SciPost Submission Page
Rational Q-systems at Root of Unity I. Closed Chains
by Jue Hou, Yunfeng Jiang, Yuan Miao
Submission summary
| Authors (as registered SciPost users): | Yuan Miao |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2310.14966v3 (pdf) |
| Date submitted: | 2024-04-05 03:46 |
| Submitted by: | Miao, Yuan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The solution of Bethe ansatz equations for XXZ spin chain with the parameter $q$ being a root of unity is infamously subtle. In this work, we develop the rational $Q$-system for this case, which offers a systematic way to find all physical solutions of the Bethe ansatz equations at root of unity. The construction contains two parts. In the first part, we impose additional constraints to the rational $Q$-system. These constraints eliminate the so-called Fabricius-McCoy (FM) string solutions, yielding all primitive solutions. In the second part, we give a simple procedure to construct the descendant tower of any given primitive state. The primitive solutions together with their descendant towers constitute the complete Hilbert space. We test our proposal by extensive numerical checks and apply it to compute the torus partition function of the 6-vertex model at root of unity.
Author comments upon resubmission
Reply to Anonymous Report 1 on 2024-2-3 (Invited Report)
Referee: 1. In the $\ell_2$-dimensional representations of the transfer matrix it is not clear how the representation depends on the complex spin $s$. Is it related to $\ell_2$? Or is it related to the highest weight as for the spin s representations?
Answer: We would like to thank the referee for the question. Complex spin $s$ is not related to $\ell_2$ in general. It can take any complex value. However, when $s = \frac{\ell_2 - 1}{2}$, the representation coincides with the highest weight representation of (half-integer) spin $s = \frac{\ell_2 - 1}{2}$.
Referee: 2. The $v$ and $v^\prime$ notation is not systematic in the paper. In (2.18) did the author mean $v$ or $v^\prime$, what is the definition of $t^\prime$? Also consider (5.1).
Answer: We would like to thank the referee for spotting the typos. We fixed the typos in Eq. (2.18) and (5.1). $t^\prime_m = \exp (v^\prime_m)$, which is also added to the main text.
Referee: 3. In (3.7) and (3.8) $u_m$ should be sent to the infinities and not $t_m$-s.
Answer: We would like to thank the referee for spotting the typos. We fixed the typos in Eq. (3.7) and (3.8).
Referee: 4. In (3.15) and (3.17) $N$ denotes, what was denoted by $M$ before.
Answer: We would like to thank the referee for spotting the typos. We have fixed the typos throughout Sec. 3.2.
Referee: 5. Some articles are not used correctly.
Answer: We would like to thank the referee for the suggestions. We try our best to fix some of the articles in the main text.
Referee: 6. Somehow one exception was Example 5.4 for me. I think the logic could be improved a bit there.
Answer: We would like to thank the referee for the comment. We added more explanations on the mirroring structure of the descendant tower in Example 5.4.
Reply to Anonymous Report 2 on 2024-2-7 (Invited Report)
Referee: 1. Key references are often missing altogether or the citations are not at the appropriate place in the text.
Answer: We would like to thank the referee for the suggestions.
Referee: 1.1) Section 4.1 - there are no references at all for Q-systems.
Answer: We cited several papers on rational Q-systems in the introduction section. We cite them again in Sec. 4.
Referee: 1.2) Page 9 paragraph 2 about FM strings and infinity pairs - there should be citations at this point in the text.
Answer: We cited papers on FM-strings and infinity pairs in the introduction section. We cite them again on page 9.
Referee: 1.3) There are no references in the background Section 2.1 apart from [12].
Answer: We add more citations on the integrability of the XXZ spin chain in Sec. 2.1.
Referee: 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.
Answer: We would like to thank the referee for the comments. We changed the sentence to "For a generic twist $\kappa$", and reduced the usage of the word "special" in the main text.
Referee: 2.3) 'For example, the scattering phase between an FM string [...] is trivial'. Why is this and where in the literature is it discussed?
