Advertisement

Determinant representations of scalar products for the open XXZ chain with non-diagonal boundary terms

  • Wen-Li Yang
  • Xi Chen
  • Jun Feng
  • Kun Hao
  • Bo-Yu Hou
  • Kang-Jie Shi
  • Yao-Zhong Zhang
Article

Abstract

The determinant representation of the scalar products of the Bethe states of the open XXZ spin chain with non-diagonal boundary terms is studied. Using the vertex-face correspondence, we transfer the problem into the corresponding trigonometric solid-on-solid (SOS) model with diagonal boundary terms. With the help of the Drinfeld twist or factorizing F-matrix, we obtain the determinant representation of the scalar products of the Bethe states of the associated SOS model. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states.

Keywords

Bethe Ansatz Lattice Integrable Models 

References

  1. [1]
    F.A. Smirnov, Form factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) 1, World Scientific, Singapore (1992).MATHGoogle Scholar
  2. [2]
    V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge U.K. (1993).CrossRefMATHGoogle Scholar
  3. [3]
    I.B. Frenkel and N.Y. Reshetikhin, Quantum affine algebras and holonomic difference equations, Commun. Math. Phys. 146 (1992) 1 [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  4. [4]
    B. Davies, O. Foda, M. Jimbo, T. Miwa and A. Nakayashiki, Diagonalization of the XXZ Hamiltonian by vertex operators, Commun. Math. Phys. 151 (1993) 89 [hep-th/9204064] [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  5. [5]
    Y. Koyama, Staggered polarization of vertex models with \( {U_q}({\widehat{sl(}n)}){\text{-}}symmetry \), Commun. Math. Phys. 164 (1994) 277 [hep-th/9307197] [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  6. [6]
    B.-Y. Hou, K.-J. Shi, Y.-S. Wang and W.-L. Yang, Bosonization of quantum sine-Gordon field with boundary, Int. J. Mod. Phys. A 12 (1997) 1711 [hep-th/9905197] [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    W.-L. Yang and Y.-Z. Zhang, Highest weight representations of \( {U_q}({\widehat{sl}( {2|1})}) \) and correlation functions of the q-deformed supersymmetric t-J model, Nucl. Phys. B 547 (1999) 599. CrossRefADSGoogle Scholar
  8. [8]
    W.-L. Yang and Y.-Z. Zhang, Level-one highest weight representation of \( {U_q}[sl({\widehat{N}} |1)] \) and bosonization of the multicomponent super t-J model, J. Math. Phys. 41 (2000) 5849. CrossRefMathSciNetADSMATHGoogle Scholar
  9. [9]
    B.-Y. Hou, W.-L. Yang and Y.-Z. Zhang, The twisted quantum affine algebra U q(A 2(2)) and correlation functions of the Izergin-Korepin model, Nucl. Phys. 556 (1999) 485. CrossRefMathSciNetADSMATHGoogle Scholar
  10. [10]
    V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  11. [11]
    A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987) 878.ADSGoogle Scholar
  12. [12]
    V.G. Drinfeld, On constant quasiclassical solution of the QYBE, Sov. Math. Dokl. 28 (1983) 667.Google Scholar
  13. [13]
    J.M. Maillet and J. Sanchez de Santos, Drinfeld twists and algebraic Bethe ansatz, Am. Math. Soc. Transl. 201 (2000) 137 [q-alg/9612012] [SPIRES].MathSciNetGoogle Scholar
  14. [14]
    N. Kitanine, J.M. Maillet and V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B 554 (1999) 647. CrossRefMathSciNetADSGoogle Scholar
  15. [15]
    S.-Y. Zhao, W.-L. Yang and Y.-Z. Zhang, Determinant representation of correlation functions for the supersymmetric t-J model, Commun. Math. Phys. 268 (2006) 505 [hep-th/0511028] [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  16. [16]
    S.-Y. Zhao, W.-L. Yang and Y.-Z. Zhang, On the construction of correlation functions for the integrable supersymmetric fermion models, Int. J. Mod. Phys. B 20 (2006) 505 [hep-th/0601065] [SPIRES].MathSciNetADSGoogle Scholar
  17. [17]
    W.-L. Yang, Y.-Z. Zhang and S.-Y. Zhao, Drinfeld twists and algebraic Bethe ansatz of the supersymmetric t-J model, JHEP 12 (2004) 038 [cond-mat/0412182] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    W.-L. Yang, Y.-Z. Zhang and S.-Y. Zhao, Drinfeld twists and algebraic Bethe ansatz of the supersymmetric model associated with U q(gl(m|n)), Commun. Math. Phys. 264 (2006) 87 [hep-th/0503003] [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  19. [19]
