Abstract
The nested off-diagonal Bethe ansatz method is proposed to diagonalize multi-component integrable models with generic integrable boundaries. As an example, the exact solutions of the su(n)-invariant spin chain model with both periodic and non-diagonal boundaries are derived by constructing the nested T − Q relations based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices.
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J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
D. Berenstein and S.E. Vazquez, Integrable open spin chains from giant gravitons, JHEP 06 (2005) 059 [hep-th/0501078] [INSPIRE].
D.M. Hofman and J.M. Maldacena, Reflecting magnons, JHEP 11 (2007) 063 [arXiv:0708.2272] [INSPIRE].
R. Murgan and R.I. Nepomechie, Open-chain transfer matrices for AdS/CFT, JHEP 09 (2008) 085 [arXiv:0808.2629] [INSPIRE].
R.I. Nepomechie, Revisiting the Y = 0 open spin chain at one loop, JHEP 11 (2011) 069 [arXiv:1109.4366] [INSPIRE].
J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [INSPIRE].
J. Dukelsky, S. Pittel and G. Sierra, Colloquium: exactly solvable Richardson-Gaudin models for many-body quantum systems, Rev. Mod. Phys. 76 (2004) 643 [nucl-th/0405011] [INSPIRE].
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].
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].
J. Cao, W.-L. Yang, K.-J. Shi and Y. Wang, Spin-1/2 XYZ model revisit: general solutions via off-diagonal Bethe ansatz, arXiv:1307.0280 [INSPIRE].
J. Cao, W.-L. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields, Nucl. Phys. B 877 (2013) 152 [arXiv:1307.2023] [INSPIRE].
H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].
F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R.J. Baxter and G.R.W. Quispel, Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models, J. Phys. A 20 (1987) 6397 [INSPIRE].
N. Crampé and É. Ragoucy, Generalized coordinate Bethe ansatz for non diagonal boundaries, Nucl. Phys. B 858 (2012) 502 [arXiv:1105.0338] [INSPIRE].
R.J. Baxter, Eight-vertex model in lattice statistics, Phys. Rev. Lett. 26 (1971) 832 [INSPIRE].
R.J. Baxter, One-dimensional anisotropic Heisenberg chain, Phys. Rev. Lett. 26 (1971) 834 [INSPIRE].
R.J. Baxter, One-dimensional anisotropic Heisenberg chain, Annals Phys. 70 (1972) 323 [INSPIRE].
R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, U.S.A. (1982).
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] [INSPIRE].
E.K. Sklyanin and L.D. Faddeev, Quantum mechanical approach to completely integrable field theory models, Sov. Phys. Dokl. 23 (1978) 902 [Dokl. Akad. Nauk Ser. Fiz. 243 (1978) 1430] [INSPIRE].
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 [Usp. Mat. Nauk 34 (1979) 13] [INSPIRE].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation function, Cambridge Univ. Press, Cambridge U.K. (1993).
E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988) 2375 [INSPIRE].
H. Fan, B.-Y. Hou, K.-J. Shi and Z.-X. Yang, Algebraic Bethe ansatz for eight vertex model with general open boundary conditions, Nucl. Phys. B 478 (1996) 723 [hep-th/9604016] [INSPIRE].
S. Belliard, N. Crampe and E. Ragoucy, Algebraic Bethe ansatz for open XXX model with triangular boundary matrices, Lett. Math. Phys. 103 (2013) 493 [arXiv:1209.4269] [arXiv:1209.4269].
N.Y. Reshetikhin, The functional equation method in the theory of exactly soluble quantum systems, Sov. Phys. JETP 57 (1983) 691.
E.K. Sklyanin, The quantum Toda chain, Lect. Notes Phys. 226 (1985) 196 [INSPIRE].
E.K. Sklyanin, Separation of variables — new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
H. Frahm, A. Seel and T. Wirth, Separation of variables in the open XXX chain, Nucl. Phys. B 802 (2008) 351 [arXiv:0803.1776] [INSPIRE].
G. Niccoli, Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and form factors, Nucl. Phys. B 870 (2013) 397 [arXiv:1205.4537] [INSPIRE].
G. Niccoli, Antiperiodic dynamical 6-vertex and periodic 8-vertex models I: complete spectrum by SOV and matrix elements of the identity on separate states, J. Phys. A 46 (2013) 075003 [arXiv:1207.1928] [INSPIRE].
G. Niccoli, Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and matrix elements of some quasi-local operators, J. Stat. Mech. 10 (2012) P10025 [arXiv:1206.0646] [INSPIRE].
