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Why scalar products in the algebraic Bethe ansatz have determinant representation

Journal of High Energy Physics volume 2019, Article number: 103 (2019) Cite this article

A preprint version of the article is available at arXiv.

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.

References

  1. L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, The Quantum Inverse Problem Method. 1, Theor. Math. Phys.40 (1980) 688 [INSPIRE].

    MATH  Google Scholar 

  2. 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].

    ADS  Google Scholar 

  3. 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].

  4. V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge U.K. (1993).

    Book  Google Scholar 

  5. 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].

    ADS  MathSciNet  Article  Google Scholar 

  6. 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].

    MathSciNet  Google Scholar 

  7. 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].

    ADS  MathSciNet  MATH  Google Scholar 

  8. 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).

  9. M. Gaudin, La Fonction d’Onde de Bethe, Masson, Paris France (1983).

    MATH  Google Scholar 

  10. V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys.86 (1982) 391 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  11. 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].

  12. 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.

    MathSciNet  Article  Google Scholar 

  13. 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].

    ADS  MathSciNet  Article  Google Scholar 

  14. 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].

    ADS  MathSciNet  Article  Google Scholar 

  15. 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].

    MathSciNet  Article  Google Scholar 

  16. 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].

    MathSciNet  MATH  Google Scholar 

  17. 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].

    MathSciNet  MATH  Google Scholar 

  18. S. Belliard and N.A. Slavnov, Scalar Products in Twisted XXX Spin Chain. Determinant Representation, SIGMA15 (2019) 066 [arXiv: 1906 .06897] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  19. N.A. Slavnov, Algebraic Bethe ansatz, 2018, arXiv:1804.07350 [INSPIRE].

  20. E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys.A 21 (1988) 2375 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  21. S. Belliard and N. Crampé, Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz, SIGMA9 (2013) 072 [arXiv:1309 . 6165] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  22. 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].

    ADS  MathSciNet  Article  Google Scholar 

  23. N. Crampé, Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries, J. Phys.A 48 (2015) 08FT01 [arXiv: 1411. 7954] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  24. 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].

    ADS  Article  Google Scholar 

  25. 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].

    ADS  MathSciNet  Article  Google Scholar 

  26. A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl.32 (1987) 878.

    ADS  MATH  Google Scholar 

  27. 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].

    ADS  Article  Google Scholar 

  28. 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].

    ADS  MathSciNet  MATH  Google Scholar 

  29. J. Cao, W. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz for exactly solvable models, Springer, Heidelberg Germany (2015).

    MATH  Google Scholar 

  30. 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].

    MathSciNet  Article  Google Scholar 

  31. S. Belliard, N.A. Slavnov and B. Vallet, Modified Algebraic Bethe Ansatz: Twisted XXX Case, SIGMA14 (2018) 054 [arXiv:1804 .00597] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  32. 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].

    MathSciNet  Article  Google Scholar 

  33. 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].

    ADS  Article  Google Scholar 

  34. A. Liashyk, S.Z. Pakuliak, É. Ragoucy and N.A. Slavnov, Bethe vectors for orthogonal integrable models, arXiv: 1906.03202 [INSPIRE].

  35. 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].

    MathSciNet  Article  Google Scholar 

  36. 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].

    ADS  MATH  Google Scholar 

  37. 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].

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Authors and Affiliations

  1. Institut Denis-Poisson, Université de Tours, Université d’Orléans, Pare de Gmmrnont, 37200, Tours, France

    S. Belliard

  2. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina str., Moscow, 119991, Russia

    N. A. Slavnov

Authors
  1. S. Belliard
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  2. N. A. Slavnov
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Correspondence to N. A. Slavnov.

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ArXiv ePrint: 1908.00032

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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

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  • DOI: https://doi.org/10.1007/JHEP10(2019)103

Keywords

  • Integrable Field Theories
  • Lattice Integrable Models