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Conserved momenta of ferromagnetic solitons through the prism of differential geometry
by Xingjian Di, Oleg Tchernyshyov
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|As Contributors:||Xingjian Di · Oleg Tchernyshyov|
|Arxiv Link:||https://arxiv.org/abs/2105.03553v1 (pdf)|
|Date submitted:||2021-05-11 02:08|
|Submitted by:||Di, Xingjian|
|Submitted to:||SciPost Physics|
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.
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Anonymous Report 2 on 2021-6-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2105.03553v1, delivered 2021-06-22, doi: 10.21468/SciPost.Report.3102
In this manuscript, the authors have revisited the long-standing problem of conserved angular momenta of a ferromagnetic system. They proposed a differential form based on the Cartan's theory to describe the dynamics of collective coordinates. The manuscript is well written I found it interesting its result interesting. I think the manuscript satisfies the journal's criteria and I support publication of this manuscript in SciPost Physics.
I have only one optional comment: I think it is possible to extend the current work to spin torsion tensor formalism, discussed in the gravity theories. I am wondering if the problem of definition of proper momenta in the magneto-elastic media can also be resolved with this approach. If yes, it should be instructive for readers to point it out in the manuscript.
Anonymous Report 1 on 2021-6-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2105.03553v1, delivered 2021-06-22, doi: 10.21468/SciPost.Report.3096
1. The paper is written clearly and methodologically.
2. The mathematical framework explained is important for many applications in magnetism.
3. The manuscript provides a better understanding of the Belavin-Polyakov solution in field theory.
A reader might be interested not only in a mathematical toolbox, but also in a more in-depth insight into physical conclusions.
The authors present a discussion about conservation laws for the topological magnetic solitons and identify a proper mathematical framework to find symmetries of the given texture and derive the respective conserved momenta. This approach is used to analyze the Belavin-Polyakov soliton. The discussed approach is well-timed because of recent interest to the complex magnetic objects, where methods of the differential geometry help to understand the physics. The Belavin-Polyakov solution itself is one of the most known fundamental models in the field theory, and a better understanding of its properties should provide an impact beyond the magnetic community. That’s why, I believe, the manuscript is of interest and can be published in SciPost Physics.
I have some comments, which are listed according to the text in this manuscript.
After Eq. (4) authors say that the gauge potential is not a physical quantity. However, its physical meaning is well discussed in literature. For example, one can mention the Feynman’s question “Is the vector potential a “real” field?”
In a discussion of the multiple-valued of $P_X$ for the domain wall, I wonder about its relation with the Berry phase. One can suggest that such quantities can obtain an additional physical sense in specific cases like motion under the action of spin-orbit torques.
If I understand correctly, the new conserved momenta are related with the finite size of the system due to the cutting radius $R$. This sounds as an additional condition which determines the conservation according to $O(X^n, Y^n)$ with some $n$ in Eq. (75). Then, the presense of these momenta can be expexted to be pronounced in the relaxation dynamics of the soliton, see DOI:10.1103/PhysRevB.90.174428 for example.
There are many works on the so-called dynamic solitons (see also Ref.  in the manuscript), stabilization of whose is reached by a steady rotation of the magnetization phase $\phi = \phi_0 + \omega t$. Is the second pair of momenta, $P_\alpha$, and $P_\Sigma$, related to this mechanism?
- sentence before Eq. (3), it seems like a word is missing: “... of a massless electrically charged [particle] in a...”
Besides misprints, I believe that the manuscript should strongly benefit from expanding the discussion on physical consequences and, especially, new pair of conserved momenta.