G.E. Andrews, R.J. Baxter and P.J. Forrester, Eight vertex SOS model and generalized Rogers-Ramanujan type identities, J. Stat. Phys. 35 (1984) 193 [INSPIRE].
V.V. Bazhanov and N.Y. Reshetikhin, Critical RSOS models and conformal field theory, Int. J. Mod. Phys. A 4 (1989) 115 [INSPIRE].
R.I. Nepomechie, Bethe ansatz solution of the open XX spin chain with nondiagonal boundary terms, J. Phys. A 34 (2001) 9993 [hep-th/0110081] [INSPIRE].
R.I. Nepomechie, Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity, Nucl. Phys. B 622 (2002) 615 [hep-th/0110116] [INSPIRE].
R.I. Nepomechie, Functional relations and Bethe ansatz for the XXZ chain, J. Stat. Phys. 111 (2003) 1363 [hep-th/0211001] [INSPIRE].
R.I. Nepomechie, Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys. A 37 (2004) 433 [hep-th/0304092] [INSPIRE].
J. Cao, H.-Q. Lin, K.-J. Shi and Y. Wang, Exact solutions and elementary excitations in the XXZ spin chain with unparallel boundary fields, Nucl. Phys. B 663 (2003) 487 [cond-mat/0212163] [cond-mat/0212163].
W.-L. Yang, Y.-Z. Zhang and M.D. Gould, Exact solution of the XXZ Gaudin model with generic open boundaries, Nucl. Phys. B 698 (2004) 503 [hep-th/0411048] [INSPIRE].
J. de Gier and P. Pyatov, Bethe ansatz for the Temperley-Lieb loop model with open boundaries, J. Stat. Mech. 03 (2004) P03002 [hep-th/0312235] [INSPIRE].
A. Nichols, V. Rittenberg and J. de Gier, One-boundary Temperley-Lieb algebras in the XXZ and loop models, J. Stat. Mech. 03 (2005) P03003 [cond-mat/0411512] [INSPIRE].
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] [INSPIRE].
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 [cond-mat/0508707].
J. de Gier and F.H.L. Essler, Exact spectral gaps of the asymmetric exclusion process with open boundaries, J. Stat. Mech. 12 (2006) 11 [cond-mat/0609645].
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. 06 (2006) P06004 [hep-th/0503019] [INSPIRE].
A. Doikou, The open XXZ and associated models at q root of unity, J. Stat. Mech. 05 (2006) P05010 [hep-th/0603112] [INSPIRE].
Z. Bajnok, Equivalences between spin models induced by defects, J. Stat. Mech. 06 (2006) P06010 [hep-th/0601107] [INSPIRE].
P. Baseilhac and K. Koizumi, Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory, J. Stat. Mech. 09 (2007) P09006 [hep-th/0703106] [INSPIRE].
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 [INSPIRE].
H. Frahm, J.H. Grelik, A. Seel and T. Wirth, Functional Bethe ansatz methods for the open XXX chain, J. Phys. A 44 (2011) 015001 [arXiv:1009.1081] [INSPIRE].
S. Niekamp, T. Wirth and H. Frahm, The XXZ model with anti-periodic twisted boundary conditions, J. Phys. A 42 (2009) 195008 [arXiv:0902.1079] [INSPIRE].
A.M. Grabinski and H. Frahm, Non-diagonal boundary conditions for gl(1|1) super spin chains, J. Phys. A 43 (2010) 045207 [arXiv:0910.4029].
Y.-Y. Li, J. Cao, W.-L. Yang, K. Shi and Y. Wang, Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic fields, Nucl. Phys. B 879 (2014) 98 [arXiv:1311.0432] [INSPIRE].
X. Zhang, J. Cao, W.-L. Yang, K. Shi and Y. Wang, Exact solution of the one-dimensional super-symmetric t-J model with unparallel boundary fields, arXiv:1312.0376 [INSPIRE].
R.I. Nepomechie, An inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin, J. Phys. A 46 (2013) 442002 [arXiv:1307.5049] [INSPIRE].
S. Belliard and N. Crampé, Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe ansatz, SIGMA 9 (2013) 072 [arXiv:1309.6165] [INSPIRE].
B. Sutherland, A general model for multicomponent quantum systems, Phys. Rev. B 12 (1975) 3795 [INSPIRE].
P. Schlottmann, Integrable narrow-band model with possible relevance to heavy-fermion systems, Phys. Rev. B 36 (1987) 5177 [INSPIRE].
