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On Scalar Products in Higher Rank Quantum Separation of Variables
by J. M. Maillet, G. Niccoli, L. Vignoli
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Submission summary
| Authors (as registered SciPost users): | Jean Michel Maillet · Giuliano Niccoli |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2003.04281v2 (pdf) |
| Date submitted: | 2020-08-28 15:41 |
| Submitted by: | Maillet, Jean Michel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models [1], we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states (transfer matrix eigenstates or some simple generalization of them). In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector, there could be more than one non-vanishing overlap with the vectors of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the $\mathcal{Y}(gl_3)$ lattice model with $N$ sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality. In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the $\mathcal{Y}(gl_2)$ rank one case.
Current status:
Reports on this Submission
Anonymous Report 2 on 2020-10-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2003.04281v2, delivered 2020-10-07, doi: 10.21468/SciPost.Report.2059
Report
In this paper an improved version of Skyanin’s SoV procedure for constructing a basis of eigenstates of the Bethe algebra is worked out in the case of a gl(3). The eigenvectors in the improved approach are obtained by repeated action of twisted transfer matrices on a reference state. Explicit formulas are given for the scalar products of the separate bra and ket states. The reported results are correct, original and significant. The paper is carefully written but it is structured so that the reader should go through the whole text in order to understand and use the results. A short summary of the most important formulas would be helpful. This of course does not diminish the qualities of the paper. I recommend publication of the manuscript it its present form.
Anonymous Report 1 on 2020-10-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2003.04281v2, delivered 2020-10-05, doi: 10.21468/SciPost.Report.2047
Strengths
1. Important results are obtained in the field of the SoV method.
2. The paper is clearly written.
Weaknesses
No obvious weaknesses.
Report
This paper is devoted to the Separation of Variables (SoV) method. The authors use the approach developed in paper [1] and apply it to integrable models with a gl_3invariant R-matrix. The main goal of this work is to construct such separated bases of co-vectors and vectors that would be convenient for calculating scalar products. The authors first construct a system of pseudo-orthogonal co-vector/vector SoV bases. Then, using the freedom in generating family of conserved charges, the authors construct orthogonal systems. These results are used to calculate the scalar product of separated states.
I am sure that the results obtained are very important and deserve to be published. The article is rather technical, but it is very clearly written. The formulas and their transformations are provided with detailed comments.
I have two comments. The first one is optional. I missed at least a small final section, despite the fact that the main results are listed in detail in Introduction. I still think that summarizing the results and describing possible perspectives would be useful. Especially considering that the article is quite technical.
Second remark: a typo in Acknowledgements (L.V. is supported by...)
To summarize, I recommend publishing the paper without re-reviewing.
Requested changes
A typo in Acknowledgements. Other changes are optional.