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 SIGMA, 2019, Volume 15, 066, 30 pp. (Mi sigma1502)

Scalar Products in Twisted XXX Spin Chain. Determinant Representation

Samuel Belliarda, Nikita A. Slavnovb

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

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.

Keywords: XXX chain, non-diagonal boundary conditions, scalar product, determinant.

 Funding Agency Grant Number Russian Science Foundation 19-11-00062 This work is supported by the Russian Science Foundation under grant 19-11-00062.

DOI: https://doi.org/10.3842/SIGMA.2019.066

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ArXiv: 1906.06897
MSC: 82B23, 81R50
Received: June 19, 2019; in final form August 27, 2019; Published online September 3, 2019
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Citation: Samuel Belliard, Nikita A. Slavnov, “Scalar Products in Twisted XXX Spin Chain. Determinant Representation”, SIGMA, 15 (2019), 066, 30 pp.

Citation in format AMSBIB
\Bibitem{BelSla19} \by Samuel~Belliard, Nikita~A.~Slavnov \paper Scalar Products in Twisted XXX Spin Chain. Determinant Representation \jour SIGMA \yr 2019 \vol 15 \papernumber 066 \totalpages 30 \mathnet{http://mi.mathnet.ru/sigma1502} \crossref{https://doi.org/10.3842/SIGMA.2019.066} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3998984} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000485987300001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85073358938} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. Belliard, N. A. Slavnov, “Why scalar products in the algebraic bethe ansatz have determinant representation”, J. High Energy Phys., 2019, no. 10, 103
2. P. Baseilhac, R. A. Pimenta, “Diagonalization of the heun-askey-wilson operator, leonard pairs and the algebraic bethe ansatz”, Nucl. Phys. B, 949 (2019), 114824
3. M. Vasilyev, A. Zabrodin, A. Zotov, “Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary”, J. Phys. A-Math. Theor., 53:49 (2020), 494002
4. M. Vasilyev, A. Zabrodin, A. Zotov, “Quantum-classical duality for Gaudin magnets with boundary”, Nucl. Phys. B, 952 (2020), 114931
5. S. Belliard, N. A. Slavnov, “Overlap between usual and modified Bethe vectors”, Theoret. and Math. Phys., 209:1 (2021), 1387–1402
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