Lifting integrable models and long-range interactions

We thank the Referees for carefully reading our paper and for their constructive comments.
We believe that we addressed all their comments/questions in the new revised version and that the paper has further improved. We hope that the paper will now be found suitable for publication in SciPost Physics.

Please find below our point-by-point responses to the report. The numbers in the replies correspond to the numbers of the questions/comments by the referees. All equation numbers and footnotes refer to the new version of the paper.

Reply to Referee 1:

1. We addressed this point in the new version after equations (2.4) and (2.16) and additionally in footnote 1.
2. We make it now clear that we refer to perturbative integrability and give a brief definition of what I mean by that.
3. The Sutherland equation, as presented in (2.19) is a system of linear ordinary differential equations on the elements of the Lax matrix $ \mathcal{L}(u) $ with specific boundary conditions. It is also an overdetermined system because for a $ N $-dimensional local Hilbert space while $ \mathcal{L}(u) $ has $ (N^2)^2 $ elements, the Sutherland equation has $ (N^3)^2 $ of them. Assuming that such matrix elements and their derivatives are differentiable (which is the case for all examples in the literature so far), the solution of Sutherland equation is indeed unique.
4. We thank the referee for these questions which allowed us to improve this part of the paper. We had used $ A $ for two operators that meant different things. We now fixed this point by calling one of them $ M $. Furthermore, we now added the correct labels in $ A $ under equation (2.23). We have opted at the moment for not adding an example of $ H^{(1)}_{k,k+1} $. The reasons behind are the following: $ H^{(1)}_{k,k+1} $ is non-zero only for functional Hamiltonians, which are the ones generated by non-difference form R-matrices (i.e. cases where $ R(u,v)\neq R(u-v) $), which are generally more complicated than the difference form cases. So, although it is easy to construct this term in Mathematica, it is not easy to write it in a simple enough way. It is also highly model dependent. With this in mind, we believe that our presentation is clearer in this way.
5. We have corrected this point by putting the Q's in all sections with same font.

Additionally:
A) All typos pointed out by the referee were corrected.
B) About the comment "The implications of the presented findings are not completely conclusive and the usefulness of the discussed results remains a bit opaque which is reflected in the very short section 8. E.g. it is stated that the results "should open the door for applications of the algebraic Bethe Ansatz" but this is not demonstrated explicitly."
Our response: The Algebraic Bethe ansatz is usually based on two ingredients: the Lax operator and the R-matrix. Until our paper those had not been available for the su(2) sector in $\mathcal{N}=4$ SYM except at one-loop order. We are working on the computation of the Bethe ansatz at the moment, but we believe that its explicit computation falls out of the scope of what proposed in this paper. In addition our work opens the door to compute the Lax operator and the R-matrix for other sectors in $\mathcal{N}=4$ SYM, like $ su(1|1) $, for example. We modified the conclusions to address these points.

Reply to Referee 2:

1. We addressed this point in the new version after equations (2.4) and (2.16) and additionally in footnote 1.
2. The definition of P_{ia} was added just after equation 2.5.
3. We have now explicitly written the dilatation operator up to three loops (see new equations (5.1-5.3)) in the notation of reference [9] and commented on the connection with our notation. Please see paragraphs around equations (5.1-5.3). In addition, to clarify the point about higher loop Lax operator and R-matrix for AdS/CFT we modified the paragraph just before the subsection "Higher range" in page 14 by writing it as "The explicit form of $\mathcal{L}$ is not completely fixed and depends on some free functions. We present its explicit form in Appendix B. The explicit form of the R-matrix was not explicitly written here because it is very long. But it can be easily computed perturbatively for up to three loops by plugging the Lax operators up to this order in the fundamental commutation relations. As mentioned before this equation is linear in the R-matrix and therefore simple to solve on Mathematica. We further checked that it perturbatively satisfies the Yang-Baxter equation up to order $ g^4 $."
In addition, we also modified the conclusions to address some of these points.

Reply to Referee 3:

1. The first question of the referee is about the recursive structure of the Lax operator and the R-matrix. Our observations in section 7 were based on our observations and explicit calculations. We found that Ansatz (7.8) is correct and works not only for the models studied in the paper but also in other examples worked out afterwards. However, we were not able to find a nice way to recursively relate the factors of the R-matrix. Take for instance, the easiest example (5.9) which is the R-matrix that includes the NNN term. Its lowest order is given by a product of 4 smaller R-matrices as explained around (5.10). In particular, we have

\begin{equation}
R_{12,34}(u,v) = R_{14}(u,0) R_{13}(u,v) R_{24}(0,0) R_{23}(0,v)
\end{equation}

It is not just a simple factorization, but also a non-trivial choice of spectral parameters. Unfortunately, when we go to higher orders in the coupling constant there seem to be no nice factorization properties and higher orders start to affect lower orders. For instance, for the NNNN term, the lowest order of the $R$-matrix is given by a product of 9 usual $R$-matrices $R_{12}$. This can not be build out of $R_{12,34}$ in an easy way. We agree that it would be important to understand the recursive structures that appear here, but this does not seem to be an easy question and would have to be addressed in future work. We have changed the text in Section 7 to reflect this discussion.

2. We have corrected equations (3.6) and (4.7) as suggested.

Extra: In addition to the points raised by the referees we also corrected a few typos
1. L ->\mathcal{L} in equation (2.5)
2. Corrected the references before equation (5.1)
3. Correct a few language typos