Spin-$s$ $Q$-systems: Twist and Open Boundaries

1. Extends the Q-system for higher spin twisted and boundary XXX spin chains and investigates the relation between physicality of the Bethe roots and polynomiality of the Q functions in the twisted case.
2. Identifies a particular boundary case, when the symmetry is enhanced and constructs the extra symmetry generators

1. The logic of the paper is not clear enough, e.g. it is not explained what are the non-physical solutions of the Bethe equations in the twisted and boundary cases.
2. There are some un-justified statements. (See report).

I do not support the publication of the manuscript in SciPost Physics. I think it does not represent a breakthrough in the field, it is not written well enough and its analysis is not complete within its framework. But after improving the manuscript it could be published in SciPost Physics Core.

The spectral problem of integrable spin chains are usually transformed into Bethe ansatz equations. These equations, however, admit more solutions than the original problem and one has to separate the physical solutions from the non-physical ones. One elegant way of doing this is the construction of a $Q$-system, since the polynomial solutions of the $QQ$ equations are in one-to-one correspondence with the physical solutions. This method was applied successfully to the periodic and some boundary XXX and XXZ spin chains before. The periodic analysis was also extended for the spin $s$ XXX chain and the aim of the present manuscript is to extend the method to its twisted and the simplest boundary versions. The manuscript starts with summarizing the existing results for the periodic spin $s$ XXX chain. Then in section 3 it generalizes to the twisted case, while in section 4 to simple diagonal boundary conditions labeled by two parameters. The main emphasis in section 3 is the construction of the P function and to show that its polinomiality is equivalent to the polynomiality of the $Q$-system together with deriving the physicality condition for the Bethe roots. These steps are missing in section 4 where the main focus is on a specific resonance case between the two boundaries, which allows for a symmetry enhancement.

The main motivation for the $Q$ system is to generate only the physical solutions of the Bethe ansatz equation as it is nicely recalled for the periodic case. In the twisted case, however no singular solutions or repeated roots are introduced at the beginning of section 3. Even more, in the introduction the authors say "Through numerical evidence, we confirm the absence of physical singular solutions and physical solutions with repeated roots in this case, and check the completeness of the rational Q-system". It is a bit confusing, if there are no nonphysical solutions in the twisted case, then what is the motivation of introducing the $Q$-system. The authors also write later in section 3 that "In the following, we also assume that twisted boundary conditions is a strong enough regularization scheme such that there is no repeated root if $\frac{Q_{0,1}}{\gamma}$ is required to be a polynomial. I think section 3 should be formulated in a conceptually clearer way. It would nice to present repeated or singular solutions at the beginning of the section (if there is any) and later show that they do not correspond to polynomial $T_{0}/\gamma$ or $Q_{0,2s+1}$ . It would be also nice to clearly point out what are the assumptions and what statements are really proven, say in a summary at the end of the section.

In section 4 nothing is mentioned about non-physical solutions, rather the authors use the already developed Q systems [25] and investigate its consequences. It is not clear why for example the polynomiality of $P$ follows from (4.9) as stated below this equation. In the $s=\frac{1}{2}$ case the paper [29] put some effort to prove it, based on the classification of singular solutions. Instead of addressing the conceptually same questions as the previous two sections, the main focus of section 4 is on a very specific situation, when the difference between the boundary parameters is an integer. They identify an extra hidden symmetry in this case, which they also support with numerical investigations. Numerical analysis also identifies solutions with extra degrees of freedom, magnetic strings and splitting of Bethe roots, but they do not elaborate on these objects, rather leave the systematic analysis for a further publication. I do not feel this section is complete in any sense.

