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Nested Algebraic Bethe Ansatz in integrable models: recent results
by Stanislav Pakuliak, Eric Ragoucy, Nikita Slavnov
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Submission summary
| Authors (as registered SciPost users): | Stanislav Pakuliak · Eric Ragoucy |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1803.00103v1 (pdf) |
| Date submitted: | 2018-03-02 01:00 |
| Submitted by: | Ragoucy, Eric |
| Submitted to: | SciPost Physics Proceedings |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
This short note summarizes the works done in collaboration between S. Belliard (CEA, Saclay), L. Frappat (LAPTh, Annecy), S. Pakuliak (JINR, Dubna), E. Ragoucy (LAPTh, Annecy), N. Slavnov (Steklov Math. Inst., Moscow) and more recently A. Hutsalyuk (Wuppertal / Moscow) and A. Liashyk (Kiev / Moscow). It presents the construction of Bethe vectors, their scalar products and the form factors of local operator for integrable models based on the (super)algebras $gl_n$, $gl_{m|p}$ or their quantum deformations. It corresponds to two talks given by E.R. and N.S. at \textsl{Correlation functions of quantum integrable systems and beyond}, in honor of Jean-Michel Maillet for his 60's (ENS Lyon, October 2017).
Current status:
Reports on this Submission
Anonymous Report 2 on 2018-6-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1803.00103v1, delivered 2018-06-30, doi: 10.21468/SciPost.Report.521
Strengths
The manuscript is the review about the netted algebraic Bethe ansatz based on already published papers of the authors and the others.
Weaknesses
Although the paper focuses on the outline of the nested algebraic Bethe ansatz without technical details, it is worth publishing as proceedings.
Report
The authors have been working on this topic for a long time and the manuscript is well-summarized. All the technical details are removed and it is hard to follow the equations, but this would help for people to understand what the nested algebraic Bethe ansatz is. There are several points I want to remark.
Requested changes
In the section 2, “The transfer matrix t(z)”
- In the equation (6), the authors use the notation (-1)^{[i]} which is not defined. This is the common notation in the field of superalgebra, but it will be helpful for non-specialists if the definition is given.
In the section 3, “In the case of higher rank n”
- Since the manuscript is mostly devoted to reviews, it is better to give references for the equations (11)-(13).
In the section 3, “Known formulas: the trace formula”
- Probably this is rather clear, but it is helpful if the authors denote the space where the elementary matrices e_{i,j} acts.
- Give the reference about a Gauss decomposition of the monodromy matrix.
In the section 4, “Sum formula”
- Although the authors say the coefficient W_{part} are rational functions of the Bethe parameters, this is true only for the non q-deformed cases. Since the q-deformed cases are also mentioned in the same subsection, some comments about the trigonometric cases are seemed to be required.
- Also, give the references for the formula (24). (They are given in the comments later, but it is better to give them at the beginning when the formula first shows up.)
In the section 4, “Determinant formula”
- The determinant formula is given under “restrict conditions” as is mentioned in the manuscript. What this condition means the special twist for the transfer matrix, but this becomes clear only after reading the manuscript at the end of this subsection. It is better to remark what is the condition the authors consider here.
In the section 5, “Bethe vectors and zero modes”
- The reason why the authors used the brackets in the equation (38) is to emphasize that one of the Bethe parameters of the family (j-1) is sent to infinity. If so, the limit in the next sentence should be t_k^{(j-1)} \to \infty instead of t_k^{(j)} \to \infty,
In the section 5, “Bethe vectors and zero modes”
- In the footnote, several problems are remarked and the authors say “we show in section 5.3 how this problem can be solved”. However, what is explained is why the method fails and I found so solution to this problem. Or, do the authors mean that the formula for the universal form factors is the solution? (As they do not depend on the behavior of the spectral parameters. )
Anonymous Report 1 on 2018-5-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1803.00103v1, delivered 2018-05-21, doi: 10.21468/SciPost.Report.458
Strengths
1- Clear and compact review of the known the determinant formulas for overlaps of Bethe vectors and form-factors in solvable models with higher rank symmetries.
2- Useful for readers interested mainly in the application of these results.
Weaknesses
1- No weaknesses
Report
This paper gives a useful concise review of the exact results obtained by the authors in the recent years concerning the Bethe vectors and their overlaps in integrable models associated with higher rank algebras, including super algebras. This is a valuable paper and I recommend publication in its present form.
Requested changes
no requested changes