Dynamics of $\mathrm{CP}^{N-1}$ skyrmions

The paper treats $CP^{N-1}$ generically with a general Hamiltonian, which is quite elegant, and as explained in the introduction, it covers several physically interesting cases: $N=2,3,4$.

Only the BPS equation is used to find solutions and it is used incompatibly with a potential in the case of the "ferromagnetic" solutions, see the report.

This paper is concerned with the dynamics of $CP^{N-1}$ skyrmions by extending the known Landau-Lifshitz (LL) or Landau-Lifshitz-Gilbert (LLG) equation to their generalizations for the $CP^{N-1}$ skyrmions (called gLL and gLLG equations in the paper).
The continuity equation, here derived as the divergence of the topological current, is satisfied for vanishing Gilbert damping.
It is also rewritten as a fractonic conservation law.
The authors find a modified continuity equation which holds also with nonvanishing Gilbert damping, but the dipole moments are not conserved.
A general solution of the $CP^2$ case is found using the BPS equation (4.4), which has two moduli, which are two complex 3-vectors ($u$ and $v$) with an orthogonality constraint as well as a size modulus, so 12 real moduli in total.
In order to fix the moduli, a potential (4.8) is introduced, which is a sum of squares of three of the fields: $n^5$, $-n^7$ and $n^2$, so the potential is
\[\kappa[(n^5)^2 + (n^7)^2 + (n^2)^2].\]
Two different minimizations of the potential are found according to
the sign of $\kappa$.
The author coin the positive $\kappa>0$ case "quadrupolar $CP^2$" and the negative $\kappa<0$ case "ferromagnetic $CP^2$".
Some speculations of the $CP^2$ skyrmions exhibiting the topological Hall effect are made.

There are some issues with this paper that the authors need to address, see the requested changes below.

1) The $CP^{N-1}$ and in particular $CP^2$ skyrmion solutions are found using the BPS equation (4.4), which is derived in [15] for the case without a potential. In principle, such solution cannot be used after a potential is introduced, although in some limiting cases, one could study infinitesimally small potentials, but one would need to be careful.

In the first case ($\kappa>0$), the potential is minimized and vanishes after minimization, so the Bogomol'nyi completion in (4.1)-(4.2) is still valid: this "quadrupolar" solution is merely a subset of the full BPS solution (a subset of the moduli space of solutions).

In the second case ($\kappa<0$), the potential does not vanish. Therefore, using first the BPS equation and then energy minimization on only the potential term, does not guarantee that the minimum of the total energy function is obtained. There may well be a non-BPS solution that has even lower energy, than the BPS solution the authors have found. The non-BPS solution may also lift some or even all of the moduli (called $\alpha$, $\beta$, $\gamma$ and $\varphi$ on page 4). Or there may exist a different Bogomol'nyi completion, which will yield the correct BPS equation for the case at hand.

A correct solution should be found or some convincing arguments should explain why the solution obtained is indeed a minimizer of the total energy functional.

2) The specific Hamiltonian is not introduced clearly in the paper, but from the (4.2) is seems that only the kinetic term (Dirichlet energy) is used. This is a problem, since the solitons are not stabilized in size, like magnetic skyrmions are (that are stabilized by a DMI term counteracting a Zeeman energy). Indeed, the solution in (4.5) contains a size modulus: these solutions are hence sigma-model lumps and not skyrmions (although they have the exact same topology).

It must be clarified, why these scale-invariant solutions are of any interest to condensed matter systems; probably the answer is that they are not. It would be better to introduce a generalized DMI term, like that of [11] as well as a potential. Such terms may change some of the physics or conclusions drawn in this paper.

3) The authors have probably not made mistakes, but it is difficult to check the signs in the equations, since the authors are too sloppy with raised and lowered Minkowski indices: $\mu$, $\nu$, ... where $\mu=0,1,2$. The affected equations are: (2.5), (2.7), $\partial_\mu J_\mu=0$ above (3.1) which leads to (3.1) and is crucial for the work, (B1), (B3), (B4). If no dynamics and only Euclidean space was used, the double lowered spacetime indices would be acceptable, but the paper is about dynamics. Indeed the current conservation in Minkowski space reads

4) The formal derivation of the Landau-Lifshitz equation in appendix A needs more detail. What are the assumptions of the auxiliary $u$-dependence of the fields? (no dependence would make (A1) vanish). Performing the variation on (A1) I get:

which, however, does not straightforwardly integrate to the expression

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