Chiral Skyrmions

  Copyright: © Lehrstuhl f. Angewandte Analysis

Topological solitons are localized field configurations in a non-trivial homotopy class which are described within the framework of a nonlinear field theory. We are particularly interested in so-called chiral skyrmions, occurring magnetic systems of broken chiral symmetry. Proving the first mathematical existence result for chiral skyrmions in the plane we kicked-off the mathematical analysis for this new class of variational problems.

Further contributions of our group concern static and dynamic stability questions and in particular the local minimality of axisymmetric skyrmions.


Christof Melcher © Copyright: Peter Winandy


+49 241 80-94585



The symmetry of minimizing skyrmions remains part of our current research as an open problem. We have developed a mathematical framework for skyrmion lattice solutions as magnetic analogues of Abrikosov vortex lattices in superconductivity, based on symmetry breaking bifurcation methods. Our analysis revealed a surprisingly large variety of patterns occurring in chiral magnetism. Beyond mathematical theory we als contributed to the exploration of new topological structures in the physics of chiral magnets. In joint work with physicists at the Forschungszentrum Jülich we have predicted the occurrence of novel skyrmionic configurations of negative degree, so-called antiskyrmions, in certain magnetic layers. Surprisingly, our analysis revealed the unexpected co-existence of skyrmions and antiskyrmions in such material systems. Our current interest is especially in dynamic states including rotating and breathing skyrmions. A joint project within the RTG EDDy concerns saddle points with the goal to understand skyrmion collapse and nucleation. A further line of research in our group concerns the occurrence knotted solitons in 3D, so-called magnetic hopfions.


  • Magnetic hopfions in solids
    FN Rybakov, NS Kiselev, AB Borisov, L Döring, C Melcher, S Blügel
    APL materials 10 (11), 111113 (2022)
  • Lattice Solutions in a Ginzburg–Landau Model for a Chiral Magnet
    X Li, C Melcher
    Journal of Nonlinear Science 30 (6), 3389-3420 (2020)
  • Curvature-stabilized skyrmions with angular momentum
    C Melcher, ZN Sakellaris
    Letters in Mathematical Physics 109 (10), 2291-2304 (2019)
  • Stability of axisymmetric chiral skyrmions
    X Li, C Melcher
    Journal of Functional Analysis 275 (10), 2817-2844 (2018)
  • Compactness results for static and dynamic chiral skyrmions near the conformal limit
    L Döring, C Melcher
    Calculus of Variations and Partial Differential Equations 56 (3), 1-30 (2017)
  • Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions
    M Hoffmann, B Zimmermann, GP Müller, D Schürhoff, NS Kiselev, C Melcher, B. Blügel
    Nature communications 8 (1), 1-9 (2017)
  • Chiral skyrmions in the plane
    C Melcher
    ​Proceedings of the Royal Society A: Mathematical, Physical and Engineering (2014)