Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard models

1-The paper provides extremely comprehensive, with few exceptions, literature review
2-The authors do a great job in providing crystal clear unambiguous definitions of all objects they study. Given notorious sign issues in supersymmetric systems, this is much appreciated
3-Derivation of the inner boundary condition
4-Reconstruction of fused transfer matrices in terms of the fundamental one

1- The authors do not attempt to interpret their results in terms of supersymmetric Q-systems although it will be potentially beneficial for several reasons. There is a good chance that various formulae simplify in terms of Baxter Q-functions, and there will be a stronger connection with the AdS/CFT integrable system whose spectrum is concisely encoded in terms of a quantum spectral curve (which is a QQ-system).
2- References to recent developments related to the AdS/CFT spectral problem and supersymmetric Q-systems are missing (some suggestions will be given below). This exaggerates to some extent the role of the Hubbard model in application to the AdS/CFT integrability (Hubbard model works in the large volume asymptotic regime only).
3- Although the authors did a great job to make their article detailed and easily understandable, it would be appreciated to have a bit more of cross-references across the text, and occasionally replace phrases like "previous theorem" and "our corollary" with more exact "theorem 2.1" etc. This will make the paper more accessible to those readers who want to understand one particular statement instead of reading the entire article. Also, there are small issues with English grammar, I would suggest proofreading the manuscript.

Separation of variables is an important tool in studies of integrable systems. This article is one of the first works devoted to the development of this tool for the case of supersymmetric systems.

The article offers two main results. The first result is the construction of an SoV basis for the case of twisted $GL(M|N)$ spin chains in fundamental representations and the Hubbard model. The idea is to systematically act with the fundamental transfer matrix on a reference state. While it is a rather straightforward generalisation of the proposal and proofs in [1] (by two of the authors), it was important to explicitly show whether and how it works in the presence of supersymmetry. The second result is to encode the spectrum of the model in terms of certain relations that are imposed directly on the fundamental transfer matrix. This result is conjectured in the general situation and proven in some particular cases (where also completeness is checked/proven for generic values of inhomogeneities).

Although a lot of work is still required in the SoV study for supersymmetric systems, this article provides one of the first stepping stones and sets up a stage for further studies.

There is no doubt that this article should be published. It is original, of scientific significance, comprehensive, and well written. I have only some minor remarks and would like to ask the authors to address them while editing the final version of the manuscript.

Although the list is quite long, all of the points are minor. It is often up to authors' judgement whether the proposed changes are worthwhile to implement

Citations:

1- References [151],[158] do $gl(3)$ case only
2- [157] has any compact representation, not only rectangular ones.
3- Given that your citations are comprehensive, citing also 0706.3327 for Analytic Bethe Ansatz is probably worthwhile.
4- For Q-operators in supersymmetric case, also 1012.6021 in addition to what you have currently
5- In your review of supersymmetric spin chains (page 5/6), functional QQ-relations got explicit practical applications for these chains: 1608.06504, 1701.03704.
6- In relation to AdS/CFT: Review paper [136] is outdated, especially in relation to the spectral problem. Let me stress that the Hubbard model is applicable in the large volume asymptotic regime only (i.e. the sentence citing [136, 143] is a bit strong). The full spectral problem is based on $psu(2,2|4)$ symmetry and was solved by means of the quantum spectral curve: 1305.1939, 1405.4857. In reviews 1708.03648, 1911.13065 you can find more citations, while detailed relation of QQ-systems and QSC which is likely to be important for SoV is discussed in 1510.02100
7- If you are aware of any work where an analogue of inner boundary condition is written, please cite it. Currently, you cite [188,189,232] but it is not clear what exactly should the works be credited for.
8- Together with [189], this one - https://doi.org/10.1023/A:1025048821756

Statements/formulae in the article:

