Abstract
Magnetic singularities known as Bloch points (BPs) present a fundamental challenge for micromagnetic theory, which is based on the assumption of a fixed magnetization vector length. Due to the divergence of the effective field at a BP, classical micromagnetics fails to adequately describe BP dynamics. To address this issue, we propose a regularized micromagnetic model in which the magnetization vector can vary in length but not exceed a threshold value. More specifically, the magnetization is treated as an order parameter constrained to an \({{\mathbb{S}}}^{3}\)-sphere. This constraint respects fundamental properties of local spin expectation values in quantum systems. We derive the corresponding regularized Landau-Lifshitz-Gilbert equation and the analog of the Thiele equation describing the steady motion of spin textures under various external stimuli. We demonstrate the applicability of our theory by modeling the dynamics of several magnetic textures containing BPs, including domain walls in nanowires, chiral bobbers, and magnetic dipolar strings. The presented results extend micromagnetic theory by incorporating a regularized description of BP dynamics.
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Source data for micromagnetic simulations and scripts for post-processing are available at https://doi.org/10.5281/zenodo.17474662.
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Acknowledgements
N.S.K. and V.M.K. acknowledge F. N. Rybakov for fruitful discussions and critical comments. V.M.K., A.H., T.L.S., and A.M. acknowledge financial support from the National Research Fund of Luxembourg (AFR/23/17951349 and C22/MS/17415246/DeQuSky). V.M.K. acknowledges financial support from the European Union’s Horizon Europe research and innovation program under the Marie Skłodowska-Curie grant agreement No. 101203692 (QUANTHOPF). N.S.K. acknowledges support from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Program (Grant No. 856538—project “3D MAGiC”).
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V.M.K. and N.S.K. conceived the project. V.M.K. developed the theoretical model and performed numerical simulations. V.M.K., A.H., A.M., T.L.S., and N.S.K. discussed the results and contributed to writing the manuscript.
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Kuchkin, V.M., Haller, A., Michels, A. et al. Regularized micromagnetic theory for Bloch points. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02565-z
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DOI: https://doi.org/10.1038/s42005-026-02565-z
