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Conserved momenta of ferromagnetic solitons through the prism of differential geometry

by Xingjian Di, Oleg Tchernyshyov

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Xingjian Di · Oleg Tchernyshyov
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-10-22
Date submitted: 2021-07-06 07:18
Submitted by: Di, Xingjian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


The relation between symmetries and conservation laws for solitons in a ferromagnet is complicated by the presence of gyroscopic (precessional) forces, whose description in the Lagrangian framework involves a background gauge field. This makes canonical momenta gauge-dependent and requires a careful application of Noether's theorem. We show that Cartan's theory of differential forms is a natural language for this task. We use it to derive conserved momenta of the Belavin--Polyakov skyrmion, whose symmetries include translation, global spin rotation, and dilation.

Published as SciPost Phys. 11, 108 (2021)

Author comments upon resubmission

Dear Editor,

We are resubmitting our manuscript in response to the referee reports.

The reports are overall positive, with both reviewers recommending publication of the manuscript in SciPost Physics. The referees raised some questions and suggested a number of improvements, for which we are grateful. We have expanded the manuscript in accordance with these suggestions. Below we outline the changes in the manuscript. Specific queries of the referees have been addressed in our replies to them.

Referee 1 wrote: "the manuscript should strongly benefit from expanding the discussion on physical consequences and, especially, new pair of conserved momenta." We have added Subsection 4.5 "Low-energy dynamics of a skyrmion," in which we spell out the physical significance of the new pair of conserved momenta---the global azimuthal angle and skyrmion spin. In the presence of perturbations beyond the ideal Heisenberg model, these two quantities evolve in time in a prescribed manner. The application of a uniform magnetic field parallel to the ground-state magnetization induces a steady precession of the global azimuthal angle, whereas the injection of a spin-polarized electric current at the center of the skyrmion leads to an expansion or contraction of the skyrmion area (spin). We have also added a few sentences mentioning these consequences to Section 5 "Conclusion."

We hope that you will find the revised manuscript suitable for publication and look forward to your reply.

With best regards,

Xingjian Di
Oleg Tchernyshyov

Reports on this Submission

Anonymous Report 3 on 2021-10-13 (Contributed Report)

  • Cite as: Anonymous, Report on arXiv:2105.03553v2, delivered 2021-10-13, doi: 10.21468/SciPost.Report.3723


The present manuscript addresses a challenge in the relation between symmetries and conservation laws in ferromagnets.
In FMs, since there is a background gauge field it is challenging to make a gauge-invariant energy-momentum tensor and apply Noether's theorem. Even though the main problem which has been considered here has been known for many years in the community of magnetism, thanks to the recent advances in spintronics, which make manipulation and interaction of nonlinear magnetic solitons and magnetic excitations possible, there is a new interest in this topic. For example these recent papers: PHYSICAL REVIEW B 88, 144413 (2013) and PHYSICAL REVIEW B 98, 224401 (2018).

Here, in this paper, the authors have developed a new approach (especially in the context of magnetism) based on Cartan's theory which resolves part of gauge-invariant difficulties, and related issues such as the relation of canonical and conserved momenta, and also possible path dependence of the conserved momenta in the collective coordinates. In addition, the rigorousness and power of Cartan's differential geometry, make the formalism developed in this paper a suitable framework for future investigations about ferromagnetic solitons. So overall, the paper has developed an approach that resolves theoretical difficulties in a more rigorous and less subtle manner. Therefore, it is a timely study in this topic that can serve as a part of theoretical developments and I suggest its publication in SciPost Physics.

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Anonymous Report 2 on 2021-7-15 (Invited Report)


The authors have addressed all questions. I believe the present version of the manuscript satisfies the SciPost Physics' criteria.

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Anonymous Report 1 on 2021-7-15 (Invited Report)


I like the updated version and recommend it for publication in SciPost Physics.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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