1- Twisted Yang-Baxter algebra can have different meanings (like twisted Yangian). So term is used a bit loosely
2- (2.9) is not a property of $gl(m|n)$ Lie superalgebra but of its fundamental (vector) representation.
3- Since dagger is often associated with Hermitian conjugation, it would be worth specifying in a footnote what l.h.s. of (2.18) means precisely.
4- Is it necessary (for immediately what follows after) that matrix (2.34) is restricted to the block-diagonal form?
5- Second paragraph of section 2.2, you presumably mean substraction of sets in the definition of a bidimensional lattice, then backslash sign.
6- “This orientation is consistent with the one used in [188]” -> This is false. Direction a is vertical in [188] and b (called s, for symmetrisation) is horizontal.
7- Strictly speaking, (2.49) is not satisfied in the (0,0) corner of the fat hook, provided convention (2.51)
8- "Let K be a (M+N )×(M+N ) square matrix solution of the gl(M|N) -graded Yang-Baxter equation of the block form (2.34),..." - from the context, it seems that K is a constant twist matrix. Why do you then say that it solves Yang-Baxter equation? It of course does, but in a very trivial way
9- In (2.90), l.h.s. is not defined
10- The authors might consider making title of section 3.2.2 more precise. Completeness is established for a specially degenerate twist, (and for generic enough inhomogeneities). Only after I read other parts of the paper could I understand that. For a reader who is skimming through, impression is that a much stronger statement is proven.
11- p.22 “in this case the transfer matrices T (K) are invertible” - please refer to a place in the text where it is demonstrated
12- In Conjecture 3.1, you say “Taken the general twisted…”. What these words mean precisely? Should we understand “the general” as “a” or “any” (hence you expect the conjecture to hold always), or “a generic” (hence you expect the conjecture to hold almost always, except for some bad values of twist, and there is no means to control which are these values, we only know that they are rare)
13- Based on the text, I understand that you give, on p.22, an argument in favour of the conjecture 3.1. Can you state precisely why is this argument not a proof yet? It seems you show that every solution (2.108), (2.109) gives $t_1$ as an eigenvalue. The other direction is obvious (Lemma 2.2). So it seems that both directions of iff in conjecture 3.1 are proven, but you don’t disclose all the details on p.22.
14- In the last paragraph of the proof to Theorem 3.1, you state that transfer matrix is diagonlizable which follows from Theorem 2.1. I presume you meant Proposition 2.1.
15- If I understood correctly, Corollary 3.1 relies on Theorem 3.1. Since Theorem 3.1 was stated a while ago in the text, could you please add “for almost any values of the inhomogeneities” in the text of the corollary?

Potential Grammar/typos noticed (it can be also a question of personal taste of course):

1 - Construction "A is something. While B is something else" with point is regularly used (sounds somewhat strange)
2- If eight-vertex model (with hyphen) then two-dimensional space, one-parameter deformation etc
3- "...of the separate variables..." -> of the separated variables
4 - "...a natural framework where to prove..."
5 - "...where the holding different representations..." (I do not understand the sentence)
6 - "Object that have a well defined parity, either even or odd, are called homogeneous" (objects or is called)
7 - "All these transfer matrices commutes" (commute)
8 - In conjecture 3.1: “define above -> defined above”
9 - In theorem 3.1: "...then the eigenvalue spectrum..." -> to drop "then"

1- The paper is very well written, and largely self-contained
2- It contains new relevant results on a topic of renewed interest in the integrable systems community. The potential applications of these methods range from condensed matter to AdS/CFT
3- The bibliography and review of the literature is overall comprehensive and balanced.

1- I feel that some references should be added to other approaches to the completeness problem, and a short discussion on how they compare to the approach of the authors (more comments on this below).
2-Often, the authors refer to arguments in their previous works to shorten the proofs of theorems. The references are very precise, however occasionally I feel that a few more words of explanation should be offered to help the reader follow the logic (I will point out one example below).
3- The review in the introduction of developments in AdS/CFT integrability is in my view incomplete (more comments below).

The paper deals with the separation of variables method in quantum integrable systems, continuing a series of important develoments in the area in the last years, by the authors and other groups.
The paper addresses the construction of separated variables bases for
supersymmetric rational spin chains at any rank, as well as the Hubbard model. This is done applying a method recently developed for the non-supersymmetric case by two of the authors in [1] , whereas the separated variables basis is built by repeated action of the transfer matrix evaluated at special points on a reference state.
After constructing the basis, the authors use it to prove some spectral properties of the transfer matrix, such as simplicity of the spectrum and diagonalizability (under some conditions on the twists).
They also emphasize the role of a nice and simple quantization condition for the transfer matrix eigenvalues (generalizing the "quantum determinant" of the bosonic case), and conjecture that it gives a complete characterization of the spectrum. They prove this statement for a particular choice of twists and superalgebra, providing an explicit example of the construction.
The paper is very well written, and the results are relevant for developing the SoV strategy to the computation of observables in supersymmetric spin chains and AdS/CFT. I have no doubt in recommending the paper for publication. However, I kindly ask the authors to consider some minor corrections detailed below.

