Spin-$s$ Rational $Q$-system

The manuscript addresses some interesting integrable models and provides for it new and relevant results toward the complete description of its spectrum by functional equations.

Different technical points require to be clarified as well as a better description of the research framework in which this work is located.

In this manuscript the authors study the homogeneous Heisenberg chain with spin s higher than 1/2. They do this analysis in a purely functional approach (i.e. without a direct construction of the eigenstates) based on the analysis of TQ-equations and QQ-equations, the so-called “spin-s rational Q-system”. Their aim is to show that this system of equations gives indeed all and only the physical solutions (i.e. the ones associated to true eigenvalues of the transfer matrix/Hamiltonian) and they in fact check it numerically for several cases. The main result is that the spin-s rational Q-system, here presented, is equivalent to the requirement that the T-function and the two associated Q-/P-solution are all polynomials for the periodic chain. This result allows the author to use Tarasov results on completeness for the spin s homogeneous XXX chains, thanks to which the completeness of the rational Q-system is derived in the periodic chain.
So, the manuscript addresses some interesting integrable models and provides for it new and relevant results toward the complete description of its spectrum by functional equations. This also allows the authors to introduce some easily formulated conditions that have to be satisfied by the physical solutions of the Bethe equations. This sheds some first light on the complete set of extra conditions to be satisfied by the physical singular solutions to the Bethe equations for the spin s cases; a priori a non-trivial task to obtain from other approaches like Algebraic Bethe Ansatz for these spin s cases in which one could have both repeated and singular physical solutions.
While I consider the manuscript relevant, I think that some modifications are required in order to publish it on SciPost.
Such modifications go from technical points to be clarified (with also better distinction between results proven and only verified) to a better description of the research framework in which this work is located.
Let me start pointing out that I agree with the first referee about the required clarifications about Theorem 1 and 2 that he has listed. About it I would add the following observations about P-function and Wronskian. Note that from the P-function and Q-function satisfying the TQ-equation with the same T-function, it is a trivial exercise to derive a first order difference equation for the Wronskian with coefficients the same coefficients of the TQ-equation. Then it is an easy exercise to prove that the Wronskian is given by equation (3.2), once it is a polynomial. As it is stated it is not clear if the proof of Theorem 1 uses this argument in the derivation of (3.2). Do the authors are supposing that the P-function in (3.22) is a solution of the TQ-equation as consequence of the fact that the Q-function is a solution and that the Wronskian formed out of P and Q is a solution of the Wronskian equation? In this case they should make this explicit. That is, they should write explicitly the first order Wronskian equation, they should comment on the fact that (3.23) is a solution of it and from it derive that the P-function (3.22) is solution of the TQ-equation.
The other central technical point is about the twisted case, as the first referee pointed out, this is not introduced at all. In fact, just the QQ-relations are modified by a twist parameter in (2.9), while this is an important case which deserve a careful analysis. So, the authors should give some references about it and methods that have been used to analyze it and they should write something about the twist matrix and twisted boundary conditions, are they considering diagonal or generic twist?
The authors have some numerical result about completeness in the twisted case, however, one should mention that in this case several steps in their spin-s rational Q-system approach are missing or even not working currently. The first missing point is about a possible derivation of completeness. Indeed, the Tarasov results for the homogeneous XXX spin s system (in reference [13]) has been derived in the case of periodic boundary conditions. Even more intrinsically, the very idea at the basis of the method, i.e. that the spin-s rational Q-system expresses the polynomiality of the two solutions P and Q of the TQ-equation associated to the same T-function is here missing. Indeed, it has been shown in the papers https://iopscience.iop.org/article/10.1088/1751-8113/49/10/104002, for spin 1/2, and https://scipost.org/10.21468/SciPostPhys.10.2.026, for higher spin s, that in the twisted case there exists only one polynomial solution to the TQ-equation for a fixed T-function. There, but for the inhomogeneous models, it is proven that in fact completeness follows just using this one polynomial Q-function for each T-function. In this inhomogeneous framework, it is also interesting to remark the simplicity on the conditions required on the Q-function roots to define physical solutions, see for example Theorem 5.1 of the second cited paper. This also brings to the natural question if the authors can with their approach start from an inhomogeneous model, get some general prescription on the physical Bethe roots and then see how to adapt them in the homogeneous limit.
A part the technical points to clarify, above listed, I think that the authors should do the effort to better introduce the integrability background and research that surround their work.
To gives some examples, let me mention that from the introduction of the Q-operator and TQ-equations since the seventies by Baxter, for which the authors only cite the Baxter book [10], there is a vast and variegate literature using these tools and their generalizations, in particular, to develop pure functionals approaches to study quantum integrable models which go far beyond the XXX and XXZ models even for higher spins. One clear example is the so-called Analytic Bethe Ansatz of Reshetikhin, already introduced in the eighties. I found bizarre the attitude of the authors that make a jump of more than 40 year from Baxter works to their references [11,12,14-17]. They should take into account some more literature in between, this should give them also the opportunity to point out that the use of these Q-systems and TQ-equation is not limited to these special representations, for example, the higher rank case and in the prospective to enlarge the use of the analysis to other interesting integrable models.
Even when the authors focus on works directly in their analysis this is not always done consistently. The main example is the paper of Tarasov [13], which is cited only in section 3, while this is just the central tool for their derivation of completeness in the periodic case that is claimed already in abstract; its role in their analysis should be cited already in the introduction.
Finally, some more proofreading is desirable, I spotted for example the repetition of article “the an” in the phrase between eq. (1.1)-(1.2).

