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Bootstrapping the $R$-matrix
by Zhao Zhang
Submission summary
| Authors (as registered SciPost users): | Zhao Zhang |
| Submission information | |
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| Preprint Link: | scipost_202601_00053v2 (pdf) |
| Code repository: | https://github.com/zz8py/R-matrix-bootstrap |
| Date submitted: | Jan. 23, 2026, 5:21 p.m. |
| Submitted by: | Zhao Zhang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
A bootstrap program is presented for algebraically solving the $R$-matrix of a generic integrable quantum spin chain from its Hamiltonian. The Yang-Baxter equation contains an infinite number of seemingly independent constraints on the operator valued coefficients in the expansion of the $R$-matrices with respect to their spectral parameters, with the lowest order one being the Reshetikhin condition. These coefficients can be solved iteratively in a self consistent way using a lemma due to Kennedy, which reconstructs the $R$-matrix after an infinite number of steps. For a generic Hamiltonian, the procedure could fail at any step, making the conditions useful as an integrability test. However in all known examples they all turn out to be satisfied whenever the lowest order condition is. It remains to be understood whether they are all implied by the Reshetikhin condition.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
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Author comments upon resubmission
The author thanks the referee and the editor for the prompt response.
List of changes
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The explanation of constant shift following Eq. (1) has been simplified according to the referee's suggestion.
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Sec. 3 has been mostly rewritten to explain the usage of Eq. (9) from three different perspectives. First as a way to find R-matrix for known integrable Hamiltonians. Then as an integrability test for generic Hamiltonians. And finally as a way to search for integrable points of a parametrized class of Hamiltonians. Also in this section, more details on Kennedy's trick is provided following Eq. (11).
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The brand-new Sec. 4 further clarifies the procedure described before by parametrizing general local Hamiltonians with n local degrees of freedom. As an example it arrives at the full classification of integrable spin-1/2 chains using the concretized version of the Reshetikhin condition, and explains that anisotropic higher-spin/multicomponent chains are non-integrable.
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The beginning and ending of Sec. 5 has been modified to make the point that higher order Reshetikhin conditions are not implied by the lowest order one, even if/though three-local charge implies infinite conserved charges. The statement on the correspondence between these two has also been made softer, lacking a derivation from one to the other.
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Eq. (46) in Sec. 8 has been made explicit within the framework of Sec. 4. Comments including in footnote 11 has been added to show how h'(0) could be solved from Eq. (46), and that it does not mean much for the purpose of getting h(\mu) or R(\mu). Particular emphasis has also been made on clarifying that Theorem 13 of Ref. 23 is not (yet) useful for claiming integrability of models like the Hubbard, although further developments using the explicit expressions in Eq. (49) may lead to some surprises.
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The manipulations in Appendix. A has been reformulated for the RHS of (A.1) and added for the LHS.
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Appendix. B has basically been reworked. B.1 is a rare example that R^(3) can be obtained by hand, which is now done that way using the language of Sec. 4. B.2 now focuses on the Takhtajan-Babujian model, which explains the subtleties of constant c. It also serves as a demonstration of how generic Hamiltonians as given by (B.5) can be put into the form of (B.7) using a complete basis of corresponding local Hilbert space dimension. I used this opportunity to show also things look different if Sec. 4 did not start with a diagonalization of matrix a. Finally, B.3 has been made slightly more general by letting the anisotropic parameter a vector, which gives very similar results. The appendix is complemented by the Mathematica code uploaded to a repository.
Current status:
Reports on this Submission
Strengths
Weaknesses
To make the story completed, a big step is missing, which is emphasized by the author in the conclusion: understand how Reshetikhin’s condition imply the YBE. However, this may be the subject of future works.
Report
Requested changes
Comments on the new version:
1) To make the presentation more transparent, I suggest stating at the beginning of Section 3, or earlier at the end of the Introduction, that the Reshetikhin condition is necessary and sufficient to guarantee the existence of an infinite number of conserved charges. However, since the connection between these charges and the R-matrix has not been worked out, the bootstrap program proposed in the paper is very important, as it provides a constructive way to derive higher-order terms in the expansion of the R-matrix from lower-order ones, and thus to reconstruct the full R-matrix.
