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

- A new approach to local conserved charges of integrable models using MPO
- Novel connection to combinatorics

- The connection between the current approach and the usual transfer matrix approach has not been fully discussed.

Report:
The authors present compact expressions for the local conserved charges of the SU(2) and, more generally, SU($N$) invariant Heisenberg chains. Although these conserved charges have been obtained in previous work, they were expressed in terms of the nested product of local operators or using the doubling notation, which may not be simple enough to make further development. The authors tackle this issue in a neat way. The idea is to use the matrix product operator (MPO) approach to express local conserved charges in a concise form. As a result, the authors have constructed the MPO components of each conserved charge explicitly.

The results are certainly original and will be of interest to the readership. However, as the other referees pointed out, they are not substantial enough for publication in SciPost Physics in their current form. Even if the authors transfer the manuscript to SciPost Core, I would suggest the authors address the following issues in the revision.

1. Connections to the standard transfer matrix
As Referee 1 pointed out, the standard transfer matrix itself is an MPO of bond dimension $N$. This should be emphasized in the introduction. I am actually wondering if the authors can obtain the recursive relation between the MPO components $\Gamma^i_k$ ($k=2,3,4,...$) using the boost operator. Naively, the boost operator can also be written in the form of MPO, at the cost of losing the translational invariance. If this is the case, the commutator between the MPO in Eq. (12) and the boost operator can also be written as an MPO. Of course, the subtlety is that the definition of the boost operator is not compatible with the periodic boundary conditions. Nevertheless, I expect that one could still formally use it in the periodic systems, as discussed in Sklyanin's lecture notes [arXiv:hep-th/9211111]. In any case, I suggest the authors consider this issue, as it might make the derivation of $\Gamma^i_k$ more transparent.

2. References for boost operator
The authors might want to cite references for the boost operator in the introduction. According to Sklyanin's review mentioned above, the boost operator was first introduced by Tetelman in [M. G. Tetelman (1982) Sov. Phys. JETP 55(2), 306–310].

3. Generalized Catalan numbers
I have tried to see the precise relation between the generalized Catalan numbers discussed in Refs. [5, 6, 7] and those in the present manuscript, but couldn't. It would be beneficial to the reader if the authors could provide some remarks on how their $C_{k,n}$ is related to $\alpha_{k,l}$ in the papers by Grabowsk and Mathieu.

Another comment: it is quite likely that the identity in Eq. (23) has been known already in combinatorics. I have the impression that $C_{n+m,n}$ can be interpreted as the number of generalized Dyck paths (the ballot number). In fact, I found the following paper: https://www.researchgate.net/publication/282737525_Generalized_Euler-Segner_formulas_of_Catalan_numbers_and_Motzkin_numbers

in which the generating function of $C_{n+m,n}$ is discussed. There is nothing wrong with including the proof of the identity in the manuscript. However, I would suggest that the authors look up several combinatorics textbooks and see if this was known already.

4. Minor comments
- Page 4, the last line: what is $k$ in $i^k$?
- Page 7, 3 lines before Eq. (21): $\Gamma^k_i$ should read $\Gamma^i_k$.
- The manuscript contains several grammatical errors that should be corrected before resubmission.

See my comments in Report.

1- Brings a new result - a disctint way to express the conserved charges of exact integrable models
2- The paper is well written

1- Although correct the addition of the result to the well established area is not
large.

The authors show in a compact from how to express the local charges of the XXX
model and its SU(N) generalization. The construction is interesting, since
the construction exhibit the general structure for distinct models. In general
it is cumbersome to express these charges, using the standard procedure coming form the the transfer matrix derivatives. In this sense the results are
interesting. But, as pointed out by the other referee, the results presented
are not enough for the publication in the SciPost format, as the present submission. Although brings an original result, its addition to the well established
area of exact integrable models is not substantial. Like the other referee I suggest the authors to submit the paper in the SciPost Core.

Suggestion for improvement: An appendix with all the conserved charges, for a small lattice (ex. L=3, or L=4), will be useful for the readers.

1- Novel idea.
2- Relatively good explanations.

1- LPU: Least Publishable Unit.

This work deals with the conserved charges of the Heisenberg XXX spin chain. This is a model whose integrability has been well known for many years. Hans Bethe solved in in 1931, and the algebraic part of the integrability was clarified in the early 80's. However, it appears that new results can be found even today! Recently there has been a renewed interest in the conserved charges in this model and its relatives, with new results emerging starting from entirely new ideas.

I strongly support this type of work: to take an old problem, and to attack it with creative new ideas. Also, I think this paper deserves to be published.

However, I don't think it should be published in SciPost Physics. It appears to be an LPU: least publishable unit. Publishing such a thing is OK if the field is very new, then it is natural that many small results are published separately. However, this field is old, well established. Now if someone wants to publish new results in good journals, then these results should have depth and extent.

I have an offer to the authors: Either they publish this in SciPost Core (which I can accept), or they do a bit of more work, and they add at least one new direction, chosen from these questions:
-Similar MPO for periodic boundary conditions, with a periodic MPO, no boundary term.
-The same result for XXZ, but perhaps even XYZ.
-A similar MPO for the open boundary case, without any extra boundary terms.
-Establishing connections to the standard transfer matrix.

I understand that all of these directions can be difficult. But I do not accept this publication in SciPost Physics, unless new results are added.

The choice is with the authors.

Other small comments:
-In the intro it should be explained that the usual transfer matrix is also an MPO. In fact, these MPO's existed even before MPO's were discovered elsewhere!
-Starting from Section 3.1 it should be explained that this MPO is an MPO with open boundaries, and it should be written in the standard MPO formulation, with boundary vectors for the auxiliary space. Actually, it becomes clear from formula (12) what the authors do, but there should be a more direct explanation. It will be enough to take the sandwhich of the operator on (12) with vectors (1 0 0 0 0) and (0 0 0 0 1) to get H^c. So this is very simple, but it should be explained.
-Also, at this point, it should be explained why this formula works. It is very simple, but it is good to add a short explanation at this point.