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Bethe vectors and recurrence relations for twisted Yangian based models
by Vidas Regelskis
Submission summary
| Authors (as registered SciPost users): | Vidas Regelskis |
| Submission information | |
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| Preprint Link: | scipost_202310_00019v3 (pdf) |
| Date accepted: | 2024-10-09 |
| Date submitted: | 2024-10-01 09:47 |
| Submitted by: | Regelskis, Vidas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study Olshanski twisted Yangian based models, known as one-dimensional "soliton non-preserving" open spin chains, by means of algebraic Bethe ansatz. The even case, when the bulk symmetry is $\mathfrak{gl}_{2n}$ and the boundary symmetry is $\mathfrak{sp}_{2n}$ or $\mathfrak{so}_{2n}$, was studied in [GMR19]. In the present work, we focus on the odd case, when the bulk symmetry is $\mathfrak{gl}_{2n+1}$ and the boundary symmetry is $\mathfrak{so}_{2n+1}$. We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and $Y(\mathfrak{gl}_n)$-type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
I'd like to thank the referees for reviewing the manuscript and for the last minor suggestions. Below are my replies to the points raised by the first referee.
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I thank the referee for noticing the overlap of transposition symbol $w$ appearing in Sections 2 with the complex parameter $w$ appearing in Section 3. The transposition symbol $w$ was replaced with $\omega$; letter $t$ is already used in Section 3.6 and Appendix A.2 to denote the usual transposition. (The symbol $\omega$ is also used in Appendix A.1 to denote weights of elements $s_{ij}[r]$; this notation appears in Appendix A.1 only and should not bring any confusion with transposition $\omega$ in Section 2.)
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It is noted above equation (2.15) that $A$, $B$, $C$ and $D$ are $\hat{n}\times {\hat{n}$ dimensional matrix operators and so $i$ and $j$ run from $1$ to $\hat{n}$.
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The notation of "overlapping" matrix operators is now illustrated in Remark 2.1 with an explicit example of the $N=3$ case; the text below the new example was also clarified. I hope this resolves the issue raised by the referee.
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I agree with the referee that the empty set notation is delicate to employ. I have now explicitly indicated the empty sets in all relevant statements and examples in Section 4 and Appendix A.3. I hope this clarifies this delicate notation.
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I thank the referee for spotting a typo in Corollary 4.2; it has been corrected. A similar typo was in Example 4.3; it was also corrected.
List of changes
- The transposition symbol $w$ was replaced with $\omega$ in lines 132, 137, 160 and equations (2.8), (2.14).
- Remark 2.1 was updated with an explicit example of the $A$, $B$, $C$ and $D$ operators for the $N=3$ case; the text below the example was clarified.
- The empty subsets were explicitly indicated in all statements and examples in Section 4 and Appendix A.3.
- Typos in Corollary 4.2 and Example 4.3 were fixed.
- Reference [Gom24] was updated with journal entry and doi.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)