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  • Open Access

General spin models from noncollinear spin density functional theory and spin-cluster expansion

Tomonori Tanaka* and Yoshihiro Gohda†

  • *Contact author: tanaka.t.da74@m.isct.ac.jp
  • †Contact author: gohda@mct.isct.ac.jp
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Phys. Rev. Research 8, 023300 – Published 15 June, 2026

DOI: https://doi.org/10.1103/qqlf-d531

Abstract

We present a data-efficient framework for constructing general classical spin Hamiltonians by combining the spin-cluster expansion (SCE) with fully self-consistent noncollinear spin density functional theory (DFT). The key idea is to fit the SCE model to magnetic torques rather than to total energies. Because torques are site-resolved vectors, each spin configuration provides many informative regression targets, improving conditioning and substantially reducing the number of required DFT calculations, especially for large supercells. Applied to the B20-type chiral magnets Mn1−xFexGe and Fe1−yCoyGe, the resulting SCE models determine full pairwise exchange tensors—including isotropic exchange, symmetric anisotropic exchange, and the Dzyaloshinskii-Moriya interaction—and predict the helical spin period via a micromagnetic mapping. The composition trends and the divergence of the period at the chirality sign-change point are well reproduced, in agreement with experiment. Moreover, the systematic nature of SCE enables controlled assessment of interaction order: As the training spin configurations become more disordered, the lowest-order model loses torque accuracy, whereas including higher-order interactions restores predictive power. These advances enable near-DFT-accurate spin models for finite-temperature magnetism and complex spin textures at modest computational cost, providing an extensible route to quantitative first-principles parametrization and predictive materials design.

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References (65)

