Magnetic structures of the Swedenborgite CaBaFe4O7 derived by powder and single-crystal neutron diffraction

The authors present new powder and single-crystal neutron diffraction data for the Swedenborgite compound CaBaFe$_4$O$_7$. Below $T_{N_1}$ = 274 K the system is found to order in a ferrimagnetic structure with spins along the $c$ axis and an additional antiferromagnetic component within the kagome planes. A further transition into a multi-q conical structure is found at $T_{N_2}$ = 202 K.

From the experimental point of view, this seems to be solid work. However, I have a problem related to the last sentence of the abstract: "The resulting ordering schemes offer valuable insight into the coupling mechanisms which serve as valuable input for further dynamical and theoretical studies of this complex system". Indeed, CaBaFe$_4$O$_7$ seems complex if not to say complicated, and I did not see a crisp message that would attract a big audience. Even worse, the work does not seem to be put properly into context. On the formal side, not only is the reference list very short (15 references), but more than half of these are publications by the present authors. Content-wise, it may be Ok to start with references for the concrete compound, but references for the general highly-frustrated magnetism context are missing, i.e., the first three sentences of the Introduction do make some comments on highly frustrated magnets, but without any references.

More on the level of detail, the magnetic susceptibility reproduced from Ref. [13] in Fig. 3(a) is remarkably anisotropic even in the high-temperature paramagnetic regime. This highlights the importance of single-ion anisotropy. While the latter is emphasized elsewhere in the manuscript, I think it would be useful to also discuss it in the context of Fig. 3(a).

At the end of section 3.1, I was surprised that the authors did not succeed to solve the inverse problem for single crystals while it works for polycrystals. I saw that there is some discussion on this point (related to twinning), but I failed to get the message. Can the authors maybe explain further? Is the single-crystal data maybe in some respects of lower quality than the poly-crystalline one (less of reciprocal space covered or such)?

Finally, I have a problem with the color shading in Fig. 5 since I see no clear signatures of phase transitions beyond a fade-out of certain reflections. Scans such as those shown in Fig. 3(b) are really much clearer such that I think that at least Fig. 3(b) should also be mentioned in the discussion of Fig. 5.

This work needs to be placed properly into context:
1- Is there a crisp message that one could convey also with title and/or abstract?
2- More than half of just 15 references being work by the present authors seems off balance.
3- Cite reviews on highly frustrated magnets and/or related systems (such as kagome ones) in the Introduction.

Further details:
4-Add comments on single-ion anisotropy to the discussion of Fig. 3(a).
5- Improve discussion/clarity of Fig. 5 (e.g., by including Fig. 3(b) in the discussion).
6-Typeset chemical formulas in the titles of Refs. [3-7,9-13] correctly (see, e.g., https://tex.stackexchange.com/questions/10772/bibtex-loses-capitals-when-creating-bbl-file for typesetting hints).

Extensive neutron study of complex magnetic structures in a potentially interesting frustrated magnetic material.

This work falls short of explaining the microscopic origin of magnetic order in CaBaFe4O7. It raises (and does not answer) multiple questions on which interactions and anisotropies are important, and how they can be rationalized with respect to the crystal structure of this material.

The authors report on the magnetic structure determination for CaBaFe4O7. This swedenborgite-type compound is potentially interesting by virtue of its coexisting triangular and kagome planes and a somewhat non-trivial magnetic structure, which has not been resolved to date. That being said, the compound was disclosed back in 2011 by some of the authors [PRB 83, 180405(R)], and the data shown in the present work seem to go back to the same time, because they are actually cited as Ref. 10 of that PRB'2011 manuscript. I appreciate that collecting neutron data on single crystals and powders may be a lengthy job compared to standard lab experiments, but the time span of 10 years is still hard to justify even for a thorough neutron study. This long delay with the publication indicates to me only a marginal interest in this compound both by the authors and by the community alike. Indeed, since 2011 only two more publications have appeared (PRB 93, 014444 (2016) and PRM 2, 054403 (2018)), and surprisingly, none of them is even cited in the present manuscript!

