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Journal of High Energy Physics

, 2019:103 | Cite as

Why scalar products in the algebraic Bethe ansatz have determinant representation

  • S. Belliard
  • N. A. SlavnovEmail author
Open Access
Regular Article - Theoretical Physics
  • 24 Downloads

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.

Keywords

Integrable Field Theories Lattice Integrable Models 

Notes

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

References

  1. [1]
    L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, The Quantum Inverse Problem Method. 1, Theor. Math. Phys. 40 (1980) 688 [INSPIRE].zbMATHGoogle Scholar
  2. [2]
    L.A. Takhtajan and L.D. Faddeev, The Quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979) 11 [INSPIRE].ADSGoogle Scholar
  3. [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. [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).CrossRefGoogle Scholar
  5. [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].ADSMathSciNetCrossRefGoogle Scholar
  6. [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].MathSciNetGoogle Scholar
  7. [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].ADSMathSciNetzbMATHGoogle Scholar
  8. [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).Google Scholar
  9. [9]
    M. Gaudin, La Fonction d’Onde de Bethe, Masson, Paris France (1983).zbMATHGoogle Scholar
  10. [10]
    V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    A. Kirillov and F.A. Smirnov, Solutions of some combinatorial problems which arise in calculating correlators in exactly solvable models, Zap. Nauchn. Sem. LOMI 164 (1987) 67 [J. Sov. Math. 47 (1989) 2413].Google Scholar
  12. [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.MathSciNetCrossRefGoogle Scholar
  13. [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].Google Scholar
  14. [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].ADSMathSciNetCrossRefGoogle Scholar
  15. [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].MathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Belliard and R.A. Pimenta, Slavnov and Gaudin-Korepin Formulas for Models without U(1) Symmetry: the Twisted XXX Chain, SIGMA 11 (2015) 099 [arXiv: 1506 .06550] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  17. [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].MathSciNetzbMATHGoogle Scholar
  18. [18]
    S. Belliard and N.A. Slavnov, Scalar Products in Twisted XXX Spin Chain. Determinant Representation, SIGMA 15 (2019) 066 [arXiv: 1906 .06897] [INSPIRE].MathSciNetGoogle Scholar
  19. [19]
    N.A. Slavnov, Algebraic Bethe ansatz, 2018, arXiv:1804.07350 [INSPIRE].
  20. [20]
    E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A 21 (1988) 2375 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    S. Belliard and N. Crampé, Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz, SIGMA 9 (2013) 072 [arXiv:1309 . 6165] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  22. [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].ADSMathSciNetCrossRefGoogle Scholar
  23. [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].MathSciNetzbMATHGoogle Scholar
  24. [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].ADSCrossRefGoogle Scholar
  25. [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].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987) 878.ADSMathSciNetzbMATHGoogle Scholar
  27. [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].ADSCrossRefGoogle Scholar
  28. [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].ADSMathSciNetzbMATHGoogle Scholar
  29. [29]
    J. Cao, W. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz for exactly solvable models, Springer, Heidelberg Germany (2015).zbMATHGoogle Scholar
  30. [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].MathSciNetCrossRefGoogle Scholar
  31. [31]
    S. Belliard, N.A. Slavnov and B. Vallet, Modified Algebraic Bethe Ansatz: Twisted XXX Case, SIGMA 14 (2018) 054 [arXiv:1804 .00597] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  32. [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].MathSciNetCrossRefGoogle Scholar
  33. [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. Surveys 72 (2017) 33 [arXiv: 1611. 09620] [INSPIRE].Google Scholar
  34. [34]
    A. Liashyk, S.Z. Pakuliak, É. Ragoucy and N.A. Slavnov, Bethe vectors for orthogonal integrable models, arXiv: 1906.03202 [INSPIRE].
  35. [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].MathSciNetCrossRefGoogle Scholar
  36. [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].Google Scholar
  37. [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].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institut Denis-PoissonUniversité de Tours, Université d’OrléansToursFrance
  2. 2.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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