Slavnov and Gaudin-Korepin formulas for models without $U(1)$ symmetry: the XXX chain on the segment

TL;DR

This paper derives modified Slavnov and Gaudin-Korepin formulas for the scalar product and norm of Bethe vectors in the XXX spin chain with non-U(1) symmetric boundary conditions, expanding integrability tools.

Contribution

It provides the first explicit formulas for scalar products and norms in the XXX chain with general integrable boundaries using a modified algebraic Bethe ansatz.

Findings

Derived a modified Slavnov formula for the scalar product.

Obtained a Gaudin-Korepin formula for the norm squared.

Extended integrability techniques to models without U(1) symmetry.

Abstract

We consider the isotropic spin$-\frac{1}{2}$ Heisenberg chain with the most general integrable boundaries. The scalar product between the on-shell Bethe vector and its off-shell dual, obtained by means of the modified algebraic Bethe ansatz, is given by a modified Slavnov formula. The corresponding Gaudin-Korepin formula, \textit{i.e.}, the square of the norm, is also obtained.