Matrix product operator representations for the local conserved quantities of the Heisenberg chain

We are very grateful to the referees for the careful review of our manuscript and also grateful to the editors for their arrangement for the peer review of our manuscript.

We made some changes based on the comments from the referees, and please find our replies to individual reports below.

Reply to Report 1 and Report 2

We express our gratitude to the referees for their recommendations regarding the publication of our paper.

Reply to Report 3 by Bernard Nienhuis

We express our gratitude to Prof. Nienhuis for his careful reading of our paper, and we are also grateful for pointing out the terminological ambiguities. We improved the manuscript following your requested changes. The reply to other comments is the following:

I fully agree that it may be useful to write the old results in a new form to make them amenable to new techiques and methods. But to say that the MPO representation is simpler than the old, known form is too much. It certainly requires more explanation.

As you noted, the explanation for the simplicity of our MPO when compared to the old known result was ambiguous. We would like to emphasize the simplicity of the coefficients appearing in the MPO. Our MPO is constructed from only the usual Catalan number. In contrast, the previously known formula for the charges, eq(4) in our manuscript, requires the generalized Catalan number. Finding the regularity of the coefficients in the local charges is generally a challenging task (please see also our reply below). Thus, we believe it is worth mentioning that the coefficients become simple in MPO representation.

We emphasized the point that coefficients are simplified in our MPO representation in the paragraph around the end of Subsection 3.3, starting with "We note that $\Gamma^{i}_{k}$ is only involved with the usual Catalan number,...", and in the sentence in the first paragraph of Section 5, starting with "Especially the coefficients appearing in the MPO are simpler ...".

The matrices are set up such that of the whole matrix product one element is the desired result, and it should still be supplemented with the terms that involve the spins near one end of the chain and also near the other end.

Thank you for pointing out the concerns related to boundary treatment, which makes the expression cumbersome. However, in the infinite chain case, the boundary terms in the local charges become irrelevant, and it is enough to consider only the bulk term.

We have added the explanation about the case of the infinite chain in the middle of subsection 3.3, in the new paragraph starting with "For the case that the Hamiltonian is defined on the infinite chain, ...".

I find the paper well written. In my opinion the paper certainly deserves publication. I find the result of the paper a significant achievement. Its usefulness is yet to be demonstrated. The expectations for SciPost Physics Core are (in my opinion) not quite met: 1) It can be argued that the rewriting of the local charges is an important problem, to make them accessible for applications. It has been addressed with appropriate methods and above average originality.
However, 2) in my opinion it does not meet the expectation of significantly advancing knowledge or underanding of the field.

Thank you for recognizing the significance of our result and considering our paper worthy of publication.

One of the advantages of our MPO representation is that it may give a new way to predict the regularity of the local charges.

For almost all interacting integrable systems, the coefficients appearing in the local charges are so complicated that we cannot infer their regularities. For example, the explicit expressions for the local charges in the XXZ chain had been a mystery for about thirty years since the progress of the isotropic XXX cases. This problem was recently solved utilizing the Temperley-Lieb algebra in your seminal paper[Nienhuis, Huijgen, 2021, J. Phys. A: Math]. We think that, in terms of the Temperley-Lieb algebra, the regularity for the coefficients becomes more straightforward than their bare expression in the spin operator, and this simplification follows the discovery of the general expressions for the charges.

In this spirit, to predict the regularity of local charges, it may be helpful to develop an alternative way to express local charges. In fact, in our case, the coefficients have been simplified, as noted in the above reply. Although, in this paper, we treat the XXX chain where the structure of the local charges has been well understood, we believe our strategy will offer a new way to analyze the regularity of unknown charges. Detailed investigation on this topic goes beyond the current paper, and we left this study as a future problem, which is stated in the summary, in the paragraph starting with "Our strategy may be useful to find the general expressions...".

Reply to Report 4

We express our gratitude to the referee for his/her careful review of our previous resubmission and helpful suggestions.

The reply to the referee's request is the following:

In fact, one can rewrite $Q_k B − B Q_k$ as

where $a$ denotes a newly introduced auxiliary space. Then $Q_kB−BQ_k$ can be thought of as an MPO with an enlarged auxiliary space. (I know the authors consider MPOs with open boundaries. But just for illustration, I used an MPO in the form of trace. Actually, one can express MPOs like equation (22) in this form by inserting a boundary matrix.) Since this procedure doubles the dimension of the auxiliary space, I am not sure if this is the best way. But perhaps, the authors might want to examine whether this trick helps simplify the derivation.

Thank you for the insight into the relation between the boost operation and MPO. We followed the method presented by the referee to obtain the MPO representation recursively using the boost operator. For simplicity, we consider the infinite chain case where the boundary effect becomes irrelevant. We added the explanation in the new subsection 3.4, starting with "One may think we can obtain the recursive relation ...". We also noted that boost-derived charges are slightly different from our $Q_k$ in the linear combination of the lower-order charges, and subsequently, the expressions of the MPO are also different.

* Added a remark on the MPO representation of the local charges of infinite chains in Subsection 3.3.
* Added a new subsection (Subsection 3.4) to explain how to construct the MPO recursively with the boost operation.
* Added an explanation about the advantage of our MPO representation of the local charges in Section 5 and Subsection 3.3.
* Some typos and confusing expressions have been fixed.