Dissipation-driven integrable fermionic systems: from graded Yangians to exact nonequilibrium steady states

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Dear Editor,

I have read the replay of the author, the modifications made and read once again the manuscript. While I can find some improvements my previous judgment of the main manuscript problem stays:

"I find that the presentation of several main objects is poor and somewhere uncertain leaving the reader missing of several operative definitions and so unable to reproduce by himself the material presented in the manuscript."

Once again for me the author has missed the opportunity to use the examples of open quantum models presented (now) in Section 5 of its manuscript to clarify its definitions.

In point I) of his replay for the definition of the "vacua state" the author use some general statement speaking of "extremal-weight states belonging to the irreducible subspaces" and arguing that "This can be considered as a definition, and is sufficient to unambiguously define the solutions.".
Well these "extremal-weight states" are not clear for me and the author should identify them explicitly for any examples of open quantum models presented in Section 5.

I think that this is a minimal clarification required to make the material presented in this manuscript understandable for a general audiance and so in the interest of SciPost and of the author himself.

the paper is technically precise
the results are correct
the paper provides an overview of the subject
the previous results are put into a nice algebraic structure

The paper is hard to read.

This is about the recent high-profile works on building exact nonequilibrium steady states in driven quantum integrable spin chains. The manuscript reviews some of the work, and puts it in a common algebraic framework. The structure identified is based on Yangians.

Again, this is an interesting work, it is important to clarify the underlying algebraic structure in order to eventually connects with advanced concepts of integrability. There is a lot of material, and a lot of interesting ideas.

However, there is no way out of the fact that the paper is somewhat hard to read. Many aspects are developed in different parts, and it is easy to get lost (in the notation, in the concepts, in what it is that the author is trying to do). Perhaps, it it were possible, if there was a section where all aspects were put together in a simple way, the fundamental ideas being explained, so that we could have a big picture, that would be helpful. But I understand that the author has made an effort to improve the text, and this is certainly improved on the first version (I think everything is defined, at least somewhere, even if sometimes we have to search a little bit for the definition).

Here are some small things that I found unclear / inaccurate:

p 7, bottom: can the author be more precise about what a "representation label" is.

p10: the dissipator acting on Omega cancels right the hand side of (20) (that is, it gives a term that equates the negative of rhs of (20); it does not annihilate rhs of (20) in the sense that acting on rhs of (20) with it does not give zero)

p 9 before eq 18: the hamiltonian density does not coincide with the permutation, as in 18 there is an extra factor of derivative of R.

p9: "the differential equation (19)" : should be (18) instead? "A general solution to eq 18" should be eq 17 instead?

p9 in paragraph after eq 18: which one is physical space and which is auxiliary space? please specify

p9 just after eq 19: in expression of ${\cal H}_{\rm aux}$ why is index now $n$ instead of $n+m$?

p10: what does it mean that two copies of a Hilbert space are "mutually conjugate" (just above eq 21)? A Hilbert space is either isomorphic or not to another Hilbert space, but I don't see how it can be conjugate.

p 11 top: "pertains to the conjugate dual" -> does it really mean "is the conjugate dual"? I don't understand the use of "pertain".

p 11 bottom: "a general solution of condition (19)"... but (19) is not a condition, it is a definition.

p12 fig 5: it is a bit strange to put 2A where it is: why not a factor 2 for this term instead?

p14: why are level-0 grassmann numbers? they seem to commute in eq 32.

p14: how is eq 33 linear in z? there is z^{-1}, so it is rather linear in z^{-1}

p 20, footnote 13: where is the Jordan-Wigner transformation used? It is not clear in the main text where it would be involved. Why is it involved in the dissipator?

If possible, a section where all aspects are put together in a simple way, the fundamental ideas and basic formulae being explained, so that we could have a big picture.

1. technically strong

1. still no clear separation between new results and the rest

The author has made modifications to his manuscript. He has reorganized his sections 1, 2 and 3,
and added diagrams to illustrate the main equations. While I find that the quality of the presentation
has improved, the main issue of the manuscript remains: it is still not clear to me what exactly is the novelty
brought by the paper.

What are the important results? What are the equations that the author wants to emphasize? This should
be made obvious to the reader. It should be explained clearly, both in the introduction and in the conclusion,
or perhaps in a separate section that could summarize the results.

For a start, the paper could easily gain clarity if the introduction was focused on the content of the paper,
instead of consisting in a long list of remotely connected topics (e.g., is it really necessary to start with the
traditional sentence about «remarkable progress in experiments in cold atoms», while the
paper is actually about graded Yangians? Wouldn’t it be more helpful and less confusing for the reader
to get to the point more quickly?).

Overall, the new results must be exposed clearly, and contrasted with the ones that are already available in the literature.
For instance, in his reply to my report, the author writes

« We disagree with the referee that having two separate sections, one for the non-graded and another for the graded algebras, would be better. The former are just a special case of the latter when the grading is trivial. We find it much better to treat everything on equal footing. Apart from this, we wish to inform the referee that the su(n) spin chains with this class of integrable dissipative boundaries have been solved before. »

But this is precisely what I am talking about! The fact that the author presents the graded and non-graded cases together,
as if the whole thing was his own contribution, is confusing because the non-expert reader cannot possibly see what is new and
what isn’t. A better, clearer, more reader-friendly presentation, would be to start by recalling the existing result for the non-graded
case, and then explain why it is possible and interesting to generalize this pre-existing result to the graded case.

Since the author did not yet make all the necessary changes to make his manuscript clearer and more accessible,
I am still unsure about the relevance of the results it contains. I am still unable to recommend its publication
on Scipost.

1. re-organize the paper, in particular with a separate section containing a short review of the existing results. Then, in later sections, contrast those results with the new ones in the paper
2. the new version contains many typos, missing words, etc. Please proofread and make corrections
3. I suggest that the introduction be rewritten. Make it less broad, get to the point more quickly.