Manipulation and creation of domain-wall skyrmions in chiral magnets

The numerical simulations explore a range of initial conditions and parameters, offering some insight into the phenomenology of skyrmion capture or annihilation.

The theoretical framework lacks clarity:

The physical validity of the analytical approximations is questionable, especially in regimes where the skyrmion and domain wall are in close proximity (if not, it needs to be clarified in the text.)

Much of the presented content overlaps with the authors' earlier work. This overlap may not be formally an overlap, but the analogues are not regarded as really new.

The writing, particularly in the introduction, is unclear, as is the construction of the logic of the formulation.

The work is phenomenological in both analytics and simulations, while their bounds and applicability are not clearly presented.

The paper shows how the domain walls and skyrmion, when combined in space, relate to each other by analyzing the equation of motion from various initial configurations.
Unfortunately, I find that the manuscript does not meet the standards of SciPost Physics or comparable journals for the following reasons.
The results may contain some new aspects like demonstrating the final states they obtain or how the energy density related to force field distributes.

However, the overall approach appears to add little to the topic extensively studied in the literature using the Landau-Lifshitz-Gilbert (LLG) equations. The phenomena involving skyrmions, impurities, and boundaries have long been explored numerically and analytically in the context of spintronics using LLG. One of the major papers are J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature Commun. 4, 1463 (2013).
The present treatment tries to rely on analytical solvability and may drop off many essential parts (if not, please clarify) that may harm the details when the two objects are not really far apart. In that aspect, I feel the ground seems fragile and find it difficult to judge to what extent the treatment is trustful or realistic.

The importance of understanding the capture of skyrmions is not so clear.
This question might become more important if the results were directly connected to experimental observations through simulations that have a one-to-one correspondence with laboratory setups.
In that case, the numerical calculations and the analytical expressions for the force could provide useful insights into experimentally observed phenomena. However, such a connection is lacking.
The authors have previously published several related studies, including, for example, Fig. 3 in Phys. Rev. B 109, 104426 (2024), which already depicts a similar scenario of a skyrmion being trapped by a domain wall. As such, the current manuscript appears more as a supplementary extension of these earlier works rather than presenting a standalone contribution.

The introduction part is particularly difficult to read. Although there is no strict rule in how to write, I think that this way of writing is not commonly accepted.
From the first three or four sentences from the top, I find some difficulties like:
"Magnetic skyrmions are extended particle-like solutions" I do not get this.
".. for a review see [5]." This kind of expression is good for lecture notes but not as formal as scientific sentences.
agnetic skyrmions were
"Chiral magnets contain also other extended solutions that are string-like, called domain walls (DW)/lines that separate magnetic domains."

The equations of motion for the domain wall and the skyrmion appear to be treated separately in Eqs. (11)-(13). However, Eq. (8), which defines the force between the two objects, involves the quantity E^{DW+sk}, whose explicit form is not provided in the text, nor at least, it is not clearly defined. When the domain wall and skyrmion are brought into close proximity, as illustrated in the figures, the classical field "u" should develop a complex, nontrivial texture. This complexity ought to be reflected in the expression for E^{DW+sk}, since the interaction energy between the two textures is central to the paper's claims.
Unfortunately, the authors do not offer a clear explanation of how this energy is calculated or approximated.
It appears they assume a superposition of the two individual textures and attempt to neglect the contributions from their overlap or boundary effects. However, this key assumption is neither properly justified nor supported by a clear derivation.
As a result, the logic remains ambiguous, and the analytical treatment, including Eq. (17), becomes essentially intractable.

Reject

1- the interplay between skyrmions and domain walls is an active field of research.
2- the paper provides both analytical and numerical results.

1- relatively limited advances compared to other works.
2- not always easy to follow what is a new result of this manuscript, and what is not.
3- discussion of the results is a bit light.

This is a theory paper on the interaction between domain walls and skyrmions. I am sorry to say that I do not think this paper meets SciPost acceptance criteria for reasons given below. After these comments have been addressed, this paper could possibly deserve publication in a more specialized journal.

1- The authors have actively worked on various aspects of the interplay between skyrmions and domain walls over the past couple of years. As a consequence, this paper sometimes reads as an incremental continuation of previous papers. For example, there are a lot of relevant references given in the introduction (section 1), but only very few in the rest of the paper. It is difficult to know which results are from previous papers and which ones are not, and which results require to check the literature in order to derive them.
For example I suspect that equations (6) and (7) have been derived previously in some forms, but where ? And I may have missed it, but what is the variable \Lambda in equation (9) ? See also comment below.



