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Introduction to the nested algebraic Bethe ansatz
by N. A. Slavnov
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Authors (as registered SciPost users):  Nikita Slavnov 
Submission information  

Preprint Link:  https://arxiv.org/abs/1911.12811v1 (pdf) 
Date submitted:  20200422 02:00 
Submitted by:  Slavnov, Nikita 
Submitted to:  SciPost Physics Lecture Notes 
for consideration in Collection: 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a $\mathfrak{gl}_3$invariant $R$matrix as the basic example, however, we also describe possible generalizations. We give recursions and explicit formulas for the Bethe vectors. We also give a representation for the Bethe vectors in the form of a trace formula.
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Submission & Refereeing History
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Reports on this Submission
Anonymous Report 4 on 2020625 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1911.12811v1, delivered 20200625, doi: 10.21468/SciPost.Report.1782
Strengths
These lecture notes provide a detailed and accessible description of the method of the Nested Algebraic Bethe Ansatz for higher rank integrable quantum chains, where different representations of Bethe vectors are shown to be equivalent.
Weaknesses
The introduction is in a draft status, some account of the mathematical and physical context should be detailed in order to motivate and ease the topic for a general audience.
Report
The manuscript contains a detailed description of the method of the Nested Algebraic Bethe Ansatz (NABA) for higher rank integrable quantum chains. Despite the algebraic complexity of the subject, I found it very wellstructured and easily accessible. This is not a surprise being the author one key player in the development of several subjects here presented. The author mainly focus his presentation of NABA method for the models associated to the fundamental representation of the rational gl(3) invariant Rmatrix also allowing for nontrivial gl(3) gauge transformations. The onshell and offshell Bethe vectors are constructed and different representations of them are shown to be equivalent and their specific interest pointed out. An interesting example is the trace form of Bethe vectors used to show their symmetry over the corresponding arguments. The gl(3) case is then used to explain the similarities or further complexity emerging for the representations of the Bethe vectors for the higher rank gl(n) cases.
Aside this very positive judgment on the manuscript (in particular on its potential to be accessible to nonexperts), I find that it misses some important explanations of the mathematical and physical context. The author has been probably motivated by the aim to keep the presentation concise. However, I found that this results in the absence of fundamental existing results in the literature (some of them derived or coauthored by the same author), whose presentation is central to motivate the relevance of the topic for a general audience.
In the following I will list some nonexhaustive examples which from my point of view should be introduced to overcome the mentioned absences.
i) Some key references should be added when the author list in the introduction the applications:
“They include condensed matter physics … supersymmetric gauge theories, etc.”
ii) I think that when referring to the qdeformed Rmatrices in section 1.3 the author should also cite the fundamental papers:
M. Jimbo. A qdifference analogue of U(g) and the YangBaxter equation. Lett. Math. Phys., 10:6369, 1985.
V. G. Drinfel’d. Quantum groups. Proceedings of the International Congress of Mathematicians Berkeley, USA, 10:798820, 1986.
which together with the paper [4] are at the origin of the quantum group mathematical subject.
iii) The introduction of section 2 is important to motivate the study of offshell Bethe vectors and NABA in general. However, it misses some fundamental elements to make it intelligible to a wide audience. I suggest that some further statement based on the existing literature should be introduced to explain why and how the action of a (quasilocal) operator over an onshell Bethe vector can be computed as linear combination of offshell Bethe vectors. The reconstruction of local operators by matrix elements of the monodromy matrix dressed by products of transfer matrices, as first derived in Nucl. Phys. B 554 (1999) 647678 and in Nucl.Phys. B575 (2000) 627644 (even for the higher rank case), is fundamental at this aim. These local operator reconstructions also allow to point out the relevance of the form of the Bethe vectors in the algebraic version of Bethe Ansatz, as being written in terms of actions of offdiagonal elements of the monodromy matrices on reference states. Indeed, they are essential to compute algebraically, by using the YangBaxter commutation relations, the action of local operators over onshell Bethe vectors.