H.J. de Vega and E. Lopes, Exact solution of the Perk-Schultz model, Phys. Rev. Lett. 67 (1991) 489 [INSPIRE].
E. Lopes, Exact solution of the multicomponent generalized six vertex model, Nucl. Phys. B 370 (1992) 636 [INSPIRE].
I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations, Commun. Math. Phys. 188 (1997) 267 [hep-th/9604080] [INSPIRE].
M. Karowski, On the bound state problem in (1 + 1)-dimensional field theories, Nucl. Phys. B 153 (1979) 244 [INSPIRE].
P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang-Baxter equation and representation theory. 1, Lett. Math. Phys. 5 (1981) 393 [INSPIRE].
P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method: recent developments, Lect. Notes Phys. 151 (1982) 61 [INSPIRE].
A.N. Kirillov and N.Y. Reshetikhin, Exact solution of the Heisenberg XXZ model of spins, J. Sov. Math. 35 (1986) 2627.
A.N. Kirillov and N.Y. Reshetikhin, Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum, J. Phys. A 20 (1987) 1565 [INSPIRE].
H. Frahm and N.A. Slavnov, New solutions to the reflection equation and the projecting method, J. Phys. A 32 (1999) 1547 [cond-mat/9810312].
W. Galleas and M.J. Martins, Solution of the SU(N) vertex model with non-diagonal open boundaries, Phys. Lett. A 335 (2005) 167 [nlin/0407027] [INSPIRE].
H.J. de Vega and A. González-Ruiz, Boundary K matrices for the XYZ, XXZ and XXX spin chains, J. Phys. A 27 (1994) 6129 [hep-th/9306089] [INSPIRE].
P.P. Kulish, Yang-Baxter equation and reflection equations in integrable models, hep-th/9507070 [INSPIRE].
M. Mintchev, É. Ragoucy and P. Sorba, Spontaneous symmetry breaking in the gl(N)-NLS hierarchy on the half line, J. Phys. A 34 (2001) 8345 [hep-th/0104079] [INSPIRE].
L. Mezincescu and R.I. Nepomechie, Fusion procedure for open chains, J. Phys. A 25 (1992) 2533 [INSPIRE].
Y.-K. Zhou, Row transfer matrix functional relations for Baxter’s eight vertex and six vertex models with open boundaries via more general reflection matrices, Nucl. Phys. B 458 (1996) 504 [hep-th/9510095] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
W.-L. Yang and Y.-Z. Zhang, Non-diagonal solutions of the reflection equation for the trigonometric \( A_{n-1}^{(1) } \) vertex model, JHEP 12 (2004) 019 [hep-th/0411160] [INSPIRE].
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] [INSPIRE].
R.I. Nepomechie, Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices, Nucl. Phys. B 831 (2010) 429 [arXiv:0911.5494] [INSPIRE].
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 [nlin/0411038].
A. Doikou, Fusion and analytical Bethe ansatz for the \( A_{N-1}^{(1) } \) open spin chain, J. Phys. A 33 (2000) 4755 [hep-th/0006081] [INSPIRE].
A. Doikou and R.I. Nepomechie, Duality and quantum algebra symmetry of the \( A_{N-1}^{(1) } \) open spin chain with diagonal boundary fields, Nucl. Phys. B 530 (1998) 641 [hep-th/9807065] [INSPIRE].
A. Doikou and R.I. Nepomechie, Bulk and boundary S matrices for the SU(N) chain, Nucl. Phys. B 521 (1998) 547 [hep-th/9803118] [INSPIRE].
H.J. de Vega and A. González-Ruiz, Exact solution of the SU q (n) invariant quantum spin chains, Nucl. Phys. B 417 (1994) 553 [hep-th/9309022] [INSPIRE].
H.J. de Vega and A. González-Ruiz, Exact Bethe ansatz solution for A n−1 chains with non-SU q (n) invariant open boundary conditions, Mod. Phys. Lett. A 09 (1994) 2207 [hep-th/9404141] [INSPIRE].
L. Mezincescu, R.I. Nepomechie and V. Rittenberg, Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms, Phys. Lett. A 147 (1990) 70 [INSPIRE].
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Cao, J., Yang, WL., Shi, K. et al. Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries. J. High Energ. Phys. 2014, 143 (2014). https://doi.org/10.1007/JHEP04(2014)143
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DOI: https://doi.org/10.1007/JHEP04(2014)143