Bootstrapping the $R$-matrix

1. Interesting submission dealing with the characterisation of integrability in one-dimensional spin chains (also applicable to fermion models) with nearest-neighbour interaction.
2. Accessible to a wider audience, not only experts in the field.

1. Improved clarity required in part of the discussion.

1. I have some reservations about the clarity of Sect. 7, and I request that modifications are made before the manuscript can be accepted. In this section, the solution of the Yang-Baxter equation is assumed to have non-difference form, satisfying

\begin{align} &R_{x,x+1}(\nu,\lambda) R_{x-1,x}(\nu,\mu) R_{x,x+1}(\lambda,\mu) \ &\qquad = R_{x-1,x}(\lambda,\mu) R_{x,x+1}(\nu,\mu) R_{x-1,x}(\nu,\lambda). \end{align} In the author's analysis, the solution is expanded as \begin{align} R_{x,x+1}(\mu,\nu)=1_{x,x+1}+\sum_{n=1}^\infty\frac{(\mu-\nu)^n}{n!} R_{x,x+1}^{(n)}(\nu). \end{align}

The author makes the comment

"If there is a solution of $h'$ that makes the LHS of (32) decompose into the difference of two bi-local operators, then the generalized Reshetikhin condition is satisfied, and the Hamiltonian would likely turn out integrable."

where Eq. (32) is given by

\begin{align} &[h_{12}(\mu)+h_{23}(\mu) ,[h_{12}(\mu),\,h_{23}(\mu)]+[h_{12}(\mu),h_{23}'(\mu)] +[h_{23}(\mu),h_{12}'(\mu)] \ &\qquad \frac{1}{2}[h_{12}(\mu),h_{12}'(\mu)] +\frac{1}{2}[h_{23}(\mu),h_{23}'(\mu)] = X_{12}(\mu)-X_{23}(\mu).
\end{align}

It has recently been proposed in

A. Hokkyo Integrability from a single conservation law in quantum spin chains, doi:10.48550/arXiv.2508.20713

that the relation above indeed implies the existence of an infinite sequence of conserved charges in involution. This reference should be included.

Then, the author goes on to make the following claim, regarding integrability of the Hubbard model, in a footnote:

"This serves as a way to determine $h'(0)$ without knowledge of the $R$-matrix, which was believed impossible according to Ref. [31]."

I am happy if the author can prove otherwise, but I do not see any reason to assume that there is a unique solution $h'(0)$ to the following equation \begin{align} &[h_{12}(0)+h_{23}(0),[h_{12}(0),\,h_{23}(0)]]+[h_{12}(0),h_{23}'(0)] +[h_{23}(0),h_{12}'(0)] \ &\qquad \qquad +\frac{1}{2}[h_{12}(0),h_{12}'(0)] +\frac{1}{2}[h_{23}(0),h_{23}'(0)]= X_{12}(0)-X_{23}(0).
\qquad\qquad () \end{align*} where $X(0)$ and the parametric form $h(\mu)$ are both unknown. The operators $X_{jk}(\mu)$ are, apparent in the generalised Reshetikhin condition derivation, functions of $h'(\mu)$, $h''(\mu)$, and $R_{x,x+1}^{(3)}(\mu)$. That these three functions could be bootstrapped from Eq. $(*)$, without explicit knowledge of the dependence on $\mu$, is very speculative.

To be clear, for the Hubbard model the spectral parameter dependence $\mu$ of the local Hamiltonian $h_{x,x+1}(\mu)$ is not in any way related to the familiar coupling parameter $U$ that appears in the Hubbard Hamiltonian; i.e., $U$ is genuinely a constant parameter in the $R$-matrix, (much like $\Delta$ in the $XXZ$ chain) and the spectral parameter $\mu$ provides a two-parameter generalisation of the Hubbard model. By contrast, for example, the two coupling parameters in the Hubbard model analogue described in

J. Links, Extended integrability regime for the supersymmetric $U$ model, J. Phys. A: Math. Gen. 32 (1999) L315 doi:10.1088/0305-4470/32/27/104

are related to two independent spectral parameters.