As well the known results on the scalar products of onshell times offshell Bethe vectors allowing to compute (2.2) should also be mentioned. These types of results are at the basis of the success of the algebraic version of Bethe Ansatz to go beyond the spectrum description and compute exactly fundamental object like correlation functions. The author himself has given central contributions to some of them and I think that their description starting from the gl(2) up to the gl(3) case can bring the required clarification on the relevance of the topic to a wide audience.
iv) To clarify that the definition of the offshell Bethe vectors admits some degree of ambiguity, the author refers to the papers [24,25] for the gl(2) case. Indeed, there is presented an example of different representations of Bethe vectors (by gauge transformed monodromy matrix elements) which are equivalent for onshell Bethe vectors. One should however mention that the paper [24] is mainly devoted to the higher rank case and that one of its main features is to predict a nonnested form of the onshell Bethe vectors. This nonnested higher rank form is still of the type (2.7) in terms of the Boperator of the Sklyanin’s quantum separation of variables for some twisted boundary conditions. The first proof for the gl(3) case that these are indeed transfer matrix eigenvectors and so onshell Bethe vectors has been done in the NABA framework by the same author with Liashyk in the 2018. While in the paper “J.Math.Phys. 60 (2019) no.3, 032701” a proof is given for the general higher rank cases in the framework of the quantum separation of variable developed in “J.Math.Phys. 59 (2018), 091417”.
As one aim of the manuscript is to present different representations of Bethe vectors occurring in the gl(3) case it seems natural to me to point out also the existence of this nonnested one and the associated literature.
Requested changes
I strongly recommend the publication of these lecture notes once the underlined missing descriptions i)iv) are covered.
Anonymous Report 3 on 2020613 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1911.12811v1, delivered 20200613, doi: 10.21468/SciPost.Report.1759
Strengths
1.) Detailed pedagogical introduction, accessible by students and beginners in the field.
2.) Written by an expert who himself contributes significantly to the development of the method.
Weaknesses
1.) No obvious weakness.
Report
The manuscript comprises lecture notes from a Les Houches summer school in 2018. The author lectured about the nested algebraic Bethe ansatz for models with rational gl(n) invariant Rmatrices. For pedagogical reasons he mostly restricted himself to the simplest nested case n = 3. The focus of the lecture was on the construction of Bethe vectors. Questions concerning the spectrum of physical models were not considered.
Being a productive contributor to the presented subject for the last ten years, the author possesses considerable insight into the details, which he tries to share with his readers. The choice of the material is rather on the mathematical side as the Bethe vectors themselves have little physical meaning. But their knowledge may be a prerequisite for calculating more physical quantities like form factors or correlation functions. Some more comments on the latter might have rendered the manuscript more readable for the general audience. A proofreading by a native speaker might also add to the readability of the manuscript.
Besides these general remarks I would only suggest very minor corrections
listed below.
1.) It should be said below (1.7) that $\Lambda (z)$ is the transfer matrix eigenvalue
2.) There should be no tilde on the right hand side of the first equation (1.29)
3.) Below (1.31) the argument of $\tilde T$ should probably rather be $u$ instead of $u$
4.) Above (1.32): as generated $\rightarrow$ is generated
4.) I guess in more mathematical terms the coloring in section 1.7 is a grading
5.) Above (2.6): over presentations $\rightarrow$ other presentations
6.) Are the indices in (3.4) correct? They look strange to me, but perhaps I simply do not understand the conventions
7.) Section 3.3: Let we have $\rightarrow$ Let us consider
8.) Above (3.18): polylinear $\rightarrow$ linear
9.) Below (3.26): get read $\rightarrow$ get rid
10.) I guess the indices in (4.1) refer to mutually distinct spaces. If yes, it may add clarity to say this
11.) Summary section: vector Beta
12.) Right hand side of (A.11): $K_1$
Requested changes
see above
Anonymous Report 2 on 202065 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1911.12811v1, delivered 20200605, doi: 10.21468/SciPost.Report.1740
Strengths
1 Comprehensive review of the mathematics of the NABA by an expert in
the field.
2 Reference for the analysis of integrable quantum chains with higher
rank symmetries
Weaknesses
1 Better motivation could ease access to the topic.
Report
The manuscript is a comprehensive presentation of the nested algebraic Bethe ansatz method (NABA) mostly using the $gl_3$invariant (rational) $R$matrix as an example. Both the 'classical' scheme of Kulish and Reshetikhin and the representation of offshell Bethe states in terms of trace formulae are covered. Explicit forms of Bethe states are obtained based on recursion relations satisfied between Bethe vectors in different color sectors. This provides the full set of tools used in the recent approaches to the computation of form factors correlation functions for quantum chains with higher rank symmetries.
