3D Magnetic Textures with Mixed Topology: Unlocking the Tunable Hopf Index

1) The paper is technically detailed. The analytical derivations and numerical simulations are carefully executed and internally consistent.

2) The flux-tube and linking-number construction is intuitive and well explained.

3) The manuscript is well structured and clearly written. The figures are of high quality.

4) The reference list is relevant.

1) The manuscript exhibits significant conceptual ambiguity regarding the nature of the quantity under discussion. The “generalized Hopf index” is not a topological invariant outside the compactifiable regime, yet it is consistently framed using topological language. The use of the term “generalized Hopf index” for a continuously tunable quantity risks confusion with the genuine topological Hopf invariant, and the same concern applies to the notion of “mixed topology.” The results presented describe smooth geometric interpolation between configurations rather than distinct or coexisting topological sectors.

2) The manuscript explicitly demonstrates that the proposed indices vary continuously with the external magnetic field. This behavior directly contradicts the defining property of topological charges, which must remain invariant under smooth deformations of the field.

3) The manuscript appears to offer limited conceptual and methodological novelty. The analysis primarily applies well-established equations and formalisms to magnetic configurations that have not been previously explored in detail, but without introducing substantially new methods or conceptual frameworks. While this point alone is not a major problem of this work, it becomes relevant in light of the broader conceptual issues mentioned above. If these issues are addressed satisfactorily, the manuscript could still merit publication.

The central claim and primary novelty of the manuscript is that the Hopf index, which is understood as a topological invariant, can take arbitrary real values. The authors refer to this quantity as a “tunable Hopf index” or “generalized Hopf index.” This claim is fundamentally incompatible with the mathematical framework of homotopy theory (the field from which this term comes). In algebraic topology, the homotopy group $\pi_n(X)$ of a topological space $X$ is defined as the set of homotopy equivalence classes of continuous maps from the $n$-dimensional sphere $\mathbb{S}^n$ into $X$, with a fixed base point, $m_0$. Two maps are considered equivalent if one can be continuously deformed into the other while keeping the base point fixed throughout the deformation. The group structure on $\pi_n(X) \equiv \pi_n(X, m_0)$ arises from a concatenation operation, defined by decomposing the domain sphere $\mathbb{S}^n$ into two hemispheres and using one map on each hemisphere. This construction induces a well-defined binary operation on homotopy classes, independent of the particular representatives chosen. For $n \ge 2$, this group operation is abelian. More importantly, the elements of $\pi_n(X)$ label discrete equivalence classes of maps rather than continuous families. The group structure reflects how these classes combine under topologically admissible deformations and does not admit any continuous dependence on a parameter. As a consequence, homotopy groups encountered in practice are discrete algebraic objects: finite groups in many cases (e.g., $\pi_1$ of compact manifolds), countable groups such as $\pi_1(S^1) = \mathbb{Z}$, $\pi_3(S^2) = \mathbb{Z}$, or finite cyclic groups such as $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2$. In all such cases, group elements form isolated equivalence classes rather than a continuum. By contrast, the additive group of real numbers $(\mathbb{R}, +)$ is uncountable, divisible, and admits continuous interpolation between its elements. It therefore cannot be isomorphic to any homotopy group. The existence of a homotopy group taking values in $(\mathbb{R})$ would contradict the foundational principles of homotopy theory, according to which topological invariants necessarily take values in discrete groups and remain constant under continuous deformations. It follows that any invariant which varies continuously and assumes arbitrary real values cannot arise from a homotopy classification and should not be interpreted as a topological invariant in the homotopy-theoretic sense. Standard references supporting these statements include: - E. H. Spanier, Algebraic Topology, Springer (1966). - A. Hatcher, Algebraic Topology, Cambridge University Press (2002). - G. E. Volovik, The Universe in a Helium Droplet, Oxford University Press (2003).

