Curvature-stabilized skyrmions with angular momentum

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We examine skyrmionic field configurations on a spherical ferromagnet with large normal anisotropy. Exploiting variational concepts of angular momentum, we find a new family of localized solutions to the Landau–Lifshitz equation that are topologically distinct from the ground state and not equivariant. Significantly, we observe an emergent spin–orbit coupling on the level of magnetization dynamics in a simple system without individual rotational invariance in spin and coordinate space.

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  1. 1.

    Belavin, A., Polyakov, A.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett. 22(10), 245–248 (1975)

  2. 2.

    Bogdanov, A., Hubert, A.: Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magnet. Magnet. Mater. 138(3), 255–269 (1994)

  3. 3.

    Bogdanov, A., Hubert, A.: The stability of vortex-like structures in uniaxial ferromagnets. J. Magnet. Magnet. Mater. 195(1), 182–192 (1999)

  4. 4.

    Bogdanov, A., Yablonskii, D.A.: Thermodynamically stable vortices in magnetically ordered crystals. The mixed state of magnets. Sov. Phys. JETP 68(1), 101–103 (1989)

  5. 5.

    Brezis, H., Coron, J.-M.: Large solutions for harmonic maps in two dimensions. Commun. Math. Phys. 92(2), 203–215 (1983)

  6. 6.

    Brezis, H., Coron, J.-M., Lieb, E.H.: Harmonic maps with defects. Commun. Math. Phys. 107(4), 649–705 (1986)

  7. 7.

    Di Fratta, G., Slastikov, V., Zarnescu, A.: On a sharp Poincaré-type inequality on the 2-sphere and its application in micromagnetics. arXiv preprint. arXiv:1901.04334 (2019)

  8. 8.

    Döring, L., Melcher, C.: Compactness results for static and dynamic chiral skyrmions near the conformal limit. Calc. Var. Partial Differ. Equ. 56(3) Art. 60, 30 (2017)

  9. 9.

    Esteban, M.J.: A direct variational approach to Skyrme’s model for meson fields. Commun. Math. Phys. 105(4), 571–591 (1986)

  10. 10.

    Gol’dshtein, E., Tsukernik, V.: Angular momentum of a Heisenberg ferromagnet with a magnetic dipole interaction. Zh. Eksp. Teor. Fiz 87, 1330–1335 (1984)

  11. 11.

    Gustafson, S., Shatah, J.: The stability of localized solutions of Landau–Lifshitz equations. Commun. Pure Appl. Math. 55(9), 1136–1159 (2002)

  12. 12.

    Hoffmann, M., Zimmermann, B., Müller, G.P., Schürhoff, D., Kiselev, N.S., Melcher, C., Blügel, S.: Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii–Moriya interaction. Nat. Commun. 8, 308 (2017)

  13. 13.

    Komineas, S., Papanicolaou, N.: Topology and dynamics in ferromagnetic media. Physica D Nonlinear Phenom. 99(1), 81–107 (1996)

  14. 14.

    Kosevich, A.M., Ivanov, B., Kovalev, A.: Magnetic solitons. Phys. Rep. 194(3–4), 117–238 (1990)

  15. 15.

    Kravchuk, V.P., Rößler, U.K., Volkov, O.M., Sheka, D.D., van den Brink, J., Makarov, D., Fuchs, H., Fangohr, H., Gaididei, Y.: Topologically stable magnetization states on a spherical shell: curvature-stabilized skyrmions. Phys. Rev. B 94(14), 144402 (2016)

  16. 16.

    Kurzke, M., Melcher, C., Moser, R., Spirn, D.: Ginzburg-Landau vortices driven by the Landau–Lifshitz–Gilbert equation. Arch. Ration. Mech. Anal. 199(3), 843–888 (2011)

  17. 17.

    Li, X., Melcher, C.: Stability of axisymmetric chiral skyrmions. J. Funct. Anal. 275(10), 2817–2844 (2018)

  18. 18.

    Lin, F., Yang, Y.: Existence of two-dimensional skyrmions via the concentration-compactness method. Commun. Pure Appl. Math. 57(10), 1332–1351 (2004)

  19. 19.

    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoam. 1(2), 45–121 (1985)

  20. 20.

    Manton, N., Sutcliffe, P.: Topological Solitons. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2004)

  21. 21.

    Melcher, C.: Chiral skyrmions in the plane. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470(2172), 20140394 (2014)

  22. 22.

    Moser, R.: Partial Regularity for Harmonic Maps and Related Problems. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005)

  23. 23.

    Papanicolaou, N., Tomaras, T.: Dynamics of magnetic vortices. Nucl. Phys. B 360(2–3), 425–462 (1991)

  24. 24.

    Papanicolaou, N., Zakrezewski, W.: Dynamics of interacting magnetic vortices in a model Landau–Lifshitz equation. Physica D Nonlinear Phenom. 80(3), 225–245 (1995)

  25. 25.

    Schütte, C., Garst, M.: Magnon-skyrmion scattering in chiral magnets. Phys. Rev. B 90(9), 094423 (2014)

  26. 26.

    Yan, P., Kamra, A., Cao, Y., Bauer, G.E.: Angular and linear momentum of excited ferromagnets. Phys. Rev. B 88(14), 144413 (2013)

  27. 27.

    Zhou, Y., Iacocca, E., Awad, A.A., Dumas, R.K., Zhang, F., Braun, H.B., Åkerman, J.: Dynamically stabilized magnetic skyrmions. Nat. Commun. 6, 8193 (2015)

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We are indebted to Stavros Komineas for pointing out the relevance of angular momenta in the context of chiral magnetism and for valuable discussions on the subject matter.

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Correspondence to Christof Melcher.

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This work is supported by Deutsche Forschungsgemeinschaft (DFG Grant No. ME 2273/3-1).

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Melcher, C., Sakellaris, Z.N. Curvature-stabilized skyrmions with angular momentum. Lett Math Phys 109, 2291–2304 (2019).

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  • Magnetic skyrmions
  • Landau–Lifshitz equation
  • Angular momentum

Mathematics Subject Classification

  • 49S05
  • 35Q60
  • 37K05
  • 82D40