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.
Similar content being viewed by others
References
F.A. Smirnov, Form factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) 1, World Scientific, Singapore (1992).
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge U.K. (1993).
I.B. Frenkel and N.Y. Reshetikhin, Quantum affine algebras and holonomic difference equations, Commun. Math. Phys. 146 (1992) 1 [SPIRES].
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].
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].
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].
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.
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.
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.
V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [SPIRES].
A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987) 878.
V.G. Drinfeld, On constant quasiclassical solution of the QYBE, Sov. Math. Dokl. 28 (1983) 667.
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].
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.
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].
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].
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].
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].
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.
N. Kitanine et al., Correlation functions of the open XXZ chain I, J. Stat. Mech. (2007) P10009 [arXiv:0707.1995] [SPIRES].
E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988) 2375 [SPIRES].
R.I. Nepomechie, Functional relations and Bethe ansatz for the XXZ chain, J. Stat. Phys. 111 (2003) 1363 [hep-th/0211001] [SPIRES].
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].
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.
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].
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].
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].
W. Galleas and M.J. Martins, Solution of the SU(N) vertex model with non-diagonal open boundaries, Phys. Lett. A 335 (2005) 167.
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.
J. de Gier and P. Pyatov, Bethe ansatz for the TemperleyLieb loop model with open boundaries, J. Stat. Mech. (2004) P002.
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.
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].
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].
J. de Gier and F.H.L. Essler, Exact spectral gaps of the asymmetric exclusion process with open boundaries, J. Stat. Mech. (2006) P12011.
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].
Z. Bajnok, Equivalences between spin models induced by defects, J. Stat. Mech. (2006) P06010 [hep-th/0601107] [SPIRES].
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].
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].
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].
A. Dikou, The open XXZ and associated models at q root of unity, J. Stat. Mech. (2006) P09010.
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].
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].
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].
R. Murgan, Bethe ansatz of the open spin-s XXZ chain with nondiagonal boundary terms, JHEP 04 (2009) 076 [arXiv:0901.3558] [SPIRES].
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].
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].
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].
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].
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].
L. Frappat, R.I. Nepomechie and E. Ragoucy, Out-of-equilibrium relaxation of the EdwardsWilkinson elastic line, J. Stat. Mech. (2007) P09008.
T.-D. Albert, H. Boos, R. Flume, R.H. Poghossian and K. Rulig, An F-twisted XYZ model, Lett. Math. Phys. 53 (2000) 201.
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].
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].
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].
R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, New York U.S.A. (1982).
G. Felder and A. Varchenko, Algebraic Bethe ansatz for the elliptic quantum group E τ,η (sl 2), Nucl. Phys. B 480 (1996) 485.
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].
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].
O. Tsuchiya, Determinant formula for the six-vertex model with reflecting end, J. Math. Phys. 39 (1998) 5946.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1011.4719
Rights and permissions
About this article
Cite this article
Yang, WL., Chen, X., Feng, J. et al. Determinant representations of scalar products for the open XXZ chain with non-diagonal boundary terms. J. High Energ. Phys. 2011, 6 (2011). https://doi.org/10.1007/JHEP01(2011)006
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2011)006