Spin-$s$ Rational $Q$-system

Reply to referee 1.

We would like to thank the referee for the careful work and positive comments on the results of our work. Below we provide answers for the points raised in the referee's report.

1. "The proof of theorem 1 is not clear."

We have rewritten the proof of theorem 1. The main modifications are : (1) The typo in eq(3.9) in the previous version is corrected. (2) More explanations are added in order to make the proof more readable. (3) By the end of the proof of theorem 1, we have added a paragraph (from eq(3.23) to eq(3.27)) to prove that the $P(u)$ which has been constructed in the previous steps is indeed a solution of the $TQ$-relation with the same transfer matrix.

1. "The proof of theorem 2 is not clear too, there are some problems with notations."

In order to eliminate confusion, we have changed the notations. All the $\bar{F}^{[k]}$ are replaced by the more explicit $F(ik/2)$. Following the suggestions by the referee, we use finite' instead ofregular' for all the quantities which are numbers.

1. "The third major point is the twisted case which is mentioned throughout the paper with any proper consideration."

Indeed the mention of the twisted case is rather brief. In the revised version, we added a subsection (section 2.3) to provide a proper introduction to the twisted case. Here we only consider the so-called diagonal twist, which preserves U(1) symmetry. Non-diagonal twists need significant modifications. We give the construction of twisted rational $Q$-system and checked that the $Q$-system indeed give all and only physical solutions also for the diagonal twist case. Note that, however, the proof of completeness for the periodic case for the rest of the paper cannot be applied directly to the twisted case. Therefore we prefer to give a more detailed discussion on the twisted case in a separated paper.

1. We also did a more careful proofread.

We hope these modifications are sufficient to address the questions and comments from the referee.

Reply to referee 2.

We thank the referee for the careful work and the detailed comments and suggestions. Below we provide our answers to the points that are raised in the referee's report.

1.Clarification of theorem 1 and 2

We have made significant modifications for the proof of theorem 1 and 2. More specific to the referee's comment, in theorem 1 what we prove is the following: From the polynomial solution $Q(u)$ of $TQ$-relation, we can construct a function called $P(u)$ which satisfies the Wronskian equation eq(3.2). It then follows that the function $P(u)$ is the other solution of the $TQ$-relation. We have made this more explicit by changing the wording of theorem 1. We also added a paragraph at the end of proof of theorem 1 to show that $P(u)$, which satisfies the Wronskian equation is the other solution of the $TQ$-relation.

1. About twisted case

Indeed the mention of the twisted case was rather brief in the draft. In the revised version, we have added a subsection (section 2.3 in the current version) to give a more proper introduction to the twisted case. Here we consider the diagonal twist, which preserves U(1) symmetry. The rational $Q$-systems for models with non-diagonal twist would require more significant modifications. For the diagonal twist, we present the construction of the rational $Q$-system, which amounts to a slight modification of the $QQ$-relation. We checked numerically that the proposed $Q$-system give all and only physical solutions of the BAE. More comments on the numerical results are added at the end of in section 2.5.

On the other hand, to prove that the proposed $Q$-system gives complete physical solutions of BAE is a different story. Indeed our proof for the periodic case which require both solutions of $TQ$-relation are polynomials cannot be adapted directly to the twisted case. As the referee has correctly pointed out, there exist only one polynomial solution for the $TQ$-function with fixed $T$. The inhomogeneous case has been discussed in the two papers mentioned by the referee, which are also cited in the revised version. Therefore we feel it is proper to give a more detailed discussion on the twisted case in a separate work which will appear soon.

1. About references

The referee suggests that we take into account more references on $TQ$-relations and $Q$-system. We are of course aware of and fully appreciate the fact that the applications of $TQ$-relations and $Q$-systems go way beyond the current context. It is precisely because of this, taking into account all works in this direction becomes a daunting task. In the current work, our main focus is on solving Bethe ansatz equations. We take the perspectives that $TQ$-relation and rational $Q$-system are reformulations of the BAE of the Heisenberg spin chain. Also we want to mention that the rational $Q$-system proposed by Marboe and Volin is related to, but different from the traditional $Q$-system. Therefore we mainly cite the papers on the rational $Q$-system and gave the referee an impression that we `make a jump of more than 40 years'. At the same time, we also understand the referee's concern and we added more references on the $TQ$-relations and $Q$-systems in the second paragraph of the Introduction. As we mentioned before, this cannot be the complete list but we feel these are the most relevant ones. If there are other papers which the referee believes is worth citing, please let us know.

1. In addition to the previous modifications, we also did a more careful proofread.

We hope these modifications are sufficient to address the questions and comments from the referee.

List of changes:

1. The proof of Theorem 1 is rewritten. The statement is made more precise, more explanations are added in the proof; A paragraph is added to prove that $P(u)$ is the solution of the same $TQ$-relation.

2. Modification of the proof of Theorem 2. Changed to more explicit and less confusing notations.

3. A subsection (section 2.3) is added to introduce the case with diagonal twisted boundary condition. More discussions on numerics are added in section 2.5.

4. More references on TQ- and QQ-relations are added in the second paragraph of the introduction.

5. Some typos are corrected.