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

SIGMA 11 (2015), 099, 12 pages      arXiv:1506.06550

Slavnov and Gaudin-Korepin Formulas for Models without ${\rm U}(1)$ Symmetry: the Twisted XXX Chain

Samuel Belliard a and Rodrigo A. Pimenta bc
a) Laboratoire de Physique Théorique et Modélisation (CNRS UMR 8089), Université de Cergy-Pontoise, F-95302 Cergy-Pontoise, France
b) Departamento de Física, Universidade Federal de São Carlos, Caixa Postal 676, CEP 13565-905, São Carlos, Brasil
c) Physics Department, University of Miami, P.O. Box 248046, FL 33124, Coral Gables, USA

Received September 02, 2015, in final form December 02, 2015; Published online December 04, 2015

We consider the XXX spin-$\frac{1}{2}$ Heisenberg chain on the circle with an arbitrary twist. We characterize its spectral problem using the modified algebraic Bethe anstaz and study the scalar product between the Bethe vector and its dual. We obtain modified Slavnov and Gaudin-Korepin formulas for the model. Thus we provide a first example of such formulas for quantum integrable models without ${\rm U}(1)$ symmetry characterized by an inhomogenous Baxter T-Q equation.

Key words: algebraic Bethe ansatz; integrable spin chain; scalar product.