I also wish to comment on a statement made in the introduction concerning the assignment of a fractional topological charge to merons. This claim is not justified from the perspective of topology. The appropriate classification of two-dimensional magnetic textures is given by the absolute homotopy group $\pi_2(S^2)=\mathbb{Z}$, or, in the presence of easy-plane anisotropy, by the relative homotopy group $\pi_2(S^2, S^2 \setminus {P_1, P_2})=\mathbb{Z}\times\mathbb{Z}$, as in Ref. [44]. In both cases, the homotopy classification admits only integer-valued topological charges. Any appeal to “immense practicality,” “physical meaning,” or “intuitive quantification” cannot alter this mathematical fact. Quantities associated with magnetic textures (such as contributions to the gyrovector or to so-called emergent magnetic fields) may take fractional values and may be physically meaningful (this itself would require explicit justification), but they are not topological invariants. Consequently, they must not be referred to as topological charges, topological indices, or related terminology.

1) The term ‘Hopf index’ is conventionally understood as a topological invariant, i.e., a quantity that remains unchanged under continuous deformations of the field. Since the quantity computed in the present manuscript does not, in general, possess this property, it should not be identified with or associated with topological invariants, in particular, not with the Hopf index. The terminology should be revised accordingly throughout the entire manuscript.

2) For the same reason, references to topology should be substantially limited. This includes, but is not limited to, the title, abstract, and section headings, which currently suggest a topological interpretation that is not generally justified. Topological terminology should be used only in regimes where a genuine homotopy-based invariant can be rigorously defined.

3) As the authors themselves acknowledge, the Whitehead formula is well defined only when the far-field magnetization is fixed, allowing for compactification of the physical space. The manuscript should consistently respect this limitation and clearly distinguish results obtained within this compactifiable regime from those that fall outside it, where no topological Hopf invariant is well defined.

4) Reference [22] presents a homotopy-group-based analysis and derives an invariant composed of two integer-valued indices, computed using the same Skyrmion charge and Hopf index integrals as employed in the present manuscript. However, because the magnetic texture is explicitly compactified after energy minimization, the resulting indices in Ref. [22] are integer-valued, most likely up to numerical precision. The authors should clarify the relationship between the approach presented in this work and that in Ref. [22]. If appropriate, the authors should apply a similar compactification approach to their configurations and, only after that, compute the skyrmion index and Hopf index for comparison.

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Investigation of 3D magnetic topological structures is an actively developing area in both numerical and experimental micromagnetism. Therefore, the authors' geometrically clear interpretation of the topological Hopf index is very timely. In addition, the authors attempted to generalize the Hopf index to structures that are periodic in one direction. Although no new physical effects are reported in the paper, the authors propose a new way to compute the Hopf index that can also be used for periodic structures with a nonuniform background. This paper advances the understanding of topological magnetic textures, so I recommend it for publication. But first, several clarifications should be provided in the text.
1. The new approach is based on the concept of “flux tubes”. However, a formal definition of a flux tube is absent in the text. It should be provided.
2. In order to use formula (1), one needs first to decompose the 3D magnetization texture into flux tubes. Is there any algorithm for how to do it in a general case? How can one identify all the flux tubes necessary to compute the Hopf index?
3. Explaining examples on Figs. 3 and 4, the authors define the flux tubes as isosurfaces m_z = m^b_z. In doing so, they implicitly made two important assumptions:
(i) The isosurfaces m_z = m^b_z have the form of tubes.
(ii) Any two field lines within a tube defined in this way must have the same self-linking number.
Neither of the statements (i) nor (ii) is obvious for the general case. If they are proven somewhere in the literature, the corresponding discussion supplemented with the references must be present in the text. If they are assumptions, this should be clearly stated.
4. What is the condition for the independence of the real-valued H on the continuum deformation of the magnetization field? Is it the frozen background, or only one background component m^b_z must be frozen?

There are also a couple of minor issues:
1. There is a typo on page 1. It is written R^3 -> S^3. It should be R^3 -> S^2.
2. The loops shown in Fig 2.b have the linking number equal to 1 for the infinitely long vertical central line only. The requirement of infinite length of one of the lines should be specified. The analogous situation takes place for Fig. 2c,d.

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