Rational Q-systems at Root of Unity I. Closed Chains

1-Tackles interesting problem using novel technqiues
2-Examines the appearance of exclusion of FM-strings in the Q-system approach
3- Works through the proposed approach in a number of simple examples
4-Main result of interest to the community

1-The statment and proof of the key result lack precision
2-The presentation style sometimes lacks clarity
3-Key references are missing, or not cited sufficiently within the text.

This paper examines the Q-system approach to the XXZ chain for q a root of unity. In particular it addresses:

1) How to handle the apparently infinite number of solutions of the Q-system resulting from Fabricus-McCoy (FM) strings.
2) After having modified the Q-system in order to remove the FM strings (and roots at infinity), how to add them back to obtain a complete set of Bethe roots.

Point 1) is achieved, at least in simple examples, by looking at the relationships between the $c_i$ coefficients that occur for FM strings (and also roots at infinity) and eliminating solutions that satisfy these relationships by adding a corresponding additional constraint into the Q-system. This is an effective, if brutal, technique, but it does of course require knowledge of the $c_i$ relations. There does not seem to be a general expression for these relations beyond the simple examples given, which is understandable but a limitation on the general applicability of the method.

Point 2) seems to me to be the key result of the paper, as expressed in the 'claim' given by Equaiton (5.3). The claim is that for a given primitive solution of the Q-system, all descendents associated with this solution are given by finding the zeros of the rational function F. If true, then despite the complexity of finding the zeros of this function, this is an important result. The authors state that the claim is proved in Appendix B, but as it stands there is insufficent detail given in Appendix B for me at least to be able to reconstruct the claimed proof. Several simple examples are given that are consistent with the claim.

Overall, this is an interesting paper that looks at extending the Q-system approach to the subtle root of unity case. It is certainly of potential interest to the community, and I believe could be worthy of ultimate publication in SciePost. However, I have the below reservations (expressed in the Requested Changes section of the report) about both the style and content of the manuscript in its current form.

While many of these points are minor (apart from the missing citations and proof), they and other similar examples add up to make the paper unnecessarily hard to read. If thIs can be satisfactorily addressed then I would recommend publication.

1) Key references are often missing altogther or the citations are not at the appropriate place in the text. For example:
1.1) Section 4.1 - there are no references at all for Q-systems.
1.2) Page 9 paragraph 2 about FM strings and infinity pairs - there should be citations at this point in the text.
1.3) There are no references in the background Section 2.1 apart from [12].
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2) The presentation style is often lacking in enough information or citations to be useful: for example
2.1) 'For special situations [...] the Hilbert space has the same structure as the generic q case.' What special situations and where is this discussed in the literature? More generally, the word special is overused in the text to avoid clear explanation.
2.3) 'For example, the scattering phase between an FM string [...] is trivial'. Why is this and where in the literature is it discussed?
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3) The mathematical presentation is sometimes confusing and at times a little sloppy. For example:
3.1) 'The QQ-relations is defined up to proportionality'. As (4.1) is your definition, this is not the case.
3.2) After the above, Equation (4.2) has a proportionality sign on the rhs. Why not make this equal in order to have a well-defined system solution (as in the existing uncited literature), or is this the proportionality you were talking about?
3.3) Define $t_j^{FM}$ in equation (4.8). I can guess what it should be but I shouldn't have to.
3.4) Beneath (4.20) the argument is very confusing. You define c_i in terms of t_i by (4.20) and then, to paraphrase, you say 'we can eliminate t_j from (4.20'. The only way I could figure out what you meant is by looking at the examples. Also, you intruduce an integer N_c in (4.21) without comment. Do you know what this is on not? Please say either way.
3.5) (4.27): Please comment on what the symbol $\omega$ represents.
3.6) Page 21: The 'Hasse diagram' section doesn't tell me either what a Hasse diagram is or how it relates to the structure of descendents. Again I had to jump forward to example to figure out what you meant. Please either cross reference to the coming explanation, or postpone the discussion until you give details.
3.7) As mentioned above, you need to decide whether your key claim given by Equation (5.3) is a conjecture or theorem. If it is a conjecture backed by examples, say so. If it is a theorem, give precise statment and a clear proof with enough details that a conscientious reader can reconstruct it.

1. Has a breakthrough result in a longstanding open problem.
2. The developed method could be applied in many related topics.
3. It is very nicely and clearly written.

1. As I list in my report, the manuscript could be a little improved.

Integrable spin chains are usually solved by the Bethe ansatz method. The number of solutions of the Bethe ansatz equations are not in one to one correspondence, however, with the eigenvalues of the transfer matrix and the Hamiltonian and it is a difficult case by case investigation to select the correct ones. The rational Q-system, which formulates the QQ equations can overcome this difficulty in the XXX spin chain and also for the XXZ spin chain if q is not a root of unity. The root of unity case is notoriously difficult as Fabricius-McCoy (FM) strings with arbitrary center as well as roots at infinities can appear.

The authors could managed to overcome this difficulty by adapting the Q-system to the root of unity case. The main trick was to eliminate the FM strings and infinite roots by extra constraints in the Q-system and figuring out the way how to add them back, which can be organised in terms of Hesse diagrams. The fusion hierarchy was also crucial in fixing the center of the FM strings. In order to test the method they performed various checks as well as calculated the torus partition function and checked its monodromy invariance.

I think the manuscript is a very nice piece of work, which fills a gap in the literature and will have significant impact on the field that is why I support its publication. I have only a few comments.

The paper is thoroughly written in a pedagogical style, with very clear linear argumentations. Somehow one exception was Example 5.4 for me. I think the logic could be improved a bit there. For instance the author write "In the meantime, we need to consider another primitive state ...". Would it be possible to argue first why do we need this other state at all.

I have also found some typos and some small points.

1. In the $l_2$-dimensional representations of the transfer matrix it is not clear how the representation depends on the complex spin s. Is it related to $l_2$? Or is it related to the highest weight as for the spin s representations?

2. The $v$ and $v'$ notation is not systematic in the paper. In (2.18) did the author mean $v$ or $v'$, what is the definition of $t'$? Also consider (5.1).

3. In (3.7) and (3.8) $u_m$ should be sent to the infinites and not $t_m$-s.

4. In (3.15) and (3.17) $N$ denotes, what was denoted by $M$ before.

5. Some articles are not used correctly.