1. New and non-trivial way to distinguish physical solutions of the Bethe equations for higher spin chains.
2. Introduction of the $Q$ system for the higher spin chains
3. New simple condition for physical singular solutions (4.6)

1. Some proofs are not clearly written and should be improved
2. Notations sometime leading to confusion
3. Twisted case is mentioned but not properly discussed
4. English grammar

The results of the paper "spin-$s$ rational $Q$ system" are clearly interesting and important, it brings significant new understanding of the Bethe equations for the higher spin chains. In particular it gives a new not straightforward property to distinguish physical singular solutions (4.6) which is quite remarkable. The authors make the effort to produce detailed proofs of their results. In general I think that ultimately these results should be published in SciPost however there are some intermediate steps which should be modified therefore I recommend a major revision. There are several points which should be improved

The proof of the Theorem 1 is not clear at all, there some crucial points that should be clarified.
In (3.9) the function $q(u)$ (without any subscript) appears for the first time without being properly introduced before. It is not clear what is this function and where it comes from. As it plays the crucial role later it should be properly defined (as far as I understand to obtain (3.7) the functions $q_\pm$ should have the form (3.11) but it is never stated).
More important remark: it is not clear from the proof why the proposed construction of $P(u)$ gives a solution of the TQ relation. In my opinion the proof should be carefully rewritten.

The proof of the Theorem 2 is not clear too, there are some problems with notations. In particular, as far as I understand the notation $F^{[k]}$ means the shift of the argument while the bar $\bar{F}^{[k]}$ means fixing $u=0$. In (3.28) these two operations are introduced together and a reader needs some guesswork to reproduce the proof, specially as the authors use the word "regular" for $\bar{P}$ while it is a number and not a function (the word finite will be more suitable in my opinion).

The third major point is the twisted case which is mentioned throughout the paper without any proper consideration. It is not properly introduced, even the Bethe equations are not written and for all the future analysis it is not treated at all (consequences of the $SU(2)$ symmetry breaking, what happens with singular solutions, which theorems work for the twisted case and what should be modified etc.). In my opinion this case deserves a more detailed explanation if it is mentioned.

The last remark is that a careful proofreading is very much recommended.

Major points:
1. Rewrite the proof of the Theorem 1
2. Fix the notations $Q^{[n]}$
3. Explain more carefully the case of twisted boundary conditions
4. Improve the English grammar
Minor points:
5. Better explain the completeness conjecture (2.4), in particular what are strange solutions
6. The figure 2.2 is a little bit misleading as it gives an impression that the box $1,2s+1$ is the last one in the upper row.