spiral magnetic order in the Weyl semimetal SmAlSi. Phys. Rev. X 13, 011035 (2023).
Dhital, C. et al. Multi-k magnetic structure and large anomalous Hall effect in candidate magnetic Weyl semimetal NdAlGe. Phys. Rev. B 107, 224414 (2023).
Lygouras, C. J. et al. Magnetic excitations and interactions in the Weyl ferrimagnet NdAlSi, arXiv. https://doi.org/10.48550/arXiv.2412.20743 (2024).
Roychowdhury, S. et al. Interplay between magnetism and topology: large topological Hall effect in an antiferromagnetic topological insulator, EuCuAs. J. Am. Chem. Soc. 145, 12920 (2023).
Soh, J.-R. et al. Weyl metallic state induced by helical magnetic order. npj Quantum Mater. 9, 7 (2024).
Nag, J. et al. GdAlSi: an antiferromagnetic topological Weyl semimetal with nonrelativistic spin splitting. Phys. Rev. B 110, 224436 (2024).
Laha, A. et al. Electronic structure and magnetic and transport properties of antiferromagnetic Weyl semimetal GdAlSi. Phys. Rev. B 109, 035120 (2024).
Xu, S.-Y. et al. Discovery of Lorentz-violating type II Weyl fermions in LaAlGe. Sci. Adv. 3, e1603266 (2017).
Chang, G. et al. Magnetic and noncentrosymmetric Weyl fermion semimetals in the RAlGe family of compounds (R = rare earth). Phys. Rev. B 97, 041104 (2018).
Suzuki, T. et al. Singular angular magnetoresistance in a magnetic nodal semimetal. Science 365, 377 (2019).
Sanchez, D. S. et al. Observation of Weyl fermions in a magnetic non-centrosymmetric crystal. Nat. Commun. 11, 3356 (2020).
Li, C. et al. Emergence of Weyl fermions by ferrimagnetism in a noncentrosymmetric magnetic Weyl semimetal. Nat. Commun. 14, 7185 (2023).
Zhang, Y. et al. Kramers nodal lines and Weyl fermions in SmAlSi. Commun. Phys. 6, 134 (2023).
Cheng, E. et al. Tunable positions of Weyl nodes via magnetism and pressure in the ferromagnetic Weyl semimetal CeAlSi. Nat. Commun. 15, 1467 (2024).
Yang, H.-Y. et al. Transition from intrinsic to extrinsic anomalous Hall effect in the ferromagnetic Weyl semimetal PrAlGe1−xSix. Apl. Mater. 8, 011111 (2020).
Yang, H.-Y. et al. Noncollinear ferromagnetic Weyl semimetal with anisotropic anomalous Hall effect. Phys. Rev. B 103, 115143 (2021).
Puphal, P. et al. Topological magnetic phase in the candidate Weyl semimetal CeAlGe. Phys. Rev. Lett. 124, 017202 (2020).
Yang, H.-Y. et al. Stripe helical magnetism and two regimes of anomalous Hall effect in NdAlGe. Phys. Rev. Mater. 7, 034202 (2023).
Bouaziz, J., Bihlmayer, G., Patrick, C. E., Staunton, J. B. & Blügel, S. Origin of incommensurate magnetic order in the RAlSi magnetic Weyl semimetals (R = Pr, Nd, Sm). Phys. Rev. B 109, L201108 (2024).
Hayami, S. & Yambe, R. Stabilization mechanisms of magnetic skyrmion crystal and multiple-Q states based on momentum-resolved spin interactions. Mater. Today Quant. 3, 100010 (2024).
Kawano, M. & Hotta, C. Phase diagram of the square-lattice Hubbard model with Rashba-type antisymmetric spin-orbit coupling. Phys. Rev. B 107, 045123 (2023).
Jain, A., Goyal, G. & Singh, D. K. Weyl semimetallic state with antiferromagnetic order in the Rashba-Hubbard model. Phys. Rev. B 110, 075134 (2024).
Sousa-Júnior, S.dA. & Mondaini, R. Weyl semimetallic, Néel, spiral, and vortex states in the Rashba-Hubbard model. Phys. Rev. B 111, 075166 (2025).
Hirschberger, S., Hayami, M. & Tokura, Y. Nanometric skyrmion lattice from anisotropic exchange interactions in a centrosymmetric host. N. J. Phys. 23, 023039 (2021).
Lovesey, S. W. & Collins, S. P. X-ray Scattering and Absorption by Magnetic Materials, Oxford series on synchrotron radiation No. 1 (Clarendon Press, Oxford University Press, 1996).
Yambe, R. & Hayami, S. Effective spin model in momentum space: toward a systematic understanding of multiple-Q instability by momentum-resolved anisotropic exchange interactions. Phys. Rev. B 106, 174437 (2022).
