On quantum separation of variables beyond fundamental representations

1- extension of SoV approach to $Y(gl_2$) quantum spin chains with mixed representations
2- explicit construction of bases suitable for SoV characterization of the spectrum
3- construction of the $Q$-operator in terms of fused transfer matrices

1- little discussion of relation / unique features of the different constructions
2- notation sometimes confusing
3- English grammar could be improved

Building on their previous work on the complete characterization of the spectrum of integrable quantum chains built on fundamental representations of the Yangians $Y(gl_n)$ and deformations thereof within the separation of variables (SoV) approach to the rational $Y(gl_2)$ spin chains with mixed local higher spin representations and quasiperiodic twisted boundary conditions.

They begin by recalling the construction of these models within the quantum inverse scattering method, in particular the fusion hierarchy of transfer matrices from monodromy matrices with different representations in auxiliary space.

In the following they present three ways to construct a basis of states suitable for the SoV.

(1) following Sklyanin's approach they obtain the quantum separate variables and introduce a covector eigenbasis of the $B$-operator from the monodromy matrix with two-dimensional auxiliary space. The action of the $A$ and $D$ operators on these basis states is given explicitly. Although this not explicitly stated, this allows to formulate the spectral problem in terms of a separable set of linear equations on a lattice with $\text{dim}\mathcal{H}$ sites -- as usual for the SoV in Sklyanin's formulation.

Using the methods developed in their earlier papers the authors construct two additional SoV bases:

(2) by action on a generic covector with the 'fundamental transfer matrices' obtained from monodromy matrices with auxiliary space isomorphic to one of the local quantum spaces. Here the spectrum of the transfer matrix is determined by a system of polynomial equations for $N$ parameters which allows to prove completeness. In fact, these equations are equivalent to the formulation of the spectral problem obtained in (1).

(3) a third 'natural' SoV basis is obtained by action on a generic covector $\langle S|$ with another combination of fused transfer matrices. For a particular choice of $\langle S|$ it coincides with the basis obtained in Sklyanin's approach (1). In this formulation the action of the transfer matrix is straightforward to compute giving the same formulation of the spectral problem as in (1).

Finally, the authors reformulate the spectral problem in terms of a 'quantum spectral curve' (i.e. Baxter type $T$-$Q$ equation) with polynomial $Q$ and reconstruct the $Q$-operator in terms of elements of the monodromy matrix. This allows to generate a SoV basis using the $Q$-operators. Again, a particular choice of a starting covector allows to reproduce the basis (1).

In summary, the authors provide interesting results which have the potential to extend the scope of the SoV approach to the investigation of integrable quantum chains.

Parts of the paper could be more accessible though if the authors would provide more links between the different approaches introduced. Furthermore, the use of notation without introducing it before or never using it again (as with the functions $\alpha$ and $\beta$ in (5.3)) can be obfuscating.

1- it should be mentioned that $P_{ab}$ in (4.4) is
the permutation operator on the spaces involved.

2- Eq. (4.3) does not make sense

3- the role of Proposition 4.2 in Section 4.1 is unclear. If
needed at all, it would be better placed in Section 2.

4- the relation of the states $\langle 0|$ (3.7) and $\langle S|$ (4.20)
used in the constructions (1) and (3) of the SoV basis should be discussed.

1- Interesting results: extension of authors' new SoV approach beyond fundamental rep, with a 'natural' SoV basis on which the transfer matrix acts in a simple way; contact made with Sklyanin's SoV and Baxter's $TQ$ equation; explicit construction of $Q$-operator
2- Contains review of the necessary background, so essentially self contained (except for relying just a few times on some proofs in, I think, especially ref [1] from the submission)
3- Quite clearly structured

1- English grammar

In 2018 the authors found a new approach to Sklyanin's separation of variables (SoV) within the framework of the quantum-inverse scattering method. The present paper, which is already the fourth in this series of works, deals with quantum-integrable lattice models that go beyond the fundamental representation: for simplicity considering the case of a rational R-matrix and rank one, the authors extend their new approach to models with higher-dimensional representations (obtained by fusion) associated to the lattice sites, where the dimensions $2s_n +1$ ($1\leq n\leq N$) may vary from site to site. To ensure a simple spectrum the inhomogeneities and twisted boundaries need to be generic to ensure a simple spectrum.

If the auxiliary space is $V_a \cong \mathbb{C}^2$ let's call $A+D$ the 'auxiliary transfer matrix' (cf the notation in ref [12] of the submission), and use 'fundamental transfer matrix' for the case $V_a \cong \mathbb{C}^{2s_n+1}$ for some $1\leq n\leq N$.