2- In the introduction, the authors wrote:
"In this paper, we discover and study the capture of a magnetic skyrmion onto a DW [...]".
This is probably the main analytical result of the paper, obtained in equation (23) and illustrated in figure 2, where they show the phase dependence (via alpha and beta) of the interaction.

However, in the paper by Ross and Nitta PRB 107, 024422 (2023), they seem to have done something very similar, to construct the low-energy effective theory between a domain wall and skyrmion in presence of DM coupling and easy-axis anisotropy. They use the same stereographic coordinates (compare equation (6) of this manuscript with equations (25,26) of Ross and Nitta) and obtain a similar expression for the function f(r) as a function of the modified Bessel function K1(r) modulo a 1/sqrt(r) prefactor (see between equations (19) and (20) of the present manuscript).

In Ross and Nitta 2023, they wrote:
"Then we can appeal to the results of Ref. [26] where the interaction of domain walls and skyrmions was studied and it was found that it is the relative orientation which is important for deciding if they attract or repel."
which is essentially the conclusion of section II.C of this manuscript. Note that Ross and Nitta also worked in the far-field regime.
Ref.[26] in Ross and Nitta 2023 includes four different papers. These papers are not cited in the present manusript and the fact that similar results have been derived in the past are not discussed in the manuscript.



3- The analytics are derived in the far-field regime when the skyrmion and domain wall are far from each other. This opens an issue when the skyrmion and domain wall merge.

The authors explain that analytics and numerics compare very well for |X_0>6|. I am a priori happy to believe it, but a figure to show it would be welcome (only |X_0|=4 and 2 are shown). In particular, if the force is the same for analytics (where alpha and beta are symmetric) and numerics, how can the numerical phase diagrams of figure 3.a and 3.b look different at |X_0|=6 ?

For |X_0>6|, almost the entire phase diagram corresponds to well separated DW and skyrmion (blue regions). It means that the analytics starts to fail precisely in the regime of interest where DW and skyrmions merge.
As a consequence, even though the analytical result of equation (23) is symmetric with respect to alpha and beta, as shown in figure 2.a,b, the numerical results are not (see figure 3.a,b). This is not discussed in the paper. More generally, the disccussion about the numerical results is relatively short.

Reject

- interplay of skyrmions and domain walls is an interesting problem
- aims to give partially an analytic answer to a complicated problem (skyrmion+domain wall)

- The paper does not sufficiently discuss the weaknesses of the various approach
- The numerics builds on minimizing the energy, not on solving LLG equations
- The paper includes numerics, but there is almost no comparison of numerics and analytics
- Central numerical results are not sufficiently discussed.

The paper discusses an interesting problem: the interplay of skyrmions and domain walls. This problem has been studied before and such structures have also been observed experimentally.
The paper has to completely separate parts: the first discusses the forces on the skyrmion using a semi-analytical approach. One approach starts from a a superposition ansatz of the solution of the domain wall and skyrmion (ignoring that they deform each other) and calculates the energy as function of distance and two additional parameters. This is problematic because the superposition rule will not hold, especially when skyrmion and domain wall get closer.
The other approach is more analytical and it was difficult for me to understand fully, but it appears to be an approximation to the first approach, valid in the limit of very large skyrmion-domain wall distance.
A second part searches numerically for local minima in the free energy starting from different initial conditions. The authors find as stable solutions either a domain-wall skyrmion or a skyrmion located at some distance from the domain wall or no skyrmion at all. There is almost no relation of the two parts of the calculation (e.g., the authors could have tracked position and energy as function of their pseudo-time to extract an energy as function of distance and thereby an effective force).

In their discussion of the numerical results, the author identify which of them are surprising but provide zero explanation of their numerical findings. To give a few examples: When placing the skyrmion close to the domain wall with both alpha and beta close to their natural value Pi/2, the authors end up with a state without skyrmion (coded in red in Fig. 3). How did the change of the topologically conserved winding number happen (it should be conserved)? Was the skyrmion squeezed to a zero-radius state and thus vanished (this would most likely depend on the discretization scheme) or did the topological charge somehow move to the edge? Why and how? Also the "fate" of the alpha and beta angles during pseudo-time evolution is not discussed, which is also important to be able to relate the two results.

For their problem, a direct solution of the LLG equation (for sufficiently large damping) would also have been much more physical compared to the used energy-minimizer.

In conclusion, I do not support the publication of the paper due to the problems outlined above. A substantially revised version might be publishable in "physics core" but in my opinion it is unlikely to meet one of the criteria for publication in the scipost physics.

Ask for major revision