Rüßmann, P. et al. DFT code, JuDFTteam/JuKKR: v3.6 (2022).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Ebert, H., Köedderitzsch, D. & Minar, J. Calculating condensed matter properties using the KKR-Green’s function method-recent developments and applications. Rep. Prog. Phys. 74, 096501 (2011).
Liechtenstein, A. I., Katsnelson, M. I. & Gubanov, V. A. Exchange interactions and spin-wave stiffness in ferromagnetic metals. J. Phys. F: Met. Phys. 14, L125 (1984).
Ebert, H. & Mankovsky, S. Anisotropic exchange coupling in diluted magnetic semiconductors: ab initio spin-density functional theory. Phys. Rev. B 79, 045209 (2009).
Solovyev, I. V. Exchange interactions and magnetic force theorem. Phys. Rev. B 103, 104428 (2021).
Nakano, R. Dataset for: Perfectly harmonic spin cycloid and multi-Q textures in the Weyl semimetal GdAlSi. Zenodo. https://doi.org/10.5281/zenodo.18228967 (2026).
Acknowledgements
We acknowledge fruitful discussions with Max T. Birch, Moritz M. Hirschmann, and Chris J. Lygouras. Support is acknowledged from the Japan Society for the Promotion of Science (JSPS) under Grant Nos. JP22H04463, JP23H05431, JP21K13873, JP22F22742, JP22K20348, JP23K13057, JP23H04862, JP24H01607, JP24H01604, JP21H01037, JP23H04869, JP23K13068, and JP25K17336. The work was partially supported by the Japan Science and Technology Agency via JST CREST Grant Numbers JPMJCR1874 and JPMJCR20T1 (Japan), JST FOREST Grant Numbers JPMJFR2238, JPMJFR2366, and JPMJFR212X, and JST PRESTO Grant Number JPMJPR259A. We are grateful for support by the Murata Science Foundation, Yamada Science Foundation, Hattori Hokokai Foundation, Mazda Foundation, Casio Science Promotion Foundation, Inamori Foundation, Kenjiro Takayanagi Foundation, Toray Science Foundation, the Marubun Exchange Grant, the Foundation for Promotion of Material Science and Technology of Japan (MST Foundation), the Yashima Environment Technology Foundation, ENEOS Toenegeneral Research/Development Encouragement & Scholarship Foundation, and Yazaki Memorial Foundation for Science and Technology. This work was supported by the RIKEN TRIP initiative (RIKEN Quantum, Advanced General Intelligence for Science Program, Many-Body Electron Systems). J.B. was supported by the Alexander von Humboldt Foundation through the Feodor Lynen Research Fellowship. P.R.B. acknowledges SNSF Postdoc. Mobility grant P500PT217697 for financial assistance. J.M. acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 547968854. M.H. is supported by JST as part of Adopting Sustainable Partnerships for Innovative Research Ecosystem (ASPIRE; Grant No. JPMJAP2426) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Transregio TRR 360-492547816. Resonant X-ray scattering at SPring-8 was carried out under proposal number 20220083. Resonant X-ray scattering at Photon Factory (KEK) was carried out under proposal numbers 2022G551 and 2023G611. The synchrotron single-crystal X-ray experiments were performed at BL02B1 in SPring-8 with the approval of RIKEN (Proposal No. 2024B2010).
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M.H. and Y.To. conceived the project. R.N., R.Y., A.K., and Y.Tag. grew and characterized the single crystals. R.N. and R.Y. performed all magnetic and electric measurements. R.N., R.Y., H.O., Y.Tan., and M.H. performed resonant X-ray scattering experiments at beamline BL19LXU of SPring-8. R.N., R.Y., M.G., H.S., H.N., and M.H. performed resonant X-ray scattering experiments at beamline BL-3A of Photon Factory. REXS experiments were designed and REXS data were analyzed under the guidance of T.-h.A. M.C. and J.M. performed Monte Carlo calculations of the real-space model. K.S., Y.O., and Y.Tak. performed SHG measurements. K.G. and O.F. performed neutron scattering measurements. R.N., Y.I., K.G., O.F., and M.H. analyzed the neutron scattering data. S.H. performed simulated annealing of the momentum space model. R.M., P.R.B., S.K., and Y.N. performed single-crystal X-ray diffraction measurements at SPring-8. J.B. performed and analyzed all first-principles calculations. J.B. and R.A. discussed the ab initio results. R.N., R.Y., and M.H. wrote the manuscript with help of J.B., J.M., and S.H.; all authors discussed the results and commented on the manuscript.
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Nakano, R., Yamada, R., Bouaziz, J. et al. Perfectly harmonic spin cycloid and multi-Q textures in the Weyl semimetal GdAlSi. Nat Commun (2026). https://doi.org/10.1038/s41467-026-69452-7
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DOI: https://doi.org/10.1038/s41467-026-69452-7