The authors review Sklyanin's SoV when the twist is not proportional to the identity, so that its off-diagonal entries can be made nonzero by $\mathfrak{gl}_2$-invariance. Then the repeated action of the $A$-operator (evaluated at paticular shifts of the inhomogeneities) gives an explicit construction of a (left) $B$-eigenbasis on which $A$ and $D$ have a simple action.

Next, following the authors' work for fundamental models, a SoV covector basis of the Hilbert space is constructed by the repeated application of fundamental transfer matrices (evaluated at the same shifts of inhomogeneities). In this setting the spectrum of the auxiliary transfer matrix can be characterised ('SoV discrete characterisation') by $N$ polynomials determined in terms of solutions to a system of polynomial equations (for the spectrum at those same shifts of inhomogeneities). Moreover, for each such polynomial there exists (up to normalisation) a unique (left) eigenvector of the auxiliary transfer matrix with factorised components with respect to the SoV covector basis.

In addition the authors construct a new, second ('natural') SoV basis, where the $h_n$-fold action of the fundamental transfer matrix (with $V_a \cong \mathbb{C}^{2s_n+1}$) is replaced by something simpler: a single application of the fused transfer matrix (evaluated at the same value) with $V_a \cong \mathbb{C}^{2s_n - h_n +1}$ of intermediate dimension. (One can't help but think here of the work of Ilievski et al on quasilocal charges, where the tower of fused transfer matrices also play an important role.) The relation of this construction with Sklyanin's SoV covectors is given. The auxiliary transfer matrix has a simple action (as is easily seen from the fusion relation). As before its eigenvalues has a 'discrete' SoV characterisation in terms of a set of polynomials depending on solutions to a system of polynomial equations; and the corresponding eigenvectors have factorised components with respect to this covector basis too.

The authors go on to reformulate their preceding 'discrete characterisation' of the auxiliary transfer-matrix spectrum in a functional equation of Baxter $TQ$-type (the 'quantum spectral curve'). The factorisation of the components of the eigenvectors features the same polynomials $Q$. Furthermore, the corresponding $Q$-operator is explicitly constructed in terms of fused transfer matrices. Finally the natural SoV covector basis is given in terms of this $Q$-operator.

***

These results are certainly very interesting. The proofs look correct. (I have not gotten around to check the proof of Theorem 5.1, or those from ref [1] in the submission that are used, in detail.)

My main point of criticism is just that the presentation would benefit from improved grammar (with some concrete examples below), perhaps with the help of feedback from a native speaker, to help the reader focus on the contents. Below I give more detailed comments.

1- Abstract: "...two different SoV basis" -> "...two different SoV bases".
2- Here "the Sklyanin's approach" looks ungrammatical to me; e.g. -> "the approach of Sklyanin" or -> "Sklyanin's approach". Likewise for various similar phrases elsewhere.
3- (Optional) p1: perhaps "secular equation" -> "characteristic equation" is more standard?
4- p4: "$Y(gl_{n\geq 2})$" looks somewhat sloppy to me; I'd prefer "$Y(gl_n)$, $n\geq 2$". Likewise for similar notation elsewhere.
5- p5: "bi-dimensional" -> "two-dimensional" (and I believe there are several occurrences of this)
6- same paragraph: to distinguish between the different transfer matrices perhaps consider introducing some name also for the non-fundamental case (which I chose to call 'auxiliary' above)
7- p6: Is the phrase "associative and commutative algebra" used for a particular reason? Something like "commutative (associative) algebra" sounds more natural to me.
8- below (2.1): "spin-$s$" -> "higher-spin" or "spin-$s_n$"
9- below (2.6): "...of rational..." -> "...of the rational..."
10- (2.7): for more uniform notation, cf (2.3), consider including a subscript 0 for the matrix
11- below (2.7): "where we have define ..." -> "where we have defined ..."
12- (2.9): perhaps superscript $K|1$ here? The current notation is really only introduced in (2.22)
13- (2.10) and elsewhere: the authors could consider using "\mathrm{qdet}" rather than the current subscript $q$, cf the (different) notation in e.g. (4.17)
14- below (2.10): "central elements" -> "central element"
15- below (2.15): "it is an" -> "is an"
16- (2.20): perhaps add "for all $\lambda, \mu$" too
17- (2.21) and elsewhere: to improve readability perhaps use parentheses around the argument of the quantum determinant
18- (2.23): mention that $k_1, k_2$ are the eigenvalues of $K$ (currently done on the top of the next page)
19- above (2.26) and elsewhere likewise: I would prefer writing out abreviations like "w.r.t."
20- (2.30): could avoid introducing $\theta$ (which I think is only used here?) by adding a few words.
21- perhaps move the sentence below (2.33) to just above (2.31)
22- below (3.1) and elsewhere: "it exists" -> "there exists"
23- bottom p10 and elsewhere: "zeros" -> "zeroes"
24- (3.5) and (3.6): is there an inverse missing on the second $W$? Also, spend at least a few words on the $\mathcal{W}$s; these are defined by (3.5)?
25- below (3.7): "opf" -> "of", "local left references states" -> "local left reference states"
26- Thm 3.2: "are verified" -> "is verified", "non proportional" -> "not proportional". Perhaps "..._{Sk}" -> "..._\text{Sk}" here and elsewhere.
27- above (3.23): "is $B^{(K)}$-eigencovector" -> "is a $B^{(K)}$-eigencovector"
28- above (3.27): "being it" -> "it being"