  1. A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu, B. Diény, P. Pirro, and B. Hillebrands, Review on spintronics: Principles and device applications, J. Magn. Magn. Mater. 509, 166711 (2020).
  2. L. Bai, W. Feng, S. Liu, L. Šmejkal, Y. Mokrousov, and Y. Yao, Altermagnetism: Exploring new frontiers in magnetism and spintronics, Adv. Funct. Mater. 34, 2409327 (2024).
  3. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  4. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  5. B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson, A method for atomistic spin dynamics simulations: Implementation and examples, J. Phys.: Condens. Matter 20, 315203 (2008).
  6. M. Y. Lavrentiev, D. Nguyen-Manh, and S. L. Dudarev, Magnetic cluster expansion model for bcc-fcc transitions in Fe and Fe-Cr alloys, Phys. Rev. B 81, 184202 (2010).
  7. M. Y. Lavrentiev, S. L. Dudarev, and D. Nguyen-Manh, Magnetic cluster expansion model for high-temperature magnetic properties of iron and iron–chromium alloys, J. Appl. Phys. 109, 07E123 (2011).
  8. R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. Chantrell, Atomistic spin model simulations of magnetic nanomaterials, J. Phys.: Condens. Matter 26, 103202 (2014).
  9. O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik, Atomistic Spin Dynamics: Foundations and Applications (Oxford University Press, Oxford, 2017).
  10. R. Yamada and T. Mohri, Conversion of magnetic freedoms into atomic configurational freedoms within the cluster variation method, Mater. Trans. 60, 915 (2019).
  11. M. Rinaldi, M. Mrovec, A. Bochkarev, Y. Lysogorskiy, and R. Drautz, Non-collinear magnetic atomic cluster expansion for iron, npj Comput. Mater. 10, 12 (2024).
  12. T. Tanaka and Y. Gohda, Impact of electron–spin coupling on exchange coupling parameters: A nonperturbative approach, arXiv:2410.11256.
  13. E. Mendive-Tapia, M. dos Santos Dias, S. Grytsiuk, J. B. Staunton, S. Blügel, and S. Lounis, Short period magnetization texture of B20-MnGe explained by thermally fluctuating local moments, Phys. Rev. B 103, 024410 (2021).
  14. M. Gutzeit, S. Haldar, S. Meyer, and S. Heinze, Trends of higher-order exchange interactions in transition metal trilayers, Phys. Rev. B 104, 024420 (2021).
  15. W. Pan, C. Xu, X. Li, Z. Xu, B. Liu, B.-L. Gu, and W. Duan, Chiral magnetism in lithium-decorated monolayer CrTe2: Interplay between Dzyaloshinskii-Moriya interaction and higher-order interactions, Phys. Rev. B 109, 214405 (2024).
  16. R. Drautz and M. Fähnle, Spin-cluster expansion: Parametrization of the general adiabatic magnetic energy surface with ab initio accuracy, Phys. Rev. B 69, 104404 (2004).
  17. R. Drautz and M. Fähnle, Parametrization of the magnetic energy at the atomic level, Phys. Rev. B 72, 212405 (2005).
  18. R. Singer, F. Dietermann, and M. Fähnle, Spin interactions in bcc and fcc Fe beyond the Heisenberg model, Phys. Rev. Lett. 107, 017204 (2011).
  19. J. W. D. Connolly and A. R. Williams, Density-functional theory applied to phase transformations in transition-metal alloys, Phys. Rev. B 27, 5169 (1983).
  20. K. Parlinski, Z. Q. Li, and Y. Kawazoe, First-principles determination of the soft mode in cubic ZrO2, Phys. Rev. Lett. 78, 4063 (1997).
  21. T. Tadano, Y. Gohda, and S. Tsuneyuki, Anharmonic force constants extracted from first-principles molecular dynamics: Applications to heat transfer simulations, J. Phys.: Condens. Matter 26, 225402 (2014).
  22. O. T. Unke, S. Chmiela, H. E. Sauceda, M. Gastegger, I. Poltavsky, K. T. Schütt, A. Tkatchenko, and K.-R. Müller, Machine learning force fields, Chem. Rev. 121, 10142 (2021).
  23. P. H. Dederichs, S. Blügel, R. Zeller, and H. Akai, Ground states of constrained systems: Application to cerium impurities, Phys. Rev. Lett. 53, 2512 (1984).
  24. S. Brinker, M. d. S. Dias, and S. Lounis, The chiral biquadratic pair interaction, New J. Phys. 21, 083015 (2019).
  25. A. Jacobsson, G. Johansson, O. I. Gorbatov, M. Ležaić, B. Sanyal, S. Blügel, and C. Etz, Efficient parameterisation of non-collinear energy landscapes in itinerant magnets, Sci. Rep. 12, 18987 (2022).
  26. R. Singer and M. Fähnle, Construction of basis functions for the spin-cluster expansion of the magnetic energy on the atomic scale in rotationally invariant systems, J. Math. Phys. 47, 113503 (2006).
  27. F. Dietermann, R. Singer, and M. Fähnle, Construction of basis functions with crystal symmetry for the spin-cluster expansion of the magnetic energy on the atomic scale, J. Math. Phys. 52, 123507 (2011).
  28. M. A. Blanco, M. Flórez, and M. Bermejo, Evaluation of the rotation matrices in the basis of real spherical harmonics, J. Mol. Struct. (THEOCHEM) 419, 19 (1997).
  29. A. Togo, K. Shinohara, and I. Tanaka, Spglib: A software library for crystal symmetry search, Sci. Technol. Adv. Mater.: Methods 4, 2384822 (2024).
  30. P.-W. Ma and S. L. Dudarev, Constrained density functional for noncollinear magnetism, Phys. Rev. B 91, 054420 (2015).
  31. R. Drautz, Atomic cluster expansion of scalar, vectorial, and tensorial properties including magnetism and charge transfer, Phys. Rev. B 102, 024104 (2020).
  32. L. Lundgren, O. Beckman, V. Attia, S. P. Bhattacherjee, and M. Richardson, Helical spin arrangement in cubic FeGe, Phys. Scr. 1, 69 (1970).
  33. B. Lebech, J. Bernhard, and T. Freltoft, Magnetic structures of cubic FeGe studied by small-angle neutron scattering, J. Phys.: Condens. Matter 1, 6105 (1989).
  34. N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and Y. Tokura, Large topological Hall effect in a short-period helimagnet MnGe, Phys. Rev. Lett. 106, 156603 (2011).