It appears there is some confusion about the subtle issue of the origin of parameters in integrable models. This needs to be clarified.

Continuing, the reason for the author's comment

"as we have not solved $h'(\mu)$ from Kennedy’s method, or even $h''(0)$ for that matter. "

is puzzling with regard to the significance of $h''(0)$. In general, given a solution $R_{x,x+1}(\mu,\nu)$ and any differentiable function $f$ then $R_{x,x+1}(\tilde{\mu},\tilde{\nu})$ with $\tilde{\mu}=f(\mu)$, $\tilde{\nu}=f(\mu)$, also solves the Yang-Baxter equation without difference property. There is no requirement that $f(0)=0$. Only when the difference property holds is it required that $f$ be a linear function in order to maintain the difference property. So I do not understand the significance in $h''(0)$, over $h''(\mu_0)$ generally for some constant $\mu_0$. I request that the author explains this aspect.

Ask for minor revision

1. The results that the author is trying to prove are particularly interesting and would be useful for the community. However, the arguments he uses are not very clear and require clarification.

1. The bootstrap program is not properly explained; more steps in the proof are needed, as well as further clarification in the examples of Appendix B.
2. The author is ignoring a recent result concerning the proof of the necessary and sufficient condition of the Reshetikhin condition (arXiv:2508.20713).

Even though the potential results of the paper would be particularly interesting, I believe that, in its current form, the paper does not prove the main claims stated in the abstract and introduction. A major revision is recommended.

I will first mention the points where I believe the proof contains gaps, and afterward suggest some minor and stylistic changes (in Requested changes).

Since Section 3 contains the main result of the paper, I recommend several clarifications here:

1) Equation (9) is the main result, however I do not understand how to obtain an expression siminal to (4) for R^{(2m+1)}. Can the author please comment on this? 2) how to see that (9) is the microscopic reason behind the conservation of higher local charges Q^{(2m+1)}? 3) It is not clear what the purpose of the bootstrapping method is. According to the abstract, it should provide a way to reconstruct the R-matrix starting from an integrable Hamiltonian (which would indeed be an interesting result). However, in the text, the author states that it can also be used as a way to check integrability. This second approach has already been addressed in several previous papers: 2206.08390, 2206.08679, 2501.18400, and 2508.20713. The author can for example explain how the R matrix of the models in Appendix B can be constructed. 4) The author mentions that the even coefficient of the R can be read from (4) and the odd from (9) by using the Kennedy's inversion formula. This method is not explained properly. Can the author provides the first several coefficients explicitly? 5) at page 5, the author claims: "These identities are expressed in terms of many new operators... input for an integrability test". How this test wold work since the expression (9) contains infinite terms? As I mentioned, according to A.Hokkyo, (6) is enough to show integrability or non integrability. But supposing this is not correct, how would you use (9)? Even if it works up to some coefficient n, how is it guaranteed that it works always? Section 4. 1) The sentence "Unlike the definitions of classical integrability..." is not clear. 2) As before, "Hence the Reshetikhin condition has been conjectured...". Recently it has been proved by A. Hokkyo. Section 7. 1) About footnote 9: In which step of ref 31. it is believed impossible to determine h'(0) without knowing R? Appendix A. 1) Can the author please comment on how to use eq. (4) to rewrite the LHS of Eq A.1 to obtain (9)? Appendix B. 1) I would put more emphasis on the examples. I think a,b operators (without the superscript) have not been defined. Can you use this relations to obtain coefficients in the R? What is the Kennedy's inversion formula explicitly?

The major changes are discussed in the Report. Here I list the minor and stylistic changes: 1. After equation (1), you could just say that the YBE remains satisfied up to a different normalization of the R-matrix (which corresponds to a shift by a constant term in the Hamiltonian) 2. In the equation (9), I would write some brackets in (a<->b) to make the expression easier to read 3. Appendix A. I believe it would be clearer to manipulate each term on the right-hand side separately and then equate them afterward.

Ask for major revision