From now on I will only suggest grammatical corrections if I believe they would prevent unclarities.

29- Prop 4.1: Presently the underline has a different meaning for the Sklyanin SoV (Sect 3) and the SoV of the authors (Sect 4). I think this is somewhat confusing, so perhaps one of these underlines can be changed to some other notation.
30- (4.3): $h$ seems to be missing (as an exponent?)
31- above (4.4): "same lines that in" -> "same lines as in". (In fact, I believe "follows the same lines" is not the standard formulation; -> "follows" or "is/goes/proceeds/... along the lines of" or "is the same as" or "is similar to". This comment applies elsewhere too.)
32- (4.5), (4.6) and following sentence: Lax operator instead of $R$-matrix?
33- (4.9): I think the first "$t(\lambda):$" and the "$\forall$" can be removed?
34- below (4.9): "solutions to" -> "that solve" or "solving". (By the way, here and elsewhere: "$N$-uple" and "$N$-upla" -> "$N$-tuple".)
35- below (4.10): perhaps start sentence with "each of which is"
36- (4.12): is this $D_{t,n}$" with superscript the same as that without?
37- Am I correct that (4.10) is the SoV analogue of the BAE? If so, are they easier to solve? (The counting of their solutions clearly is.) How would they compare for the TBA?
38- Top of p17: reference should be in the references, not a footnote
39- Also, what is a 'component' of a polynomial?
40- below (4.15) and elsewhere: I think that the $\bigotimes$ should be a Cartesian product?
41- Thm 4.2: "The set" -> "The set consisting of"; after all, (4.19) is not a set
42- (4.19) and elsewhere: can the parentheses in the power of $k_2$ be dropped? Looks like a superscript now.
43- p19: current notation is a bit confusing to me; I don't think there's any need to have $\bar{h}$s, $h$s, as well as $\hat{h}$s. Can't we just assume the result holds for some $h_1,\cdots,h_N$ (or all with bars) and consider $h_1,\cdots,h_n-1,\cdots,h_N$ (then also with bars) for arbitary $1\leq n\leq N$, and then with $h_1,\cdots,2s_n,\cdots,h_N$ (or $h_n=2s_n$ if you prefer, and use bars elsewhere).
44- Top of p21: "invariant w.r.t." -> "independent of"
45- (4.41): perhaps good to repeat the meaning of the notation $t^{(2s_n-h_n)}$ (with the superscript) from the proof of Thm 4.1, see the line below (4.16).
46- (5.1) is a bit unclear to me: should the "such that ..." go perhaps to the next line, or "for any $(a,b)\in ...$" to the (4.5)?
47- (5.2) perhaps it would be useful to give the explicit form, using (2.11), too, as this might be more familiar to the reader (and is used in the proof)
48- below (5.13): "trigonal" -> "tridiagonal"
49- p25 and elsewhere: "unicity" -> "uniqueness"
50- above Cor 5.1: "monodromy matrix" -> "transfer matrix"?
51- Cor 5.1: I assume that the meaning of "it is a $Q$-operator family" is what's explained at the end of the corrolary?
52- Sect 5.3: the first sentence is a bit hard to parse at the moment.
53- Around (5.45): what are the precise conditions on $\langle L|$ and other assumptions needed to construct the SoV basis from the $Q$-operator?
54- (5.46): perhaps refer to (4.20) and/or write $\langle 2s_1 ,\cdots 2s_n|$ instead?
55- Final paragraph: is this statement new, or can it already be found in the literature?
56- If I'm not mistaken, cf ref [12] in the submission, in the ABA for higher spin the auxiliary transfer matrix gives the BAE, while the fundamental transfer matrix the Hamiltonian and energy. What about the ('discrete' and 'functional') SoV characterisation of the spectrum of the fundamental transfer matrix, especially in the case where all $s_n$ are equal to some fixed spin $s$?