2) I am still convinced that the author should emphasize the result of A. Hokkyo in the Introduction, perhaps after the sentence “It has hence been conjectured to be both a sufficient and a necessary condition for quantum integrability.” So far, this remains the only proof that the Reshetikhin condition is both sufficient and necessary to guarantee the existance an infinite number of translational invariant conserved charges.
3) In the Introduction, the part beginning with “Secondly and more importantly, …” is not entirely clear in its current form. I understand the author’s point and agree with it, but the wording could be improved. When mentioning the Takhtajan-Babujian spin-1 model, the author could add a footnote indicating that this point is clarified in Appendix B.2.
4) After Eq (1) maybe a footnote can be added saying that: one can either consider h+c1 or consider h and modify the initial term in the expansion expansion \checkR=\alpha 1 + …
5) Beginning of section 3: “The fact that an independent…” this cannot be said since it has not been proven the relation between higher order terms in the expansion of R matrix and the charges.
6) At page 5: “Before detailing this standard procedure in the Sec. 5, I first explain how Eq. (9) can be used to solve b_{2m−1}”. Is this a typo? It should be b_{2m+1}. Also, what is a_2m+1? I think it is not specified in the main text but only in the Appendix A.
7) Page 5 “When this happens, there is also hope to obtain the 2D classical statistical mechanical model dual to the quantum Hamiltonian. This could be a future direction worth exploring for those quantum integrable models that do not currently have a known classical counterpart.” Could the author clarify this point?
8) Page 5: “After repeating these steps an infinite number of times” I think at this point (or maybe in section 4) it will be good to mention the finding of https://arxiv.org/pdf/1904.12005. For models of difference form and XYZ type interaction, it has been observed that the coefficients of R^(n) are polynomial of H.
9) After Eq. (11), I believe the argument concerning a constant shift in c is not correct. As the author notes in the Appendix, the role of c is not relevant in the Reshetikhin condition, so D and X do not depend on c. The constant c only appears in higher commutators.
10) Page 6. After “Surprisingly, …” I would add that this is a further indication that it is indeed possible to find a formula connecting higher powers of the R-matrix with the conserved charges (and hence with the Reshetikhin condition).
11) Footnote 6: “This is because the support of the densities do not grow as fast as one would expect from order to order due to similar cancellations as in (8).” I think that the support of densities grows linearly with the log derivative of the transfer matrix, but indeed, if one does not use the cancellation as in (8), it is hard to see it.
12) In section 4, I think the author should emphasize that in section 4 the Hamiltonian is produced and that, due to Hokkyo’s proof, this guarantees the existence of infinitely many translationally invariant conserved charges. However, to obtain a full proof of Yang-Baxter integrability, it is still necessary to use the bootstrap approach proposed in the paper to construct the R-matrix (or to solve the R-matrix from the Sutherland equation).
13) Beginning of page 10: The number of commutators is not O(N), it is one. Again from the proof of Hokkyo, having [H,Q_3]=0, with Q_3=[B,Q_2], guarantee that it exhists n translational invariant conserved quantities such that [H,Q_n]=0 and [Q_n,Q_m]=0. Hence, only [H,Q_3] is the independent one.
14) About the section 8, I recommend the author to add the reference 2206.08390 . In that paper, the authors explain how to “lift” a constant Hamiltonian, and to obtain the Lax operator and R-matrix where the parameter of the Hamiltonian has been fixed to a constant.
15) Pag 16 appendix B1, “R-matrix for the Heisenberg Hamiltonian is proportional to the identity operator,” I think R and R check have to be exchanged.
16) Ref 40 have 2 authors
Previous report
5) About point 5) of the previous report. Thank you for the clarification, now I understand better. Hokkyo’s proof also works for non-difference form model, hence for Hubbard, see also 2501.18400. I think that the main point is the following (which is probably better to add in the text): the connection between the charges and the expansions of the R-matrix has not been worked out so (even if I strongly believe it is true), in principle, it could fail at any step.
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