Answer: The scattering phase between any FM string and other Bethe roots is trivial because Eq. (3.10), the scattering phase $\prod_{n=1}^{\ell_2} S(u, u_n) = 1$. This is a straightforward exercise, and it is well-known in the literature. We added a few citations here.
Referee: 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?
Answer: We would like to thank the referee for the comments. We would like to remark here that the proportionality when defining the QQ-relations is quite common in the previous literature. Nevertheless, we deleted the sentence "The QQ-relations is defined up to proportionality", and fixed the notation such that all the formulae are defined as equalities.
Referee: 3.3) Define $t^{\rm FM}_j$ in equation (4.8). I can guess what it should be but I shouldn't have to.
Answer: We would like to thank the referee for the comment. We could not find $t^{\rm FM}_j$ in Eq. (4.8), but we presumed that the referee was referring to Eq. (4.18). We now added the definition of $t^{\rm FM}_j$ as the exponential of the FM-string Bethe roots in Eq. (3.9).
Referee: 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 introduce an integer $N_c$ in (4.21) without comment. Do you know what this is on not? Please say either way.
Answer: We would like to thank the referee for the comment. We added a sentence below Eq. (4.20) to explain the relation between $t_j$ and $c_j$ from the elimination theory. We added an explanation on the meaning of $N_c$ below Eq. (4.21).
Referee: 3.5) (4.27): Please comment on what the symbol $\omega$ represents.
Answer: We would like to thank the referee for the comment. In Eq. (4.27) we used the variable $w$ to ensure the constraint $\left(|R_1|^2+\ldots+|R_{N_c}|^2\right)$ remains non-zero. We added the comment below Eq. (4.27).
Referee: 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 descendants. 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.
Answer: We would like to thank the referee for the comment. We added a citation of the definition of the Hasse diagram from Wikipedia.
Referee: 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 statement and a clear proof with enough details that a conscientious reader can reconstruct it.
Answer: We would like to thank the referee for the comment. We confirm that Eq. (5.3) is indeed a theorem. We added the details of the derivation below (B.8) on the structure of the zeros of $P(u)$ and how to obtain (5.3). We hope that now we elucidate the method to obtain the zeros of FM-strings in the main text.
Referee: 1. In the $\ell_2$-dimensional representations of the transfer matrix it is not clear how the representation depends on the complex spin $s$. Is it related to $\ell_2$? Or is it related to the highest weight as for the spin s representations?
Answer: We would like to thank the referee for the question. Complex spin $s$ is not related to $\ell_2$ in general. It can take any complex value. However, when $s = \frac{\ell_2 - 1}{2}$, the representation coincides with the highest weight representation of (half-integer) spin $s = \frac{\ell_2 - 1}{2}$.
Referee: 2. The $v$ and $v^\prime$ notation is not systematic in the paper. In (2.18) did the author mean $v$ or $v^\prime$, what is the definition of $t^\prime$? Also consider (5.1).
Answer: We would like to thank the referee for spotting the typos. We fixed the typos in Eq. (2.18) and (5.1). $t^\prime_m = \exp (v^\prime_m)$, which is also added to the main text.
Referee: 3. In (3.7) and (3.8) $u_m$ should be sent to the infinities and not $t_m$-s.
Answer: We would like to thank the referee for spotting the typos. We fixed the typos in Eq. (3.7) and (3.8).
Referee: 4. In (3.15) and (3.17) $N$ denotes, what was denoted by $M$ before.
Answer: We would like to thank the referee for spotting the typos. We have fixed the typos throughout Sec. 3.2.
Referee: 5. Some articles are not used correctly.
Answer: We would like to thank the referee for the suggestions. We try our best to fix some of the articles in the main text.
Referee: 6. Somehow one exception was Example 5.4 for me. I think the logic could be improved a bit there.
Answer: We would like to thank the referee for the comment. We added more explanations on the mirroring structure of the descendant tower in Example 5.4.
