# Bethe vectors and recurrence relations for twisted Yangian based models

## TL;DR

This paper develops explicit Bethe vectors and recurrence relations for twisted Yangian models, focusing on the odd case with $rak{gl}_{2n+1}$ bulk symmetry and $rak{so}_{2n+1}$ boundary symmetry, extending previous work on even cases.

## Contribution

It introduces new explicit constructions of Bethe vectors and recurrence relations for twisted Yangian models with odd bulk symmetry, enhancing understanding of their algebraic structure.

## Key findings

- Constructed explicit Bethe vectors for odd case
- Presented symmetric trace formula for Bethe vectors
- Derived recurrence relations for Bethe vectors

## 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 $\mathfrak{gl}_{2n}$, was studied in arXiv:1710.08409. 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.