Traveling skyrmions in chiral antiferromagnets

1. The paper contains new original results which help to understand the dynamic properties of antiferromagnetic skyrmions.

1. It would be nice to have a comparison of numerical calculation with spin lattice simulations.
2. Almost all results are obtained by means of numerical calculations without any analytical predictions.

The problem of magnetic skyrmion dynamics is interesting and well worth discussing in leading journals.

In the current MS authors study by means of numerical calculations the dynamics of antiferromagnetic (AFM) skyrmions. The authors show that AFM skyrmion behaves as a particle and it has relativistic properties. Authors also show that AFM skyrmions can move without any external driving. These results are interesting and deserve to be published.

I suggest the publication with minor revision.

1. In the Sec. II.A after Eq. (2) it is mentioned that authors consider material $K_{2}V_{4}O_{8}$ from Ref. [14]. However in Ref. [14] it is considered material $K_{2}V_{3}O_{8}$. Please fix it. Additionally, the AFM skyrmion in such material were studied in [A] earlier than in Ref. [14].

2. $K_{2}V_{3}O_{8}$ is predicted to have the weak ferromagnetism [B]. Fig. 3 shows that the traveling skyrmion has non-zero magnetization $\vec{m}$. Does AFM skyrmion have its non zero magnetization $\vec{m}$ in statics in your calculations?

3. It would be nice to have the dependence $v_c(\lambda)$ in Fig.4.

4. In the second paragraph of Sec. I please fix "LandauLifshitz"$\to$"Landau-Lifshitz".

5. Fig.5 discussed in the Sec. II.B earlier then Fig. 4. Please fix it.

[A] A. N. Bogdanov, U. K. R{\"o}{\ss}ler, M. Wolf, and K.-H. M{\"u}ller, PRB 66, 214410 (2002)
[B] M. D. Lumsden, B. C. Sales, D. Mandrus, S. E. Nagler, and J. R. Thompson, PRL 86, 159 (2001)

1. The paper contains a number of results important for the understanding of the dynamics of antiferromagnetic (AFM) skyrmion.
2. The paper is well written and is easy to read.

1. Most results are obtained by means of numerical calculations. It would be nice to get some analytical estimations, e.g. dependence of skyrmion mass on velocity.
2. Previous works on AFM skyrmions are not properly cited.

This is a timely and well-written paper which provides a number of results necessary for the proper understanding of the antiferromagnetic AFM skyrmion dynamics: (i) AFM skyrmion demonstrates the inertial motion (without external driving), (ii) the moving AFM skyrmion gains elliptical deformation and (iii) it has properties of a relativistic particle: there exists the maximal velocity and skyrmion mass diverges when reaching the critical velocity.

I suggest the publication with minor revision.

1. The statement about "the vast difference between the dynamical sectors of the $\sigma$ model and the Landau-Lifshitz equation" made at the end of Section II is at least unclear. The main dynamical equation (9) of the proposed "$\sigma$ model" can be directly obtained from the set of two Landau-Lifshitz equations written for each of the sublattices in the so-called exchange approximation. It was demonstrated (for the first time) in [I], well explained in [II, III] and used for AFM skyrmion description in [IV, V].

2. AFM skyrmions in the material K$_2$V$_3$O$_8$ were studied in [IV] three years earlier than in [14]. Thus, the paper [IV] should be cited at least together with [14].

3. Elliptical deformation of the moving AFM skyrmion was first predicted in [VI]. This paper must be cited.

4. The expression (11) for the magnetization is misleading. It looks like the magnetization vanishes in the continuos limit when $\epsilon \to 0$. On the other hand, the expression for $\mathbf{m}$ in (11) is the so-called dynamical magnetization which generally does not vanish [I, II]. The point is that $\epsilon$ appears in the magnetization (11) because of the time renormalization (A2). Thus, the dynamical magnetization is finite what is consistent with the previous findings. I believe it should be explained after formula (11).

5. Fig. 5 is discussed in the text earlier than Fig. 4 (page 5).

6. The large-radius AFM skyrmion was recently considered [VII] within the model of the circularly closed domain wall. Thus [VII] should be cited together with [19,20] on page 5.

[I] I. Bar’yakhtar and B. Ivanov, Sov. J. Low Temp. Phys. 5, 361 (1979).
[II] O. Gomonay, V. Loktev, Phys. Rev. B, 81, 144427 (2010).
[III] E. Tveten, A. Qaiumzadeh, O. Tretiakov, et al., Phys. Rev. Lett., 110, 127208 (2013).
[IV] A. Bogdanov, U. Rößler, M. Wolf, et al., Phys. Rev. B, 66, 214410 (2002).
[V] H. Velkov, O. Gomonay, M. Beens, et al., New J. Phys. 18, 075016 (2016).
[VI] C. Jin, C. Song, J. Wang, et al., APL 109, 182404 (2016).
[VII] V. Kravchuk, O. Gomonay, D. Sheka, et al., Phys. Rev. B 99, 184429 (2019).