This work is of excellent technical quality and certainly merits publication. On the other hand, it falls short of making a compelling case for SciPost Physics with regard to the acceptance criteria of the journal. The authors conclude their manuscript with a fairly scarce outlook where they claim that their results will "serve as valuable input for further dynamical and theoretical studies". However, such a statement could apply to essentially any work on the magnetic structure determination. It was not clear to me in which way the insights provided by this magnetic structure will be useful to understand peculiarities of the magnetoelectric response reported in PRB'2016, or how they fundamentally (on the level of individual magnetic interactions and anisotropies) distinguish this Fe-based system from its Co-based sister compounds. Moreover, the discussion of magnetic couplings in Section 4 seems rather superficial (see below) and does not provide sufficient information that a theoretical study could build upon.

Overall, I believe that not simply a revision but a significant extension of this work (for example, by an ab initio calculation of magnetic interactions and anisotropies) would be required to meet the acceptance criteria of SciPost Physics. On the other hand, the manuscript seems to be in a rather good shape to be published in SciPost Phys. Core or Physical Review B. I would fully support such a publication as soon as the following points are addressed:

1. In my opinion, the main peculiarity of this magnetic structure lies in the coexisting FM and AFM orders for the out-of-plane and in-plane spin components, respectively. Where do the ab-components come from? Why do they even appear if the spins tend to point along 'c'? The crystal structure is a derivative of the trigonal one, so I would expect a leading anisotropy term that chooses either an in-plane or the out-of-plane spin direction. Why does it not happen? How does the trigonal-orthorhombic distortion affect the crystal structure and the local environment of the Fe atoms? Is the Jahn-Teller effect at play? I understand that the data at hand may not be sufficient to address these questions, but the authors could at least formulate them and lay down explicit directions for further research.

2. Likewise, I could not understand where the difference between the T1 and T2 triangles comes from. Why do the T1 triangles show the 120-deg order for the ab-component of the spin, whereas the T2 triangles do not? Is it due to a difference between the triangles themselves, or due to the fact that T1's are capped with the Fe1 sites from the triangular layers? Which feature of the crystal structure is mainly responsible for the magnetic structure observed in the present experiments? Is it a deformation of the kagome planes, or the mutual stacking of the kagome planes and triangular layers?

3. In Section 4, the authors also argue that the "weaker Dzyaloshinskii-Moriya interaction... certainly plays a role". I could not understand the reasoning behind this statement. One does not need the DM interaction to form the 120-deg order. The DM term will only choose the chirality of this order. However, it was not clear from the manuscript whether the symmetry of the DM terms on the T1 triangles is in agreement with the experimental magnetic chirality. Is it possible that the interlayer interactions (to Fe1) stabilize same chirality of the T1 triangles above and below Fe1, and the DM term is not even needed?

4. Please, explain how you treated magnetic reflections from the different twin domains in the 220 K refinement. Was the twin model from the RT refinement used with the fixed populations of the twin domains?

5. Please, give a proper credit to the PRB'2016 and PRM'2018 publications. The latter reports the trigonal high-T crystal structure, which you refer to as "presumably hexagonal" (Sec. 3.1.1).

6. Please, label the T1 and T2 triangles in one of the figures.

7. Figures 4b and 7b would be easier to understand if only the ab-spin components were shown.

8. It would be helpful if mcif-files with the refined magnetic structures were included in the eventual publication. Readers will benefit from the direct access to the magnetic structures without the need to decipher the irreps.

9. In the Introduction, I stumbled upon the statement "Due to the Heisenberg-like nature of the involved spins" that seems very misleading in the light of the magnetization data shown in Fig. 3a. The huge uniaxial anisotropy displayed by these data (and also by the field-dependent measurements in Fig. 2 of PRB'2016) reflects strong deviations from the Heisenberg model. I suggest that the statement in the Introduction should be revised accordingly.

10. I was wondering why none of the data sets have been referenced in the manuscript. Is it in line with the ILL data policy?