Reply to Anonymous Report 2 on 2024-2-7 (Invited Report)
Referee: 1. Key references are often missing altogether or the citations are not at the appropriate place in the text.
Answer: We would like to thank the referee for the suggestions.
Referee: 1.1) Section 4.1 - there are no references at all for Q-systems.
Answer: We cited several papers on rational Q-systems in the introduction section. We cite them again in Sec. 4.
Referee: 1.2) Page 9 paragraph 2 about FM strings and infinity pairs - there should be citations at this point in the text.
Answer: We cited papers on FM-strings and infinity pairs in the introduction section. We cite them again on page 9.
Referee: 1.3) There are no references in the background Section 2.1 apart from [12].
Answer: We add more citations on the integrability of the XXZ spin chain in Sec. 2.1.
Referee: 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.
Answer: We would like to thank the referee for the comments. We changed the sentence to "For a generic twist $\kappa$", and reduced the usage of the word "special" in the main text.
Referee: 2.3) 'For example, the scattering phase between an FM string [...] is trivial'. Why is this and where in the literature is it discussed?
Answer: The scattering phase between any FM string and other Bethe roots is trivial because Eq. (3.10), the scattering phase $\prod_{n=1}^{\ell_2} S(u, u_n) = 1$. This is a straightforward exercise, and it is well-known in the literature. We added a few citations here.
Referee: 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?
Answer: We would like to thank the referee for the comments. We would like to remark here that the proportionality when defining the QQ-relations is quite common in the previous literature. Nevertheless, we deleted the sentence "The QQ-relations is defined up to proportionality", and fixed the notation such that all the formulae are defined as equalities.
Referee: 3.3) Define $t^{\rm FM}_j$ in equation (4.8). I can guess what it should be but I shouldn't have to.
Answer: We would like to thank the referee for the comment. We could not find $t^{\rm FM}_j$ in Eq. (4.8), but we presumed that the referee was referring to Eq. (4.18). We now added the definition of $t^{\rm FM}_j$ as the exponential of the FM-string Bethe roots in Eq. (3.9).
Referee: 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 introduce an integer $N_c$ in (4.21) without comment. Do you know what this is on not? Please say either way.
Answer: We would like to thank the referee for the comment. We added a sentence below Eq. (4.20) to explain the relation between $t_j$ and $c_j$ from the elimination theory. We added an explanation on the meaning of $N_c$ below Eq. (4.21).
Referee: 3.5) (4.27): Please comment on what the symbol $\omega$ represents.
Answer: We would like to thank the referee for the comment. In Eq. (4.27) we used the variable $w$ to ensure the constraint $\left(|R_1|^2+\ldots+|R_{N_c}|^2\right)$ remains non-zero. We added the comment below Eq. (4.27).
Referee: 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 descendants. 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.
Answer: We would like to thank the referee for the comment. We added a citation of the definition of the Hasse diagram from Wikipedia.
Referee: 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 statement and a clear proof with enough details that a conscientious reader can reconstruct it.
Answer: We would like to thank the referee for the comment. We confirm that Eq. (5.3) is indeed a theorem. We added the details of the derivation below (B.8) on the structure of the zeros of $P(u)$ and how to obtain (5.3). We hope that now we elucidate the method to obtain the zeros of FM-strings in the main text.
List of changes
1. We fixed typos in the previous version of the draft. For example, in Eqs. (2.18), (3.7), (3.8), (3.15), (3.17) and (5.1).
2. We added citations in Sec. 2 and 4 according to the referee's suggestions.
3. We the definition of $t^{\rm FM}_j$ in Eq. (4.18).
4. We added the comment below Eq. (4.27) on the additional constraints in the Q-system.
5. We changed several equations in Sec. 4 to remove the ambiguity that "The QQ-relations is defined up to proportionality" (which sentence has been removed from the article too).
6. We added further explanation in Example 5.4.
7. We added the details in Appendix B to explain the derivation of Eq. (5.3).
8. Small changes regarding the misused articles has been performed.
Current status:
Voting in preparation