An Ising model on a 3D honeycomb zigzag-ladder lattice: a solution to the ground-state problem and application to the SrRE$_2$O$_4$ and BaRE$_2$O$_4$ magnets

The paper by Dublenych and Petrenko reports on a theoretical study of an Ising model aimed to explain experimental properties of Sr(Ba)Re2O4 magnetic materials with so-called honeycomb zigzag-ladder lattice.The member compounds exhibit a variety of unusual properties including magnetization jumps, multiple phases, partial long-range order etc. To my knowledge there was little if any effort to understand these behaviors at the microscopic level. Thus the manuscript addresses a potentially interesting problem and may be relevant to SciPost. However, I find that the paper is poorly written to the extent it's hard to judge its scientific merit and relevance to the family of honeycomb zigzag-ladder materials.

Here is the list of specific remarks and comments.
1) the formulated Ising model is not justified in relation to specific SrRe2O4 or
BaRe2O4 compounds. For example, Er, Dy, Tb and Ho ions may have quite different properties, Kramers vs non-Kramers, Ising vs XY. Furthermore, there are two positions for the rare-earth ions. What are the differences -- known or/and supposed, do they have different local environments, g-factors, crystal-field level structures the reader is left in dark about all that. What is the physical meaning of the effective fields h1 and h2. Are these different components of an applied magnetic field or it's the same component multiplied by the different g-factors?

2) Figure 1 showing the lattice structure is not sufficiently clear. It's just an adopted poor quality version of the Figure from Ref.[8]. Authors can show several projections along different directions and have more discussion of the choice of exchange parameters.

3) The most important comment is that the authors give, in my view, an unacceptable presentation of their theoretical approach. There are only vague and short explanations in the end of Sec.2.1, which are compensated with very long and perhaps not so necessary pictures for all configurations in the supplemental material. For details of their approach the authors refer to the previous works of one of them. This is not sufficient. Furthermore, in his early works PRE 2011, Dublenych cites the article by Kaburagi and Kanamori, JPSJ 44, 718 (1978) as the origin of the used theoretical method. The authors have to cite this work in their paper, explain it and contrast with their own approach.

To summarize, the manuscript is not worth of publishing in the present form. Still the authors have chances to improve it by making substantial revisions responding to the above remarks.

1- Detailed ground-state investigation.
2- Lots of nice figures.

1- Lengthy but nevertheless incomplete.
2- Oversimplified model for the compounds mentioned in the title.

An incomplete exact study is performed of the ground-state problem for an Ising model on zigzag-ladders arranged in a 3D honeycomb geometry. 5 coupling constants and 2 independent magnetic fields are considered, resulting in a correspondingly rich (again incomplete) 7-dimensional phase diagram.

I have two major concerns.

The first one is the motivation drawn from the SrRE$_2$O$_4$ and BaE$_2$O$_4$ family of compounds. These are complicated compounds resulting in the corresponding complex 7-parameter model. Given that one can get an infinite number of phases already in a relatively simple one-dimensional Ising model [see, e.g., Per Bak and R. Bruinsma, Phys. Rev. Lett. 49, 249 (1982)], it is not a surprise that a complicated 3D model gives rise to a complex phase diagram. More importantly, however, none of the SrRE$_2$O$_4$ and BaE$_2$O$_4$ compounds is probably accurately described by the model (1). Either the degrees of freedom are actually not $\pm 1$ Ising degrees of freedom, or if they are, the local axes on the "red" and "blue" sites are actually not parallel, but rather orthogonal (as stated, e.g., at the end of the second paragraph of the Introduction). Thus, the physically relevant model should include at least a certain amount of quantum fluctuations that might either smear out certain features, or stabilise ground states by a quantum order-by-disorder mechanism. By contrast, the purely classical Ising model investigated in the present work is probably not appropriate for any of the SrRE$_2$O$_4$ or BaE$_2$O$_4$ compounds.

Secondly, this manuscript is long, but still not self-contained. For example, the triangular plaquettes as building blocks are well explained, but I got the impression that the "basic rays" are not completely explained even though they are a central tool of the present investigation, nor is the "${\mathbf r}$" notation properly introduced.

There is obviously an incredible amount of work in the many nice figures of the present manuscripts. Nevertheless, the authors should not try to publish their notebooks, at least not as a journal article. There is probably a core of interesting material in the present manuscript, but it would have to be presented in a more concise and accessible manner. Furthermore, SciPost Physics Core may be a more appropriate venue even for such a rewritten manuscript.

