Abstract
We show that the scalar products of on-shell and off-shell Bethe vectors in the algebralic Bethe ansatz solvable models satisfy a system of linear equations. We find solutions to this system for a wide class of integrable models. We also apply our method to the XXX spin chain with broken U(l) symmetry.
Article PDF
Similar content being viewed by others
References
L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, The Quantum Inverse Problem Method. 1, Theor. Math. Phys.40 (1980) 688 [INSPIRE].
L.A. Takhtajan and L.D. Faddeev, The Quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys34 (1979) 11 [INSPIRE].
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic gravitation and gravitational radiation. Proceedings of School of Physics, Les Houches France (1995), A. Connes et al. eds., North Holland, Amsterdam The Netherlands (1996), pg. 149 [hep-th/9605187] [INSPIRE].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge U.K. (1993).
N. Kitanine, J.M. Maillet and V. Terras, Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field, Nucl. Phys.B 567 (2000) 554 [math-ph/9907019] [INSPIRE].
N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov and V. Terras, Form factor approach to dynamical correlation functions in critical models, J. Stat. Mech.1209 (2012) P09001 [arXiv:1206.2630] [INSPIRE].
F. Gohmann, A. Klumper and A. Seel, Integral representations for correlation functions of the XXZ chain at finite temperature, J. Phys.A 37 (2004) 7625 [hep-th/0405089] [INSPIRE].
M. Gaudin, Modèles exacts en mécanique statistique: la méthode de Bethe et ses généralisations, Preprint, Centre d’Etudes Nucléaires de Saclay, CEA-N-1559:1 (1972).
M. Gaudin, La Fonction d’Onde de Bethe, Masson, Paris France (1983).
V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys.86 (1982) 391 [INSPIRE].
A. Kirillov and F.A. Smirnov, Solutions of some combinatorial problems which arise in calculating correlators in exactly solvable models, Zap. Nauchn. Sem. LOMI164 (1987) 67 [J. Sov. Math.47 (1989) 2413].
N.A. Slavnov, Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz, Theor. Math. Phys.79 (1989) 502.
N. Kitanine, J.M. Maillet and V. Terras, Form factors of the XXZ Heisenberg spin- \( \frac{1}{2} \)finite chain, Nucl. Phys.B 554 (1999) 647 [math-ph/9807020] [INSPIRE].
Y.-S. Wang, The scalar products and the norm of Bethe eigenstates for the boundary XXX Heisenberg spin-1/2 finite chain, Nucl. Phys.B 622 (2002) 633 [INSPIRE].
N. Kitanine, K.K. Kozlowski, J.M. Maillet, G. Niccoli, N.A. Slavnov and V. Terras, Correlation functions of the open XXZ chain I, J. Stat. Mech.0710 (2007) P10009 [arXiv: 0707 .1995] [INSPIRE].
S. Belliard and R.A. Pimenta, Slavnov and Gaudin-Korepin Formulas for Models without U(1) Symmetry: the Twisted XXX Chain, SIGMA11 (2015) 099 [arXiv: 1506 .06550] [INSPIRE].
S. Belliard and R.A. Pimenta, Slavnov and Gaudin-Korepin formulas for models without U(1) symmetry: the XXX chain on the segment, J. Phys.A 49 (2016) 17LT01 [arXiv: 1507 .03242] [INSPIRE].
S. Belliard and N.A. Slavnov, Scalar Products in Twisted XXX Spin Chain. Determinant Representation, SIGMA15 (2019) 066 [arXiv: 1906 .06897] [INSPIRE].
N.A. Slavnov, Algebraic Bethe ansatz, 2018, arXiv:1804.07350 [INSPIRE].
E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys.A 21 (1988) 2375 [INSPIRE].
S. Belliard and N. Crampé, Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz, SIGMA9 (2013) 072 [arXiv:1309 . 6165] [INSPIRE].
S. Belliard, Modified algebraic Bethe ansatz for XXZ chain on the segment – I: Triangular cases, Nucl. Phys.B 892 (2015) 1 [arXiv: 1408 . 4840] [INSPIRE].
N. Crampé, Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries, J. Phys.A 48 (2015) 08FT01 [arXiv: 1411. 7954] [INSPIRE].
S. Belliard and R.A. Pimenta, Modified algebraic Bethe ansatz for XXZ chain on the segment – II: General cases, Nucl. Phys.B 894 (2015) 527 [arXiv: 1412. 7511] [INSPIRE].
J. Avan, S. Belliard, N. Grosjean and R.A. Pimenta, Modified algebraic Bethe ansatz for XXZ chain on the segment – III: Proof, Nucl. Phys.B 899 (2015) 229 [arXiv: 1506 .02147] [INSPIRE].
A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl.32 (1987) 878.
J. Cao, W. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz and exact solution of a topological spin ring, Phys. Rev . Lett.111 (2013) 137201 [arXiv:1305. 7328] [INSPIRE].
J. Cao, W.-L. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions, Nucl. Phys.B 875 (2013) 152 [arXiv: 1306 . 1742] [INSPIRE].
J. Cao, W. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz for exactly solvable models, Springer, Heidelberg Germany (2015).
S. Belliard, N.A. Slavnov and B. Vallet, Scalar product of twisted XXX modified Bethe vectors, J. Stat. Mech.1809 (2018) 093103 [arXiv:1805.11323] [INSPIRE].
S. Belliard, N.A. Slavnov and B. Vallet, Modified Algebraic Bethe Ansatz: Twisted XXX Case, SIGMA14 (2018) 054 [arXiv:1804 .00597] [INSPIRE].
S. Belliard, S. Pakuliak, É. Ragoucy and N.A. Slavnov, Bethe vectors of GL(3)-invariant integrable models, J. Stat. Mech.1302 (2013) P02020 [arXiv: 1210 .0768] [INSPIRE].
A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, É. Ragoucy and N.A. Slavnov, Current presentation for the super- Yangian double DY \( \left(\mathfrak{gl}\left(m\left|n\right.\right)\right) \)and Bethe vectors, Russ. Math. Surveys72 (2017) 33 [arXiv: 1611. 09620] [INSPIRE].
A. Liashyk, S.Z. Pakuliak, É. Ragoucy and N.A. Slavnov, Bethe vectors for orthogonal integrable models, arXiv: 1906.03202 [INSPIRE].
S. Belliard et al., Algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models, J. Stat. Mech.1210 (2012) P10017 [arXiv: 1207 .0956] [INSPIRE].
A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, É. Ragoucy and N.A. Slavnov, Scalar products of Bethe vectors in models with \( \left(\mathfrak{gl}\left(2\left|1\right.\right)\right) \)symmetry 2. Determinant representation, J. Phys.A 50 (2017) 034004 [arXiv:1606.03573] [INSPIRE].
N.A. Slavnov, Scalar products in GL(3)-based models with trigonometric R-matrix. Determinant representation, J. Stat. Mech.1503 (2015) P03019 [arXiv:1501.06253] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1908.00032
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Belliard, S., Slavnov, N.A. Why scalar products in the algebraic Bethe ansatz have determinant representation. J. High Energ. Phys. 2019, 103 (2019). https://doi.org/10.1007/JHEP10(2019)103
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)103