The paper's focus is on mathematical aspects of the NABA (as stated in the introduction) and it certainly provides a reference for anybody working in the field. However, it would be more accessible to those trying to learn the methods if more motivation e.g. by relating to the mathematical challenges in obtaining insights into physical problems would be given.
Author: Nikita Slavnov on 20200619 [id 858]
(in reply to Report 2 on 20200605)I am grateful to Referee for careful reading of my notes and valuable remarks. I have somewhat expanded the physical interpretation of the NABA in Introduction and added some references in a new version of the manuscript. However, I prefer not to touch this topic too deeply, because I am not an expert in this field.
Anonymous Report 1 on 2020525 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1911.12811v1, delivered 20200525, doi: 10.21468/SciPost.Report.1711
Strengths
1. The most complete detailed and rigorous presentation of the Nested Bethe Ansatz construction of the Bethe states.
2. Focus on the notion of onshell and offshell Bethe states
Weaknesses
The presentation in its current form is more suitable for a review article than lecture notes
Report
This is the most complete and the most detailed presentation of the nested algebraic Bethe ansatz construction, and for this reason it is perfectly suitable for the ``les Houches series''. The lecture course is built around the concepts of onshell and offshell Bethe states. This point of view is the main originality and an evident strength of this lecture course as it opens a way for further study of form factors and correlation functions.
The only important improvement I would recommend is to add more explanation in the beginning of each section. In the current form the manuscript ressembles more to a review article than lecture notes. It is important to better explain the motivation and logic of each section (specially for the sections 4, 5 and 6). This motivation is clear for an experienced researcher working in the field, however it is much less evident for a new PhD student studying the subject and it should be kept in mind that it is the target audience of the ``les Houches series''. I would also suggest to move the appendices into the main body of the lectures, as it seems to me that this paper structure with 4 appendices once again is more suitable for a research paper than lecture notes (taking into account the fact that all the four appendices are not too long and perfectly fit each into corresponding section). In particular I would suggest to do it for the appendix B as it is crucial for all the following constructions.
There are also few minor remarks and typos listed below
Requested changes
1 Typo, line after (1.13): unit AT the intersection
2 In the first line of (1.29), there should be no tilde over T.
3 Also in (1.29) the bar notation for the indices can lead to a confusion with the set notations used throughout the manuscript
4 The phrase after (1.40) can lead to a confusion (using of either and or) as both terms always appear.
5 The fact stated between (2.1) and (2.2) that a local operator acting on an onshell Bethe state produces a linear combination of offshell states is not that evident, it should be better explained and appropriate citations added.
6 Typo in the phrase before (2.6), should be probably: there exist OTHER representation
7 Remark 2 in the section 2.2 is very important for the future presentation, maybe it can be better illustrated with examples (1.22) and (1.26) already introduced in the first section?
8 Beginning of 3.3 "Let we" should be replaced
9 May be it is better to reproduce (3.38) after (3.60) to have all the nested Bethe equations together
10 Before (3.65): may be a little bit more comments on the difference with the restricted case here would be useful.
11 Definition 3.1, typo: AN offshell vector
12 After (3.78), typo: becomes AN onshell state
13 Last phrase before the Conclusion, typo: OPEN question
14 Careful proofreading would be welcome
Author: Nikita Slavnov on 20200619 [id 857]
(in reply to Report 1 on 20200525)I am grateful to Referee for careful reading of my notes and valuable remarks. I believe that Referee found serious disadvantages in my manuscript. This primarily concerns the lack of motivation and explanation in the beginning of the sections. A very important remark is the fact that the action of operators on the onshell Bethe vectors, generally speaking, is not necessarily expressed through the offshell Bethe vectors. The suggestion to move the appendices to the main text is not obvious for me. I agree that such a construction of the text of the article is more consistent with the lecture notes. However, sometimes this strongly violates the presentation logic. I think that appendices B and D can be moved to the text. At the same time, I prefer to keep appendices A and C in their present form (that is, as appendices).
Author: Nikita Slavnov on 20200619 [id 859]
(in reply to Report 3 on 20200613)I am grateful to Referee for careful reading of my notes and valuable remarks. I would only like to give a comment on the second remark 4. The coloring in the Yangian of the gl_n algebra is related to the eigenvalues of the Cartan generators of the gl_n algebra (see page 6 of https://arxiv.org/pdf/1704.08173.pdf). This has nothing to do with what is commonly called grading (see the beginning of section 2 of the same paper).