Modelling the Elliptical Instability of Magnetic Skyrmions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the peer-reviewed manuscript "Modelling the Elliptical Instability of Magnetic Skyrmions", the Author considers several numerical approaches to modeling the elliptic instability of magnetic skirmions. It considers two new scenarios: the tilted ferromagnetic phase of chiral magnets dominated by light-plane anisotropy, and the general case of a chiral magnet with a tilted applied field and arbitrary uniaxial anisotropy. The results obtained in the manuscript may be useful in liquid crystal or ferroelectric systems where there are a large number of parameters describing the system: a single analytical calculation can include all of them, giving a range of material parameters for which solitons can be stable and thus guide future numerical studies.  I believe that the manuscript can be accepted for publication provided that it undergoes minor revision and addresses a few of my questions and comments:

1. What impact do assumptions made in analytical models (e.g., neglecting stray field interactions) have on the generalizability of results to real materials?

2. The manuscript indicates that analytical predictions align with previous numerical results in certain instances but not in others. Could you please provide a more detailed explanation of the specific conditions or parameters under which the discrepancies occur?

3. I believe the introduction would benefit from a more detailed description of the applicability of skirmions and magnetic solitons, as well as the inclusion of additional relevant works on this subject. 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In the manuscript, a complete map of stability of stationary magnetic skyrmions for a large set of parameters (including Dzyaloshinskii vector and different orientations and amplitudes of external field) is created. It is done based on briliant application of a method of Lyapunov stability analysis to stationary problems of spatial distribution of fields. Though the method and the problem are not new, the author completes the current knowledge by widening the stability map in the parameter space (compared to e.g. ref. [13]). 

However, several statements make the paper unclear to the reader and require explanation/clarification.

1. The stability analysis is a methodology developed for autonomous (dynamical) systems of ordinary differental equations. Use of this methodology to static problems is non-standard (though performed earlier) and it requires an introductory information to the general reader. Also, the term "ellipic instability" is known for the researchers involved into fluid dynamics, though, for wide community, it is exotic term and it should be introduced somehow. 

2. It seems a questionable statement (on page 5) that the 2pi domain wall gets better approximation of the cross-sectional profile of the magnetic soliton as the soliton extends. For instance, large skyrmion (a magnetic bubble) is better represented by a separate magnetic domain than 2pi domain wall. The 2pi domain wall is an object of a constant width which is determined by material parameters. Is not use of the name 2pi domain wall a nomenclature misunderstanding?

3. The distinction between 'long', 'short' and '2pi' domain wall is unclear. At least a draw or maybe a strict mathematical definition would be in order.

Minor:

1. The parameter "k"  should be clearly defined in please it is first used.

2. The title does not sound well since the 'instability' is a property to be checked out not 'modelled'.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper under consideration presents a theoretical study on the instability of magnetic skyrmions. Two  particular cases with dominating easy-plane anisotropy or tilted magnetic field and arbitrary easy-axis anisotropy were studied by using two different approaches, yielding reasonable results. The paper is well written and deserves to be published in my opinion. There are a few issues I would like to ask the author to comment on. 

It is mentioned in the paper that the dipolar energy was neglected in the calculations. What is the justification for this approximation? How would the dipolar interaction affect the results?

Form the point view of dynamics, the instability is usually accompanied by the occurrence of a soft mode at the critical point. Would the methods employed in the work provide any information about the soft modes?

Finally, some symbols used in the equations were not explicitly defined. 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf