# Scalar Products in Twisted XXX Spin Chain. Determinant Representation

## TL;DR

This paper derives a determinant formula for scalar products and norms of Bethe vectors in the twisted XXX spin chain, demonstrating orthogonality of eigenstates with different eigenvalues.

## Contribution

It provides a new determinant representation for scalar products and norms of Bethe vectors in the twisted XXX spin chain with non-diagonal boundaries.

## Key findings

- Determinant formula for scalar products of Bethe vectors.
- Explicit norm formula for on-shell Bethe vectors.
- Proof of orthogonality of eigenstates with different eigenvalues.

## Abstract

We consider XXX spin-$1/2$ Heisenberg chain with non-diagonal boundary conditions. We obtain a compact determinant representation for the scalar product of on-shell and off-shell Bethe vectors. In the particular case when both Bethe vectors are on shell, we obtain a determinant representation for the norm of on-shell Bethe vector and prove orthogonality of the on-shell vectors corresponding to the different eigenvalues of the transfer matrix.