The fabrication of materials with a chiral magnetic interaction (Dzyaloshinskii-Moriya interaction) and the subsequent experimental observation of solitons in films (so-called “chiral skyrmions”) has provided a system for the study of static and dynamical properties of topological magnetic solitons.

The size of a chiral skyrmion can vary from very small to very large depending on system parameters.
The profile of an axially-symmetric skyrmion is what makes it different from the previously studied magnetic bubbles.
The skyrmion solutions can be studied by asymptotic methods.
For small DM parameter ε, the skyrmion radius R is implicitly given by ε=-R ln(R).
The skyrmion profile is that of a Belavin-Polyakov solution over and well-beyond the core before it assumes an exponential decay.
[Nonlinearity, 2020].
For dimensionless DM parameter ε close to the critical value ε_{0}=2/π, the skyrmion radius diverges according to
ε = 2/π - 0.3057/R^{2} - 0.8792/R^{4} + ...
[arXiv, 2019]

- A small DM parameter supports skyrmions of small radius, and the relation involves a logarithm, ε=-R ln(R).
- A chiral skyrmion is born out of a Belavin-Polyakov solution with an infinitesimally small radius, as the DM parameter is increased from zero.
- This critical value ε
_{0}=2/π where the skyrmion radius diverges coincides with the transition point from the uniform phase to the helical phase.

- DM parameter ε vs skyrmion radius R, from skyrmion profiles obtained by a shooting method. (See [Nonlinearity, 2020], [arXiv, 2019].)

DM materials are especially interesting because they support both, robust topological solitons (“chiral skyrmions”) and also topologically trivial (loosely called “nontopological”) skyrmions. Especially the dynamics of topologically trivial solitons was only partialy described by the existing formalism - it is essentially an open question.

We have taken this opportunity to revisit skyrmion dynamics. We find that the dynamics of topological and topologically trivial solitons differ in a profound way [PRB, 2015]. Newtonian dynamics is found for topologically trivial solitons and we could rigorously attribute a mass to them. That elevates magnetic solitons to particle-like objects. The dynamics of topologically trivial solitons turns relativistic for larger velocities. Topological skyrmions undergo Hall motion under external forces, as anticipated, i.e., they exhibit non-Newtonian kinetics where the skyrmion moves in a direction perpendicular to the applied force.

We extended the theory to the dynamics induced by spin-transfer torques [PRB, 2015]. The Landau-Lifshitz equation (including damping and spin-torque) can be brought to a form with known solutions in some cases. This gives a significant insight for the usually complicated behaviour of the skyrmions under external forces.

Skyrmion and skyrmionium under in-plane spin transfer torque.

- Talk at the seminar "Solitons at Work", Leeds, 25/11/2020.
- "Dynamics of skyrmions in chiral ferromagnets", AIM Conference, Bormio, Italy, 14-16/3/2016.

Chiral symmetry breaking has the consequence that the total magnetization is not conserved. This has profound consequences for the dynamics of solitons in the chiral magnets. We have shown that chiral symmetry breaking enables traveling domain wall solution for the conservative Landau–Lifshitz equation of a uniaxial ferromagnet with Dzyaloshinskii–Moriya interaction. They cannot be found in closed form. For the construction we have followed a topological approach and have provided details of solutions by means of numerical calculations.

By a scaling of the equation, we could show that the attainable domain wall velocity scales proportionally to the anisotropy or the DM interaction, if we keep their ratio constant. There is no theoretical limit to the velocity and this is set only by the availability of materials with high anisotropy and DM parameters. Furthermore, the width of the domain wall scales inversely proportional to the anisotropy which means that faster and narrower walls are obtained for increasing anisotropy. The above dynamical behavior is very different than that of the standard Walker domain walls which become slower as the anisotropy increases and a theoretical limit exists for small anisotropies. [Nonlinearity 2019]