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  1. Avan J., Belliard S., Grosjean N., Pimenta R.A., Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof, Nuclear Phys. B 899 (2015), 229-246, arXiv:1506.0214.
  2. Baseilhac P., Koizumi K., Exact spectrum of the XXZ open spin chain from the $q$-Onsager algebra representation theory, J. Stat. Mech. Theory Exp. 2007 (2007), P09006, 27 pages, hep-th/0703106.
  3. Batchelor M.T., Baxter R.J., O'Rourke M.J., Yung C.M., Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions, J. Phys. A: Math. Gen. 28 (1995), 2759-2770, hep-th/9502040.
  4. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  5. Belliard S., Modified algebraic Bethe ansatz for XXZ chain on the segment - I: Triangular cases, Nuclear Phys. B 892 (2015), 1-20, arXiv:1408.4840.
  6. Belliard S., Crampé N., Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe ansatz, SIGMA 9 (2013), 072, 12 pages, arXiv:1309.6165.
  7. Belliard S., Crampé N., Ragoucy E., Algebraic Bethe ansatz for open XXX model with triangular boundary matrices, Lett. Math. Phys. 103 (2013), 493-506, arXiv:1209.4269.
  8. Belliard S., Pakuliak S., Ragoucy E., Slavnov N.A., The algebraic Bethe ansatz for scalar products in ${\rm SU}(3)$-invariant integrable models, J. Stat. Mech. Theory Exp. 2012 (2012), P10017, 25 pages, arXiv:1207.0956.
  9. Belliard S., Pakuliak S., Ragoucy E., Slavnov N.A., Form factors in ${\rm SU}(3)$-invariant integrable models, J. Stat. Mech. Theory Exp. 2013 (2013), P04033, 16 pages, arXiv:1211.3968.
  10. Belliard S., Pimenta R.A., Modified algebraic Bethe ansatz for XXZ chain on the segment - II - General cases, Nuclear Phys. B 894 (2015), 527-552, arXiv:1412.7511.
  11. Bethe H., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen Atomkette, Z. Phys. 71 (1931), 205-226.
  12. Cao J., Yang W., Shi K., Wang Y., Off-diagonal Bethe ansatz and exact solution a topological spin rin, Phys. Rev. Lett. 111 (2013), 137201, 5 pages, arXiv:1305.7328.
  13. Crampé N., Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries, J. Phys. A: Math. Theor. 48 (2015), 08FT01, 12 pages, arXiv:1411.7954.
  14. de Vega H.J., Families of commuting transfer matrices and integrable models with disorder, Nuclear Phys. B 240 (1984), 495-513.
  15. Derkachev S.É., The $R$-matrix factorization, $Q$-operator, and variable separation in the case of the XXX spin chain with the ${\rm SL}(2,{\mathbb C})$ symmetry group, Theoret. and Math. Phys. 169 (2011), 1539-1550.
  16. Derkachov S.É., Korchemsky G.P., Manashov A.N., Separation of variables for the quantum ${\rm SL}(2,{\mathbb R})$ spin chain, J. High Energy Phys. 2003 (2003), no. 7, 047, 30 pages.
  17. Faddeev L.D., Takhtadzhyan L.A., Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model, J. Sov. Math. 24 (1984), 241-267.
  18. Galleas W., Functional relations from the Yang-Baxter algebra: eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions, Nuclear Phys. B 790 (2008), 524-542, arXiv:0708.0009.
  19. Gaudin M., La fonction d'onde de Bethe, Collection du Commissariat à l'Énergie Atomique: Série Scientifique, Masson, Paris, 1983.
  20. Gaudin M., McCoy B.M., Wu T.T., Normalization sum for the Bethe's hypothesis wave functions of the Heisenberg-Ising chain, Phys. Rev. D 23 (1981), 417-419.
  21. Kitanine N., Kozlowski K.K., Maillet J.M., Niccoli G., Slavnov N.A., Terras V., Correlation functions of the open XXZ chain. I, J. Stat. Mech. Theory Exp. 2007 (2007), P10009, 37 pages, arXiv:0707.1995.
  22. Kitanine N., Kozlowski K.K., Maillet J.M., Niccoli G., Slavnov N.A., Terras V., Correlation functions of the open XXZ chain. II, J. Stat. Mech. Theory Exp. 2008 (2008), P07010, 33 pages, arXiv:0803.3305.
  23. Kitanine N., Maillet J.M., Niccoli G., Terras V., On determinant representations of scalar products and form factors in the SoV approach: the XXX case, arXiv:1506.02630.
  24. Kitanine N., Maillet J.M., Terras V., Form factors of the $XXZ$ Heisenberg spin-$\frac 12$ finite chain, Nuclear Phys. B 554 (1999), 647-678, math-ph/9807020.
  25. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  26. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.
  27. Mukhin E., Tarasov V., Varchenko A., Bethe algebra of homogeneous XXX Heisenberg model has simple spectrum, Comm. Math. Phys. 288 (2009), 1-42, arXiv:0706.0688.
  28. Nepomechie R.I., Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys. A: Math. Gen. 37 (2004), 433-440, special issue on recent advances in the theory of quantum integrable systems, hep-th/0304092.
  29. Nepomechie R.I., Wang C., Algebraic Bethe ansatz for singular solutions, J. Phys. A: Math. Theor. 46 (2013), 325002, 8 pages, arXiv:1304.7978.
  30. Niccoli G., Terras V., Antiperiodic XXZ chains with arbitrary spins: complete eigenstate construction by functional equations in separation of variables, Lett. Math. Phys. 105 (2015), 989-1031, arXiv:1411.6488.
  31. Pakuliak S., Ragoucy E., Slavnov N.A., Zero modes method and form factors in quantum integrable models, Nuclear Phys. B 893 (2015), 459-481, arXiv:1412.6037.
  32. Ribeiro G.A.P., Martins M.J., Galleas W., Integrable ${\rm SU}(N)$ vertex models with general toroidal boundary conditions, Nuclear Phys. B 675 (2003), 567-583, nlin.SI/0308011.
  33. Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ, 1992, 63-97, hep-th/9211111.
  34. Sklyanin E.K., Takhtadzhyan L.A., Faddeev L.D., Quantum inverse problem method. I, Theoret. and Math. Phys. 40 (1979), 688-706.
  35. Slavnov N.A., Calculation of scalar products of wave functions and form-factors in the framework of the algebraic Bethe ansatz, Theoret. and Math. Phys. 79 (1989), 502-508.
  36. Takhtadzhan L.A., Faddeev L.D., The quantum method for the inverse problem and the XYZ Heisenberg model, Russian Math. Surveys 34 (1979), no. 5, 11-68.
  37. Wang Y., Yang W., Cao J., Shi K., On the inhomogeneous T-Q relation for quantum integrable models, arXiv:1506.02512.

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