    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. CrossRefADSGoogle Scholar
  20. [20]
    N. Kitanine et al., Correlation functions of the open XXZ chain I, J. Stat. Mech. (2007) P10009 [arXiv:0707.1995] [SPIRES].
  21. [21]
    E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988) 2375 [SPIRES].MathSciNetADSGoogle Scholar
  22. [22]
    R.I. Nepomechie, Functional relations and Bethe ansatz for the XXZ chain, J. Stat. Phys. 111 (2003) 1363 [hep-th/0211001] [SPIRES].CrossRefMathSciNetMATHGoogle Scholar
  23. [23]
    R.I. Nepomechie, Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys. A 37 (2004) 433 [hep-th/0304092] [SPIRES].MathSciNetADSGoogle Scholar
  24. [24]
    J. Cao, H.-Q. Lin, K.-J. Shi and Y. Wang, Exact solution of XXZ spin chain with unparallel boundary fields, Nucl. Phys. B 663 (2003) 487. CrossRefMathSciNetADSGoogle Scholar
  25. [25]
    W.L. Yang and R. Sasaki, Exact solution of Z n Belavin model with open boundary condition, Nucl. Phys. B 679 (2004) 495 [hep-th/0308127] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    W.L. Yang and R. Sasaki, Solution of the dual reflection equation for A n−1 (1) SOS model, J. Math. Phys. 45 (2004) 4301 [hep-th/0308118] [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  27. [27]
    W.-L. Yang, R. Sasaki and Y.-Z. Zhang, Z n elliptic Gaudin model with open boundaries, JHEP 09 (2004) 046 [hep-th/0409002] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    W. Galleas and M.J. Martins, Solution of the SU(N) vertex model with non-diagonal open boundaries, Phys. Lett. A 335 (2005) 167. MathSciNetADSGoogle Scholar
  29. [29]
    C.S. Melo, G.A.P. Ribeiro and M.J. Martins, Bethe ansatz for the XXX-S chain with non-diagonal open boundaries, Nucl. Phys. B 711 (2005) 565. CrossRefMathSciNetADSGoogle Scholar
  30. [30]
    J. de Gier and P. Pyatov, Bethe ansatz for the TemperleyLieb loop model with open boundaries, J. Stat. Mech. (2004) P002. Google Scholar
  31. [31]
    A. Nichols, V. Rittenberg and J. de Gier, The effects of spatial constraints on the evolution of weighted complex networks, J. Stat. Mech. (2005) P05003. Google Scholar
  32. [32]
    J. de Gier, A. Nichols, P. Pyatov and V. Rittenberg, Magic in the spectra of the XXZ quantum chain with boundaries at Δ = 0 and Δ = −1/2, Nucl. Phys. B 729 (2005) 387 [hep-th/0505062] [SPIRES].ADSGoogle Scholar
  33. [33]
    J. de Gier and F.H.L. Essler, Bethe ansatz solution of the asymmetric exclusion process with open boundaries, Phys. Rev. Lett. 95 (2005) 240601 [SPIRES].CrossRefMathSciNetGoogle Scholar
  34. [34]
    J. de Gier and F.H.L. Essler, Exact spectral gaps of the asymmetric exclusion process with open boundaries, J. Stat. Mech. (2006) P12011. Google Scholar
  35. [35]
    W.-L. Yang, Y.-Z. Zhang and M. Gould, Exact solution of the XXZ Gaudin model with generic open boundaries, Nucl. Phys. B 698 (2004) 503 [hep-th/0411048] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  36. [36]
    Z. Bajnok, Equivalences between spin models induced by defects, J. Stat. Mech. (2006) P06010 [hep-th/0601107] [SPIRES].
  37. [37]
    W.-L. Yang and Y.-Z. Zhang, Exact solution of the A n−1 (1) trigonometric vertex model with non-diagonal open boundaries, JHEP 01 (2005) 021 [hep-th/0411190] [SPIRES].CrossRefADSGoogle Scholar
  38. [38]
    W.-L. Yang, Y.-Z. Zhang and R. Sasaki, A n−1 Gaudin model with open boundaries, Nucl. Phys. 729 (2005) 594 [hep-th/0507148] [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  39. [39]
    A. Doikou and P.P. Martin, On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary, J. Stat. Mech. (2006) P06004 [hep-th/0503019] [SPIRES].
  40. [40]
    A. Dikou, The open XXZ and associated models at q root of unity, J. Stat. Mech. (2006) P09010.Google Scholar
  41. [41]
    R. Murgan, R.I. Nepomechie and C. Shi, Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity, J. Stat. Mech. (2006) P08006 [hep-th/0605223] [SPIRES].