1- Shorten the core manuscript such as to focus on the essential findings.
2- Discuss shortcomings of the model (1) as applied to the SrRE$_2$O$_4$ and BaE$_2$O$_4$ compounds. Removal of emphasis on these compounds (e.g., from the title) and/or focus on results that are really expected to be relevant to these compounds would also be appropriate.
3- Many details are difficult to grasp by the reader. This is particularly true for the "SUPPLEMENT" (Ref. [29]) that has a long list of tables without any explanatory text. Note also that SciPost itself does not publish Supplementary Material, but that authors should make any such material available themselves, e.g., via institutional repositories, see https://scipost.org/submissions/author_guidelines . Complementary, but non-essential information can be provided in this form.
4- Add proper explanation of basic rays.
5- The "(incomplete)" after "exact solution" at the beginning of the abstract is honest, but not very convincing. The authors might try to present this differently.
6- Fix incorrect cross-references such as to "Section H" on page 3, Figs. 9-13 instead of 9-14 (?) in the first paragraph of section 2.7 on page 14 and "Section II, Subsection F" at the beginning of the SUPPLEMENT.

Reviewer report on the manuscript scipost_202107_00007v1: "Ising model on a 3D honeycomb zigzag-ladder lattice: a solution to the ground-state problem and application to the SrRE2O4 and BaRE2O4 magnets" by Yu. I. Dublenych and O. A. Petrenko

The submitted paper brings an incomplete exact solution for a ground state of the spin-1/2 Ising model on a 3D honeycomb zigzag-ladder lattice, which is treated by a method of basic rays and sets of cluster configurations. The investigated model is inspired by rare-earth magnetic compounds SrRE2O4 and BaRE2O4, which are definitely worthwhile to study. However, the manuscript has following serious insufficiencies and drawbacks:

1. The solution for a ground-state problem of the spin-1/2 Ising model on a 3D honeycomb zigzag-ladder lattice is incomplete and hence, one cannot exclude possibility of overlooking some additional ground state(s) and overall incompleteness of the established ground-state phase diagrams.
2. The used method has been developed by one of the present authors more than 10 years ago and because of its obscurity it did not find any followers yet (at least to the best of my knowledge). From this perspective, it seems to be of marginal interest for potential readers from the methodological point of view.
3. The investigated model was developed for rare-earth magnets SrRE2O4 and BaRE2O4, however, the paper contains only a little new information about these compounds. The authors merely predict character of the ground states, but their stability (field range) is not comprehensively compared with available experimental data. The true nature of ground state is also questionable.
4. The appropriateness of the suggested model for rare-earth magnets SrRE2O4 and BaRE2O4 was not justified by any density-functional theory (DFT), which would provide insight into all underlying coupling constants, crystal-field parameters as well as local anisotropy axes. The studied model thus represents just a very crude and oversimplified description of physical reality. The high total angular momentum and strong crystal-field anisotropy are not sufficient for the Ising-model description.
5. Two crystallographically inequivalent positions of rare-earth ions in SrRE2O4 and BaRE2O4 imply two inequivalent local anisotropy axes, which means that the application of the magnetic field along one of them will have longitudinal as well as transverse projection with respect to the second one. This fact is not accounted for within the present model at all.
6. An infinite degeneracy of some ground states does not immediately imply their disordered character, see for instance the book by R. Liebmann, Statistical Mechanics Of Periodic Frustrated Ising Systems. For instance, the ground states with "one-dimensional disorder" often exhibit a spontaneous long-range order with nonzero critical temperature, so the "disordered" acronym is not appropriate.
7. The results presented in this work are not truly exact and they should verified by some independent numerical method (e.g. Monte Carlo simulations).
8. Some statements are too oversimplified such as:

"However, among frustrated magnets, there are many compounds with large-moment magnetic atoms. These magnets can be well described with classical Heisenberg spin models." - it is not true, even the magnetic compounds involving spin-7/2 Gd3+ ions cannot be satisfactorily described by classical (vector) spins.

"If, in addition, there are easy axes of magnetization, then Ising-type models could be applied." - it is not true, the crystal-field effects are usually deciding whether or not the Ising description is applicable.

In summary, the manuscript brings incomplete solution for a ground-state problem of the spin-1/2 Ising model on a 3D honeycomb zigzag-ladder lattice, whereas the studied model represents just an oversimplified description of rare-earth magnets SrRE2O4 and BaRE2O4. In my opinion, the present manuscript does not meet general acceptance criteria of a highly reputable journal such as Scipost Physics and I cannot recommend its publication.