In summarizing, the work formulates important steps in the generalizations of the rational $Q$ systems of the spin $s$ XXX spin chains. However, I do not think that it details a "groundbreaking theoretical discovery "or "presents a breakthrough on a previously-identified and long-standing research stumbling block" as expected from a publication in SciPost Physics. It is more like a natural evolution of the field. For example, the results of the twisted model is a natural generalization of the periodic case. This part does not have a proof for the completeness of the Bethe ansatz, rather a few numerical confirmations. Also it has assumptions about absence of repeated roots for polynomial $T_{0}/\gamma$ in the deformed case. The boundary part does not even address these relevant questions, rather it focuses on a very specific parameter resonance $\alpha-\beta\in\mathbb{Z}$, although even that case is not analyzed systematically either. Concerning the impact of the paper, the higher spin version of the XXX model does not seem to be relevant in the problems listed in the motivation of the paper, i.e. in calculating the spectral problem of the N=4 super Yang Mills theory. Based on these, I do not support the publication of the manuscript int SciPost.

I have also some small technical problems:

1. Concerning citations:

"and the correlators are related to open string states in the N = 4 super-Yang-Mills theory [42–47]". I think [44-47] are not relevant for the open spin chain spectral problem, they are rather related to boundary states and overlaps. But the Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries by Konstantinos Zoubos, Lett.Math.Phys. 99 (2012) 375-400, and many other papers dealing with the spectral problem could be.

Boundary string solutions of spin chains appeared already, e.g. in Boundary bound states and boundary bootstrap in the sine-Gordon model with Dirichlet boundary conditions, Sergei Skorik, Hubert Saleur, J.Phys.A 28 (1995) 6605-6622. It would be interesting to relate the appearing strings to those (if there is any relation).

2. Some typos: Some definite articles are missing in section 4 and there are some typos: In (4.18) there is an extra \beta. Below (4.19) .. here satisfies following relation (the following relation). Below (4.25) condition are satisfied.

to improve the manuscript

3. section 3 should be rewritten in a conceptually clearer way along with the lines of section 2.

4. Section 4 should address also the same questions about non-physical solutions and polinomiality of the Q-functions with possible proofs.

Having done all these changes I could support the publication in SciPost Physics Core.

Accept in alternative Journal (see Report)

1-- The article provides a systematic treatment of spin-$s$ Q-systems in the presence of a diagonal twist and with open boundaries. It solves a technical problem of the field.

2-- New phenomena such as a hidden symmetry are identified in the open boundary case.

3-- The analytical and numerical evidences are in details, which would be helpful for other researchers in the field.

1-- Some formulae (such as the explicit form of the spin chain Hamiltonians) are missing.

2-- The discussions on the "ambiguity solutions" in Sec. 4.3 and 4.5.2 need to be improved. I provide the details in the Report part.

3-- Some important references are missing, cf. Report part.

There are a few places in the article that need to be improved significantly, which are collected below:

1-- There is no formula about the Hamiltonians of spin-$s$ XXX spin chain with/without twist. It would be helpful for readers who are not very familiar with the previous results. I suggest that the authors write down the spin chain Hamiltonians (at least for the spin-$1/2$ and spin-$1$, i.e. Babujian-Takhtajan) in Sec. 2.1 and 3.1. The derivation of the Hamiltonians with transfer matrix fusion might be given too, as the authors did for the open case in the Appendix. By given the explicit formulae of the Hamiltonian, Eq. (2.2) can be fixed properly.

2-- The last paragraph of Sec. 2.1. I don't follow the authors' conclusion that "Bethe equations typically give too many solutions". In fact, in the [16] of the article, the other authors showed this via numerical solutions. The authors should arrive at the conclusions by consulting the numerical results of [16] in my opinion.

3-- In Sec. 3.3, the authors defined the so called "$T$-series". This is actually very confusing, because the T-functions here are NOT the ones in T-systems, which have clear physical meaning as the eigenvalues of the higher-spin transfer matrices. I would suggest the authors to use other letters. But if the authors still want to keep the notation, at least a remark that they are NOT the T-functions in T-system is required.

4-- Sec. 4.3 and Table 6. The authors seem to imply that by solving Q-system it is not enough to obtain all physical solutions, since some of them are defined ambiguously. However, this is a quite common phenomenon when solving Q-system with certain symmetry. For example, if one tries to solve the Q-system for spin-1/2 XXX chain with periodic boundary and $M>L/2$, this phenomenon appears. But we all know that the solution is to add Bethe roots at $\infty$, i.e. $SU(2)$ descendants. Obviously here $\alpha - \beta \in \mathbb{Z}_+$, the $SU(2)$ symmetry is broken. But at least the authors should try some other ways to elucidate the ambiguity, e.g. adding infinity roots. I think that those "ambiguous solutions" should be degenerate with some of the states below the equator.

