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Skyrmion Jellyfish in Driven Chiral Magnets
by Nina del Ser, Vivek Lohani
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Submission summary
Authors (as registered SciPost users):  Nina del Ser 
Submission information  

Preprint Link:  scipost_202303_00034v1 (pdf) 
Date accepted:  20230524 
Date submitted:  20230326 14:29 
Submitted by:  del Ser, Nina 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Chiral magnets can host topological particles known as skyrmions which carry an exactly quantised topological charge $Q=1$. In the presence of an oscillating magnetic field ${\bf B}_1(t)$, a single skyrmion embedded in a ferromagnetic background will start to move with constant velocity ${\bf v}_{\text{trans}}$. The mechanism behind this motion is similar to the one used by a jellyfish when it swims through water. We show that the skyrmion's motion is a universal phenomenon, arising in any magnetic system with translational modes. By projecting the equation of motion onto the skyrmion's translational modes and going to quadratic order in ${\bf B}_1(t)$ we obtain an analytical expression for ${\bf v}_{\text{trans}}$ as a function of the system's linear response. The linear response and consequently ${\bf v}_{\text{trans}}$ are influenced by the skyrmion's internal modes and scattering states, as well as by the ferromagnetic background's Kittel mode. The direction and speed of ${\bf v}_{\text{trans}}$ can be controlled by changing the polarisation, frequency and phase of the driving field ${\bf B}_1(t)$. For systems with small Gilbert damping parameter $\alpha$, we identify two distinct physical mechanisms used by the skyrmion to move. At low driving frequencies, the skyrmion's motion is driven by friction, and $v_{\text{trans}}\sim\alpha$, whereas at higher frequencies above the ferromagnetic gap the skyrmion moves by magnon emission and $v_{\text{trans}}$ becomes independent of $\alpha$.
List of changes
List of changes
1. Provided more justification for perturbative ansatz in Eq.(8), Eq.(9).
2. Added discussion of inertia terms & validity of effective Thiele equation to last paragraph of Sec. III.
3. Specified that the free energy F in Eq. (20) and App. E is rescaled by a factor $D^2/J$ with respect to the physical free energy defined in Eq. (1). Also that in the process of obtaining the various free energy terms in App. E we also rescale the length scales by a factor $D/J$
4. Clarified rescaling of external $B$field static and oscillating components above Eq. (5) and in App. E.
5. Last sentence of Sec.II discussing setting $b_0=1$ modified to reflect fact that we are using this for our plots and numerics but actually leaving $b_0$ as a free parameter in all our equations.
6. Text above and below Eq.(21) modified to better explain why we do not list the full nonlinear equation of motion for $a$/$a^*$ in the main text and only need the linearised version, Eq.(22).
7. Modified and added text in the first two paragraphs of Sec IV.2. to better explain the necessity of the $k=0$ mode and the difficulties encountered when calculating it numerically. A few sentences also added after Eq.(39) to justify and explain the given ansatz.
8. Added details about how skyrmion coordinate $\bf R (t)$ is calculated from the numerical data at the end of Sec. V, just after Fig. 5.
9. Corrected typos with numbering of Eqs.(14) and (44) in Sec. V.
10. Added discussion of skyrmionium and skyrmion bubbles in second paragraph of Conclusion.
11. Added references [9], [2326], [30].
12. Small typos corrected and minor changes to wording made throughout the paper.
Published as SciPost Phys. 15, 065 (2023)
Reports on this Submission
Anonymous Report 1 on 2023426 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202303_00034v1, delivered 20230426, doi: 10.21468/SciPost.Report.7107
Report
1. Perturbative expansions often use some guesswork. This seems to be the case also here. The explanations in combination with the success of the approach can be thought of as a justification of the initial Ansatz, i.e., Eq. (9).
2a) The approach used in the paper assumes that the implicit assumptions of the Thiele formalism (e.g., the definition of the skyrmion coordinate, the achievement of a steady state, etc) are consistent and these work so as to "integrate out" fluctuations. This is likely, but I cannot find a justification beyond doubt.
One could mention that, depending on the definition of the skyrmion coordinate, fluctuations can be either introduced or "integrated out". I do not see why the Thiele formalism is the best one for this case. It might well be, but there is the risk that it might give misleading results.
2b) If P in Eq. (C7) is the canonical momentum and this is proportional to the skyrmion position (e.g., P_x proportional to R_y), then one should note that the Thiele approach does not, in general, use the same definition for the skyrmion position.
Thus, I can still not find a complete justification that the calculations in the paper are fully consistent.
One could argue that the different approaches used (Thiele or the one in the Appendix) are likely to give only small differences that are equivalent to small fluctuations of the skyrmion position. It might be worth exploring further whether small fluctuations might be of importance and whether they might affect significantly the particular dynamics studied in the paper.
3) I would not question the final result and the argument that longtime dynamics are correctly given.
But, the approach of a detailed calculation of oscillations (probably introduced by the choice of formalism) still seems to obscure rather than reveal the apparent simplicity of the longtime dynamics.
On "Some other comments"
1) The paper results are valid for some cases of skyrmions (b_0=1 or similar) but this is not certain for other skyrmions (maybe for b0<<1, or b>>1). The paper would only be improved if this would be explained.
The paper presents an interesting phenomenon in a detailed way and it contains a number of interesting results. It also uses various techniques. It will be very useful that these results to be communicated to the community and further discussed. Therefore I recommend accepting the paper for publication.