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Bethe vectors and recurrence relations for twisted Yangian based models
by Vidas Regelskis
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Authors (as registered SciPost users):  Vidas Regelskis 
Submission information  

Preprint Link:  scipost_202310_00019v3 (pdf) 
Date accepted:  20241009 
Date submitted:  20241001 09:47 
Submitted by:  Regelskis, Vidas 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study Olshanski twisted Yangian based models, known as onedimensional "soliton nonpreserving" 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.
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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.

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.)

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 $\hat{n}$.

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.

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.

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.
Published as SciPost Phys. 17, 126 (2024)