  42. [42]
    P. Baseilhac and K. Koizumi, Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory, J. Stat. Mech. (2007) P09006 [hep-th/0703106] [SPIRES].
  43. [43]
    W. Galleas, Functional relations from the Yang-Baxter algebra: eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions, Nucl. Phys. B 790 (2008) 524 [SPIRES].MathSciNetADSGoogle Scholar
  44. [44]
    R. Murgan, Bethe ansatz of the open spin-s XXZ chain with nondiagonal boundary terms, JHEP 04 (2009) 076 [arXiv:0901.3558] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  45. [45]
    W.-L. Yang and Y.-Z. Zhang, On the second reference state and complete eigenstates of the open XXZ chain, JHEP 04 (2007) 044 [hep-th/0703222] [SPIRES].CrossRefADSGoogle Scholar
  46. [46]
    W.-L. Yang and Y.-Z. Zhang, Multiple reference states and complete spectrum of the Z n Belavin model with open boundaries, Nucl. Phys. B 789 (2008) 591 [arXiv:0706.0772] [SPIRES].CrossRefADSGoogle Scholar
  47. [47]
    R.I. Nepomechie and F. Ravanini, Completeness of the Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys. A 36 (2003) 11391 [Addendum ibid. A 37 (2004) 1945] [hep-th/0307095] [SPIRES].MathSciNetADSGoogle Scholar
  48. [48]
    W.-L. Yang, R.I. Nepomechie and Y.-Z. Zhang, Q-operator and T-Q relation from the fusion hierarchy, Phys. Lett. B 633 (2006) 664 [hep-th/0511134] [SPIRES].MathSciNetADSGoogle Scholar
  49. [49]
    W.-L. Yang and Y.-Z. Zhang, T-Q relation and exact solution for the XYZ chain with general nondiagonal boundary terms, Nucl. Phys. B 744 (2006) 312 [hep-th/0512154] [SPIRES].CrossRefADSGoogle Scholar
  50. [50]
    L. Frappat, R.I. Nepomechie and E. Ragoucy, Out-of-equilibrium relaxation of the EdwardsWilkinson elastic line, J. Stat. Mech. (2007) P09008.Google Scholar
  51. [51]
    T.-D. Albert, H. Boos, R. Flume, R.H. Poghossian and K. Rulig, An F-twisted XYZ model, Lett. Math. Phys. 53 (2000) 201.CrossRefMathSciNetMATHGoogle Scholar
  52. [52]
    W.-L. Yang and Y.-Z. Zhang, Drinfeld twists of the open XXZ chain with non-diagonal boundary terms, Nucl. Phys. B 831 (2010) 408 [arXiv:1011.4120] [SPIRES].CrossRefADSGoogle Scholar
  53. [53]
    H.J. de Vega and A. Gonzalez Ruiz, Boundary K matrices for the six vertex and the n(2n − 1)A n−1 vertex models, J. Phys. A 26 (1993) L519 [hep-th/9211114] [SPIRES].Google Scholar
  54. [54]
    S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. A 9 (1994) 4353] [hep-th/9306002] [SPIRES].MathSciNetADSGoogle Scholar
  55. [55]
    R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, New York U.S.A. (1982).MATHGoogle Scholar
  56. [56]
    G. Felder and A. Varchenko, Algebraic Bethe ansatz for the elliptic quantum group E τ,η(sl 2), Nucl. Phys. B 480 (1996) 485. CrossRefMathSciNetADSGoogle Scholar
  57. [57]
    B.-Y. Hou, R. Sasaki and W.-L. Yang, Algebraic Bethe ansatz for the elliptic quantum group E τ,η(sl n) and its applications, Nucl. Phys. B 663 (2003) 467 [hep-th/0303077] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  58. [58]
    B. Hou, R. Sasaki and W.-L. Yang, Eigenvalues of Ruijsenaars-Schneider models associated with A n−1 root system in Bethe ansatz formalism, J. Math. Phys. 45 (2004) 559 [hep-th/0309194] [SPIRES].CrossRefMathSciNetADSMATHGoogle Scholar
  59. [59]
    O. Tsuchiya, Determinant formula for the six-vertex model with reflecting end, J. Math. Phys. 39 (1998) 5946. CrossRefMathSciNetADSMATHGoogle Scholar
  60. [60]
    W.-L. Yang et al., Determinant formula for the partition function of the six-vertex model with an non-diagonal reflecting end, Nucl. Phys. B 844 (2011) 289. CrossRefADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Wen-Li Yang
    • 1
  • Xi Chen
    • 1
  • Jun Feng
    • 1
  • Kun Hao
    • 1
  • Bo-Yu Hou
    • 1
  • Kang-Jie Shi
    • 1
  • Yao-Zhong Zhang
    • 2
  1. 1.Institute of Modern PhysicsNorthwest UniversityXianP.R. China
  2. 2.The University of Queensland, School of Mathematics and PhysicsBrisbaneAustralia

Personalised recommendations