SciPost Submission Page
Spin-$s$ $Q$-systems: Twist and Open Boundaries
by Yi-Jun He, Jue Hou, Yi-Chao Liu, Zi-Xi Tan
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Jue Hou · Zixi Tan |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2502.15636v2 (pdf) |
| Date submitted: | March 4, 2025, 10:23 a.m. |
| Submitted by: | Jue Hou |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
In integrable spin chains, the spectral problem can be solved by the method of Bethe ansatz, which transforms the problem of diagonalization of the Hamiltonian into the problem of solving a set of algebraic equations named Bethe equations. In this work, we systematically investigate the spin-$s$ XXX chain with twisted and open boundary conditions using the rational $Q$-system, which is a powerful tool to solve Bethe equations. We establish basic frameworks of the rational $Q$-system and confirm its completeness numerically in both cases. For twisted boundaries, we investigate the polynomiality conditions of the rational $Q$-system and derive physical conditions for singular solutions of Bethe equations. For open boundaries, we uncover novel phenomena such as hidden symmetries and magnetic strings under specific boundary parameters. Hidden symmetries lead to the appearance of extra degeneracies in the Hilbert space, while the magnetic string is a novel type of exact string configuration, whose length depends on the boundary magnetic fields. These findings, supported by both analytical and numerical evidences, offer new insights into the interplay between symmetries and boundary conditions.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-4-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2502.15636v2, delivered 2025-04-02, doi: 10.21468/SciPost.Report.10958
Strengths
- 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.
- Identifies a particular boundary case, when the symmetry is enhanced and constructs the extra symmetry generators
Weaknesses
- 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.
- There are some un-justified statements. (See report).
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.
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.
Requested changes
I have also some small technical problems:
- 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).
- 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
-
section 3 should be rewritten in a conceptually clearer way along with the lines of section 2.
-
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.
Recommendation
Accept in alternative Journal (see Report)
Report #1 by Anonymous (Referee 1) on 2025-3-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2502.15636v2, delivered 2025-03-26, doi: 10.21468/SciPost.Report.10911
Strengths
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.
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.
Weaknesses
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.
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.
Report
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.
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.
Requested changes
Please see the Report part.
Recommendation
Ask for major revision
Category:
answer to question
We would like to express our gratitude to the Referee #1 for the feedback and many constructive suggestions. We have made the following modifications to the article based on the suggestions.
- We have added the explicit formulae of the Hamiltonian of Heisenberg spin-1/2 XXX chain with the periodic boundary conditions and twisted boundary conditions at the beginning of Section 2.1 and Section 3.1. And the fusion of transfer matrix has been given in the Appendix.A (A.7)(A.8) .
- We have supplied some numerical evidence in Table.1 and Table.2 in Section 2.1 and Table 3 in Section 3.1. They are notes to the conclusion of "Bethe equations typically give too many solutions".
- The previous definition of "T-series" might lead to ambiguity, so we have changed our notation as "$\mathcal{T}$-series".
- Thank the referee for the helpful comment! Yes, if we consider the spin-flip operator, when the magnon number $M$ is beyond the equator, the solutions will become clearer. Like the $SU(2)$ cases, there are two kinds of "strange solutions". One contains infinite roots and corresponds to descendant states. Another has the form like $Q+const.P$, which does not directly correspond to Bethe states, since the number of C operators, $L-M$, is not equal to the zeros of $Q+const.P$, $L-M+1$. As the referee said, we can eliminate this problem via the spin-flip. It is interesting that for the $SU(2)$ case, the $Q+const.P$ phenomenon appears once $M$ is beyond the equator and for the $U(1)$ case, the phenomenon doesn't exist at all. However for the open chain in our paper, we find that only when the parameters take some special values and $M$ is beyond the equator, the phenomenon appears (Table 6 in the last version of the manuscript and Table 9 in the revised version). So we say the symmetry here is between $SU(2)$ and $U(1)$. We add the discussion in the paragraph below Table 9.
- The fusion of $R$ matrices has been added in Appendix.A.
- For terminology of "magnetic strings", the reason why we call it "magnetic strings" is that the string emerges when specific constraints on the boundary parameters $\alpha$, $\beta$, spin-$s$, the length of chains $L$ and the magnon number $M$ are satisfied. "magnetic string" is a novel type of exact string configuration related to boundary magnetic fields and this explanation has been added in the introduction.
- Due to technical limitations, the numerical program cannot directly handle boundary parameters $(\alpha, \beta)=(\infty, \infty)$. Therefore, our numerical results are presented as $(\alpha, \beta)$ increases to sufficiently large finite values, allowing us to observe the behavior approaching the free boundary limit. We rewrite our numerical results in Table.6,7 in Section 4.5.1.
- We have added all the references mentioned and corrected all the typographical errors pointed out.
Author: Jue Hou on 2025-06-14 [id 5571]
(in reply to Report 2 on 2025-04-02)We would like to thank Referee #2 for the thorough and constructive review of our manuscript. We have carefully considered each of points in the report and outline our responses and revisions below:
Reply to Weaknesses and Report
We hope to kindly point out that in Section 1 (Page 3) of the first version, we have given the definition about the non-physical solutions of the Bethe equations in the twisted and boundary cases, which is "'Non-physical' here means that......" To make it clearer, we add the footnote 6 on Page 10 and the footnote 14 on Page 21 to emphasize the definition.
As we emphasize in our abstract and the paper's overall narrative, the main thrust of this work is to utilize the rational $Q$-system as a tool to uncover and investigate interesting physical phenomena, such as hidden symmetries and the novel magnetic string configurations. While mathematically rigorous proofs of these fundamental equivalences are indeed crucial for a complete theoretical framework, providing such an exhaustive derivation for higher spins and general open boundary conditions is an extremely complex task that would necessitate a substantial expansion of the manuscript, extending beyond the scope of its current focus on phenomenological discoveries.