REQUESTS OF CLARIFICATIONS AND BIBLIOGRAPHICAL SUGGESTIONS:
1. When discussing completeness, in particular in the Introduction and section 2.5, I would ask to include a reference to the works on completeness based on the Wronskian QQ relations. In particular, to the best of my knowledge the work math.QA/1303.1578 is considered as a proof of completeness of the Bethe Ansatz for the gl(N) XXX spin chain. I also mention the related recent work math-ph/2004.02865 which presents a proof of completeness for the supersymmetric case (of course, this work appeared after the work of the authors, therefore I leave it up to them whether to include this reference).
In general, I would really appreciate seeing a short discussion of links and differences of the authors approach with the QQ relations approach.
2. In the second paragraph on page 6 , when talking about generating states with a single, non-nested B operator evaluated at the zeros of the Q function, I would ask to add a citation to [150], where this property was first discovered and the form of B conjectured for any rank.
3. I would like some points to be clarified in the discussion of AdS/CFT integrability, and I kindly ask the authors to take into accounts the following comments. First, the AdS/CFT S matrix is not equivalent to the Hubbard R-matrix, but to two copies of the latter, multiplied by a (nontrivial) dressing phase. Second, the direct relation between the dilatation operator and the Hubbard Hamiltonian is only valid at weak coupling (and in a special sector). Third, spin chain approaches such as those in [148]-[149] are nowadays known to miss one important part of the result for the AdS/CFT spectrum, the so-called wrapping corrections, and have been surpassed by approaches based on the TBA. Finally, I think the authors should include in this discussion the fact that a full solution for the spectral problem of AdS/CFT has been proposed in hep-th/1305.1939 (this has been tested extensively and is regarded as one of the main results in this area).
4.- The study of the Hubbard model in a paper dealing mostly with supersymmetric spin chains could be motivated more strongly. I believe an explicit link exists since the R matrix of the Hubbard model has a su(2|2) invariance (this is e.g. discussed in reference [132] or in math-ph/1401.7691). It could be useful to make this comment.
5. Since the authors cite the fusion hierarchy for the Hubbard model, I point out to them the paper hep-th/1501.04651 where the hierarchy of functional relations and the related quantization conditions are discussed in the finite temperature case.
6. In the proof of Th. 2.1, when mentioning "special limits" used in the proof, it would help the reader if the authors recalled explicitly the limit of large inhomogeneities.
7. Can the authors elaborate on the name "quantum spectral curve" for eq. (3.26)?
8. I feel it would be better to name the zeros of \phi in (3.25) \mu rather than \lambda, consistently with (3.38) and (3.58).
9. This is a simple suggestion. To make contact with other SoV approaches using the B operator, it may be relevant to remark that (3.29) is a rewriting of the wave functions in terms of Q functions.

SOME TYPOS AND NOTATION ISSUES:
10. In (2.90) since this is a covector, should it belong to V_a^* ?
11. I believe there is a typo in (2.68), namely the signs of the shifts in the arguments of the the T functions should be reversed.
12. For notation consistency, in formula (2.36) should it be M^(K) ? Otherwise, I believe the superscript ^(K) comes in (2.43) with no previous explanation.
13. The notation \xi^(n) is used only in two equations. For clarity I would advise writing the shift explicitly in (3.59), since the definition of this notation is difficult to spot.
14. Some typos: "ortogonality" should read "orthogonality" in a couple of places. Another typo is "antisymetric" on page 7. Finally, a typo in one of the names in ref. [135].
15. Below (3.54), the degree of the polynomial \bar \phi is given as M \leq N- m_h. However for consistency with (3.25) , it looks like it should be M- m_h instead.
16. The footnotes 10 and 11 are identical.
17. A power of \lambda is missing in eq. (A.7)
18. It looks like (B.9) and (B.18) are repetitions of (B.8) and (B.17).

The separation of variables is potentially the most powerful method for investigation the off-shell behaviour of the quantum integrable models. In the paper by J.M.Maillert, G. Nicolli and L. Vignoli the problem of separation of variables is solved for models with SUSY. This case is important for studying the Hubbard model, the case considered in details in the paper, another important application can be provided by AdS/CFT correspondence. This is a highly professional and well-written paper, I do recommend it for publication.