Basic Aspects of Ferroelectricity Induced by Noncollinear Alignment of Spins

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Reviewer comments

This manuscript presents a detailed theoretical review of ferroelectricity induced by non-collinear spin alignments in magnetic insulators and focuses on symmetrical breaking mechanisms which are responsible for magnetically driven polarization. The author critically reevaluates the Katsura-Nagaosa-Balatsky (KNB) model and extends it within the framework of super-exchange theory and modern polarization theory based on Winner functions. The study offers microscopic insight into the origins of electric polarization, especially in centrosymmetric systems, and discusses symmetry-dependent tensor properties and constraints that govern spin-polarization coupling. However, the following points will increase clarity, readability, and broader impact, and finally, it will be accepted for publication to Condensed Matter:

• Typographical errors are minimal, but a final check is still recommended for readability, such as in line 17: “in the from”à should be “in the form.”
• How can ferroelectricity emerge in non-collinear antiferromagnets, where both time-reversal and inversion symmetry are broken? Are such mechanisms discussed or generalized across other AFM materials?
• The manuscript is rich in theory but lacks experimental studies. The authors might consider incorporating experimental studies (e.g., M. Sheeraz et al. Phys. Rev. Materials 3, 084405 (2019); C. W. Ahn et al., New Phys.: Sae Mulli 71: 991-1003 2021), which theoretically predicted and experimentally demonstrated the anionic effect on inversion symmetry breaking in oxides via sulfurization. These works align well with the review’s discussion on lattice distortion, single-ion contributions, and symmetry-based tuning of polarization, particularly in perovskite-type oxides.
• The authors are encouraged to add recent and relevant studies to strengthen the discussion around generalized symmetry-breaking, such as (doi:10.1016/j.cap.2024.11.011) and (doi.org/10.1103/PhysRevLett.112.017205), which complement the review’s discussion on Berry curvature and symmetry-breaking mechanisms in noncollinear antiferromagnets like Mn3Sn and Mn3Ge.
• If possible, can the authors discuss how this framework could be extended using first-principles simulations or machine learning approaches for multiferroic discovery?

Author Response

I would like to thank the Referee for the positive evaluation of my work and recommendation for publication as a review in MDPI: Condensed Matter. I would also like to thank the Referee for the careful reading of the manuscript and interesting suggestions. My responses are summarized below:

Typographical errors are minimal, but a final check is still recommended for readability, such as in line 17: “in the from” it should be “in the form.”

I would like to thank the Referee for careful reading. I used this opportunity to carefully proofread the text again and correct typographical error. I have also reformulated several sentences to make them more readable. The changes are marked by the blue color in the revised PDF file.

How can ferroelectricity emerge in non-collinear antiferromagnets, where both time-reversal and inversion symmetry are broken? Are such mechanisms discussed or generalized across other AFM materials?

If the inversion symmetry is broken by lattice distortions, we deal with the so-called Type-I multiferroics, where the ferroelectricity and magnetism coexist as two independent phenomena and often emerge in two different sublattices: the inversion symmetry breaking occurs in one sublattice, while the time-reversal symmetry breaking, resulting in an antiferromagnetism, occurs in another sublattice. The typical example is BiFeO₃. These materials are typically characterized by high ferromagnetic Curie temperature and magnetic Néel temperature, but weak magnetoelectric coupling.  Such materials are well studied and discussed in many review articles: see, for example, Refs. [4,6,8,9]. I simply do not know whether I can add anything to this knowledge (as I clearly sate in the Introduction). In my view, the scenario when the inversion symmetry is broken by magnetic degrees of freedom, which I choose for this review, is much more unusual and, therefore, more interesting. Moreover, there are still many controversies and confusions existing in this field. Another possibility is when the noncollinear spin-spiral texture is induced Dzyaloshinskii-Moriya interactions in materials with broken inversion symmetry and this spin-spiral texture induces the polarization via generalized KNB mechanism. Such situation is realized in Ba₂CuGe₂O₇, as discussed in Section 10.6 and Ref. [24]. Finally, even if the spatial inversion remains to be one of the symmetry operations, it can be broken in individual bonds, resulting in several interesting possibilities summarized in Table 1.