Another way of seeing this is by acting a spin-flip operation to the Hamiltonian,

$H(\alpha^\prime , \beta^\prime) = H(-\alpha, -\beta) = \prod_n \sigma^z_n H(\alpha, \beta) \prod_n \sigma^z_n$,

The two Hamiltonians $H(\alpha^\prime , \beta^\prime)$ and $H(\alpha, \beta)$ have the same spectra, but the eigenstates are related by a spin flip (therefore the rôles of $M$ and $2sL-M$ are reversed).

There is actually no ambiguity for the new Hamiltonian $H(\alpha^\prime , \beta^\prime)$, since $\alpha^\prime - \beta^\prime= \beta - \alpha \in \mathbb{Z}_-$. I believe that the authors can use this approach to investigate the "ambiguous solutions" in the original Hamiltonian. At least there should be a NON-ambiguous way to construct the eigenstates in terms of ABA, thus fixed the Bethe roots without any ambiguity.

5-- In the Appendix, the authors mentioned the fusion of boundary K matrices. Actually, the fusion of R matrices should be mentioned. The R matrix for spin-$s$ XXX should coincide with the notation in Eq. (A9) and Eq. (A10), i.e.
$\mathbf{R}_{an} (u) = \mathbf{R}_{an}^{(s,s)} (u)$.

6-- Terminology. The authors used "magnetic strings" to refer the fixed Bethe roots in some solutions. I don't see how the usage of "magnetic" is justified.

7-- Sec. 4.5.1. I do not understand the logic of that section by the authors. The authors seem to suggest that by increasing $\alpha$ and $\beta$, some Bethe roots will go to infinity, recovering the free boundary results. But the authors only offer one data point $(\alpha , \beta) = (10^{10},10^{10})$, which cannot demonstrate what the authors want to say. I suggest that the authors to just show the results of $(\alpha , \beta) \to (\infty, \infty)$ results. Actually, in the last paragraph of the section, the authors wrote "we designate as descendant states those [...] infinity" (there is a small grammatical error). But we know that those states are descendants of the $SU(2)$ symmetry at the free boundary limit. I'm confused by that sentence, which needs to be improved.

8-- References. The authors should mention the renowned papers of Takhtajan and Babujan in the introduction and when mentioning the Hamiltonian:

[R1] L. A. Takhtajan, The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins, Phys. Lett. A 87, 479 (1982).
[R2] H. Babujian, Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S, Phys. Lett. A 90, 479 (1982).

Moreover, in the Appendix, when discussing fusion, the classics should be mentioned among a vast of other papers:

[R3] P. P. Kulish, N. Yu. Reshetikhin and E. K. Sklyanin, Yang-Baxter equation and representation theory: I, Lett. Math. Phys. 5, 393–403, (1981).
[R4] I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum Integrable Models and Discrete Classical Hirota Equations, Commun. Math. Phys. 188(2), 267 (1997).

At the beginning of page 3, the authors mentioned that the spin-$1/2$ Heisenberg spin chain "describe(s) phenomena in the real world". Maybe the authors should cite at least one or two experimental papers that justify the statement.

There are some typos that need to be corrected.

1-- The last paragraph of page 6, the authors wrote "one can proof...", which should read "one can prove...".

2-- In the caption of Fig. 1, the authors used "node". I would prefer to refer that as the "vertices of each square".

3-- Page 8, bullet point 4. The authors used "the zero reminder equation". It should read "zero remainder equations". (I think that one should use the plural form here. This comment applies to other instances when the authors use the singular form.)

4-- Page 11. Below Eq. (3.13), the authors used "which is called the Wronskian relation". Actually the authors have introduced the Wronskian relation previously. I would suggest "satisfy the Wronskian relation, [equation]." and delete the sentence afterwards.

Please see the Report part.

Ask for major revision