Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 066, 30 pages      arXiv:1906.06897      https://doi.org/10.3842/SIGMA.2019.066

Scalar Products in Twisted XXX Spin Chain. Determinant Representation

Samuel Belliard a and Nikita A. Slavnov b
a) Institut Denis-Poisson, Université de Tours, Université d'Orléans, Parc de Grammont, 37200 Tours, France
b) Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow, 119991, Russia

Received June 19, 2019, in final form August 27, 2019; Published online September 03, 2019

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.

Key words: XXX chain; non-diagonal boundary conditions; scalar product; determinant.

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