  35. X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe, Nat. Mater. 10, 106 (2011).
  36. S. V. Grigoriev, N. M. Potapova, S.-A. Siegfried, V. A. Dyadkin, E. V. Moskvin, V. Dmitriev, D. Menzel, C. D. Dewhurst, D. Chernyshov, R. A. Sadykov, L. N. Fomicheva, and A. V. Tsvyashchenko, Chiral properties of structure and magnetism in Mn1−xFexGe compounds: When the left and the right are fighting, who wins? Phys. Rev. Lett. 110, 207201 (2013).
  37. K. Shibata, X. Z. Yu, T. Hara, D. Morikawa, N. Kanazawa, K. Kimoto, S. Ishiwata, Y. Matsui, and Y. Tokura, Towards control of the size and helicity of skyrmions in helimagnetic alloys by spin-orbit coupling, Nat. Nanotechnol. 8, 723 (2013).
  38. N. Kanazawa, K. Shibata, and Y. Tokura, Variation of spin–orbit coupling and related properties in skyrmionic system Mn1−xFexGe, New J. Phys. 18, 045006 (2016).
  39. S.-A. Siegfried, A. S. Sukhanov, E. V. Altynbaev, D. Honecker, A. Heinemann, A. V. Tsvyashchenko, and S. V. Grigoriev, Spin-wave dynamics in the helimagnet FeGe studied by small-angle neutron scattering, Phys. Rev. B 95, 134415 (2017).
  40. C. S. Spencer, J. Gayles, N. A. Porter, S. Sugimoto, Z. Aslam, C. J. Kinane, T. R. Charlton, F. Freimuth, S. Chadov, S. Langridge, J. Sinova, C. Felser, S. Blügel, Y. Mokrousov, and C. H. Marrows, Helical magnetic structure and the anomalous and topological Hall effects in epitaxial B20 Fe1−yCoyGe films, Phys. Rev. B 97, 214406 (2018).
  41. E. Turgut, H. Paik, K. Nguyen, D. A. Muller, D. G. Schlom, and G. D. Fuchs, Engineering Dzyaloshinskii-Moriya interaction in B20 thin-film chiral magnets, Phys. Rev. Mater. 2, 074404 (2018).
  42. Y. Guang, Y. Fujishiro, A. Tanaka, L. Peng, Y. Kaneko, N. Kanazawa, Y. Tokura, and X. Yu, Topological stability of spin textures in Si/Co-doped helimagnet FeGe, J. Phys. Mater. 7, 025009 (2024).
  43. J. Gayles, F. Freimuth, T. Schena, G. Lani, P. Mavropoulos, R. A. Duine, S. Blügel, J. Sinova, and Y. Mokrousov, Dzyaloshinskii-Moriya interaction and Hall effects in the skyrmion phase of Mn1−xFexGe, Phys. Rev. Lett. 115, 036602 (2015).
  44. T. Kikuchi, T. Koretsune, R. Arita, and G. Tatara, Dzyaloshinskii-Moriya interaction as a consequence of a Doppler shift due to spin-orbit-induced intrinsic spin current, Phys. Rev. Lett. 116, 247201 (2016).
  45. T. Koretsune, T. Kikuchi, and R. Arita, First-principles evaluation of the Dzyaloshinskii–Moriya interaction, J. Phys. Soc. Jpn. 87, 041011 (2018).
  46. S. Grytsiuk, M. Hoffmann, J.-P. Hanke, P. Mavropoulos, Y. Mokrousov, G. Bihlmayer, and S. Blügel, Ab initio analysis of magnetic properties of the prototype B20 chiral magnet FeGe, Phys. Rev. B 100, 214406 (2019).
  47. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  48. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  49. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  50. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  51. B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, A first-principles theory of ferromagnetic phase transitions in metals, J. Phys. F: Met. Phys. 15, 1337 (1985).
  52. E. Mendive-Tapia, J. Neugebauer, and T. Hickel, Ab initio calculation of the magnetic Gibbs free energy of materials using magnetically constrained supercells, Phys. Rev. B 105, 064425 (2022).
  53. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120, 91 (1960).
  54. M. Bornemann, S. Grytsiuk, P. F. Baumeister, M. Dos Santos Dias, R. Zeller, S. Lounis, and S. Blügel, Complex magnetism of B20-MnGe: From spin-spirals, hedgehogs to monopoles, J. Phys.: Condens. Matter 31, 485801 (2019).
  55. A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and V. A. Gubanov, Local spin density functional approach to the theory of exchange interactions in ferromagnetic metals and alloys, J. Magn. Magn. Mater. 67, 65 (1987).
  56. T. Oguchi, K. Terakura, and N. Hamada, Magnetism of iron above the Curie temperature, J. Phys. F: Met. Phys. 13, 145 (1983).
  57. S. Mankovsky, S. Polesya, and H. Ebert, Extension of the standard Heisenberg Hamiltonian to multispin exchange interactions, Phys. Rev. B 101, 174401 (2020).
  58. A. V. Ruban and O. E. Peil, Impact of thermal atomic displacements on the Curie temperature of 3d transition metals, Phys. Rev. B 97, 174426 (2018).
  59. L. Mauger, M. S. Lucas, J. A. Muñoz, S. J. Tracy, M. Kresch, Y. Xiao, P. Chow, and B. Fultz, Nonharmonic phonons in α-iron at high temperatures, Phys. Rev. B 90, 064303 (2014).
  60. T. Tanaka and Y. Gohda, Prediction of the Curie temperature considering the dependence of the phonon free energy on magnetic states, npj Comput. Mater. 6, 184 (2020).
  61. T. Tanaka and Y. Gohda, First-principles study of magnetism-dependent phonons governed by exchange ligand field, J. Phys. Soc. Jpn. 89, 093705 (2020).
  62. X. Z. Lu, X. Wu, and H. J. Xiang, General microscopic model of magnetoelastic coupling from first principles, Phys. Rev. B 91, 100405(R) (2015).
  63. S. Mankovsky, H. Lange, S. Polesya, and H. Ebert, Spin-lattice interaction parameters from first principles: Theory and implementation, Phys. Rev. B 107, 144428 (2023).
  64. I. P. Miranda, M. Pankratova, M. Weißenhofer, A. B. Klautau, D. Thonig, M. Pereiro, E. Sjöqvist, A. Delin, M. I. Katsnelson, O. Eriksson, and A. Bergman, Spin-lattice couplings in 3d ferromagnets: Analysis from first principles, Phys. Rev. Mater. 9, 024409 (2025).
  65. T. Tanaka, Julia package, GitHub, 2026, https://github.com/Tomonori-Tanaka/Magesty.jl.

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Phys. Rev. Research 8, 023300– Published 15 June, 2026
  • Received 24 February 2026
  • Accepted 27 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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