The manuscript is rich in theory but lacks experimental studies. The authors might consider incorporating experimental studies (e.g., M. Sheeraz et al. Phys. Rev. Materials 3, 084405 (2019); C. W. Ahn et al., New Phys.: Sae Mulli 71: 991-1003 2021), which theoretically predicted and experimentally demonstrated the anionic effect on inversion symmetry breaking in oxides via sulfurization. These works align well with the review’s discussion on lattice distortion, single-ion contributions, and symmetry-based tuning of polarization, particularly in perovskite-type oxides.

These are certainly interesting studies (which will certainly motivate me to study Korean language to read the second publication). However, I am afraid they are too far from topic of this review. The authors basically study nonmagnetic d⁰ ferroelectrics. The problem is briefly mentioned in Introduction, as one of possible mechanisms of breaking the inversion symmetry in nonmagnetic systems. I do not feel that I need to expand this discussion.

The authors are encouraged to add recent and relevant studies to strengthen the discussion around generalized symmetry-breaking, such as (doi:10.1016/j.cap.2024.11.011) and (doi.org/10.1103/PhysRevLett.112.017205), which complement the review’s discussion on Berry curvature and symmetry-breaking mechanisms in noncollinear antiferromagnets like Mn3Sn and Mn3Ge.

This is certainly an interesting suggestion and very interesting subject. The Referee means the phenomenon of weak ferromagnetism (which in the light of more recent developments is called altermagnetism: https://arxiv.org/abs/2503.23735 ). Nevertheless, the problem is that the magnetoelectricity and weak ferromagnetism are two fundamentally different phenomena, which belong to two different symmetry classes and, therefore, cannot coexist: simply, the magnetoelectricity requires the inversion symmetry to be combined with the time reversal, while for the weak ferromagnetism, the spatial inversion must be alone – see, for example, E. A. Turov, Can the magnetoelectric effect coexist with weak piezomagnetism and ferromagnetism?, Uspekhi Fizicheskikh Nauk 164, 325 (1994) [Physics-Uspekhi 37, 303-310 (1994)]. Therefore, I do not feel that I need to expand the discussion also in this direction (but, again, this is an interesting subject, which could be a good subject of another review). Regarding this point, I have added a new paragraph on page 24 (Section “Summary and outlook”) and new Refs. [68-71]. These changes are indicated by red color.

If possible, can the authors discuss how this framework could be extended using first-principles simulations or machine learning approaches for multiferroic discovery?

Thank you for this suggestion. I see it from a different perspective: there are plenty of first-principles simulations of noncollinear multiferroics. Basically, if there was a chance to realize a noncollinear magnetic order in certain magnetic insulator, it was immediately studied and simulated. For instance, this is how the MnWO₄ and CuO multiferroics have been discovered [55,59]. Nowadays, it is basically a routine procedure to calculate the electric polarization induced by a noncollinear (or any other) magnetic order using modern packages for first-principles calculations. Nevertheless, in my view, what is missing is a clear understanding of basic principles which stand behind many of such studies. These principles are rather simple. This was my main motivation to write this review.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is of review type where besides of comprehensively commenting the work by KNB additional symmetry considerations and constraints are reviewed. The authors analysis of the origin of multiferrroicity is entirely based on the basis that spins are the primary "order parameter" and thus entirely misses the aspect that polarization could be the leading parameter thereby triggering the spin ordering.

In addition, from the viewpoint of the author the aspect of magnetic order is intimately related to the fact that a transition metal is involved in it. The aspect of spin-phonon coupling is omitted whereas this could be very relevant.

However, the title of the article is in line with the articles basic discussions, and I suggest acceptance of it.

Author Response

I would like to thank the Referee for the positive evaluation of my work and recommendation for publication as a review in MDPI: Condensed Matter.