We apologize that the motivation for introducing the $Q$-system in the twisted case might have appeared confusing. To clarify: in the general Bethe Ansatz framework for the spin-s XXX chain, singular solutions and solutions with repeated roots can appear as both physical solutions and non-physical solutions. The rational $Q$-system's fundamental role is to provide a systematic and efficient method to precisely identify and remove all the non-physical solutions from the complete set of Bethe solutions.
Our numerical and analytical findings, specifically discussed in Section 3.6 ("Numerical Results") and building upon the general definitions in Section 2.1, confirm that when the polynomiality conditions of the rational $Q$-system are enforced, there are no physical singular solutions and no physical solutions with repeated roots in the twisted case. We should stress that in the twsited case the $Q$-system also filters out all non-physical solutions, resulting in a set of only physical, non-singular, and non-repeated solutions. According to the referee's suggestions, we add Table 1 and Table 2 to show that Bethe ansatz equations give too many solutions and so $Q$-systems are needed. And we add Table 3 to show the repeated, singular, regular solutions exist for the twisted case.
In the revised manuscript, we will significantly clarify this motivation and distinction in Section 3.6 to avoid any ambiguity regarding the $Q$-system's purpose.
We completely agree with the referee's valuable suggestion to provide a clearer distinction between our assumptions and proven results. Our assumption in Section 3.5 is indeed that "twisted boundary conditions is a strong enough regularization scheme such that there is no repeated root if $Q_{0,1}/\gamma$ is required to be a polynomial." Our derived result is the exact physical condition for singular solutions, given at the end of Section 3.5, as its title "Physical Conditions" implies.
In the revised manuscript, as the referee suggested, we will add a dedicated summary at the end of Section 3.5 to clearly list all assumptions made and the key theoretical and numerical results obtained, thereby enhancing the conceptual clarity of this section.
Explain the polynomiality of $P$ follows from (4.9).
The referee is correct that the derivation of the $P$-function's polynomiality from equation (4.9) in the open boundary case requires a more detailed explanation. As discussed in point 2, a fully rigorous and general proof is beyond the scope of this paper's primary focus.
The referee said that "the main focus of Section 4 is on a very specific situation, when the difference between the boundary parameters is an integer", but actually our main objective in Section 4 is indeed to unveil and analyze the hidden symmetries and magnetic strings, these phenomena are only accessible in the case where the difference between boundary parameters is an integer. In other words, the reason we discuss this case is that it is the uniquely non-trivial case. We mention this point in the introduction, at the first paragraph of Section 4, and stress it at the beginning of Section 4.5, "We verify that for parameters $\alpha, \beta, s, L, M$ which do NOT permit a hidden symmetries, the number of solutions is coincident with the formula (3.43), since the apparent symmetry of the Hamiltonian is $U(1)$.".
For the completeness proof for twisted boundaries, we agree this is an important open question. However, our focus in this paper is not on rigorously proving completeness but on applying the rational $Q$-system as a powerful tool to uncover novel physical phenomena. We acknowledge the immense difficulty of such completeness proofs, as highlighted by the decades of effort required to resolve completeness for periodic boundaries (e.g., Kirillov's foundational work, Mukhin et al.'s significant contributions, and the recent conclusive proof by our co-author, Jue). Given the distinct nature and complexity of these mathematical proofs, we believe that a comprehensive treatment of completeness would constitute a separate, major theoretical work that falls outside the primary scope of the present study, which is more focused on the physical implications and applications of the $Q$-system.
Reply to Requested changes
On the citations related to the open spin chain spectral problem We appreciate the referee's observation. However, we respectfully disagree with the assertion that references [44–47] are irrelevant. From the perspective of string theory in the AdS/CFT correspondence, the study of correlators involving determinant operators is related to the study of open spin chains. In the open channel, the problem is indeed related to the spectral problem of open spin chains, while in the closed channel, it corresponds to boundary overlaps. These two perspectives are related via a change of channel. Therefore, we believe it is appropriate to cite these papers.
Following the referee's suggestion, we will include the reference to the review by Konstantinos Zoubos as well as other relevant literature as additional citations.
Typos and minor technical issues We thank the referee for pointing out the typos. We have corrected them accordingly.
On the conceptual clarity of Section 3 We understand the referee's suggestion for conceptual clarity, but we believe that this request arises from a misunderstanding of the structure and purpose of our sections. Section 2 is meant as a pedagogical review of the known rational $Q$-system under periodic boundary conditions, which lays a foundational framework for general readers.
In contrast, Section 3 has a different purpose: it presents a generalization of the rational $Q$-system to twisted boundary conditions, and highlights the change of singular/repeated solutions after turning on the twist, as well as a detailed numerical completeness check. Therefore, it is structured to reflect these priorities, which do not align with the pedagogical style of Section 2.
On non-physical solutions and polynomiality in Section 4 The referee implies that Section 4 is incomplete regarding non-physical solutions and the polynomiality of $Q$-functions and suggest to add these contents. We thank for the suggestion and agree that these contents are important for the $Q$-system method. However, our aim in Section 4 is to uncover new structures such as magnetic strings and hidden symmetries, which are unique to the open boundary setup. These phenomena do not have analogs under periodic or twisted boundaries and thus justify a distinct structure and treatment in Section 4. We hope the section focuses on these new phenomena.
We thank the referee once again for the constructive feedback. We believe the revisions and clarifications discussed above improve the presentation and strengthen the manuscript.