Abstract
The off-diagonal Bethe ansatz method is generalized to the high spin integrable systems associated with the su(2) algebra by employing the spin-s isotropic Heisenberg chain model with generic integrable boundaries as an example. With the fusion techniques, certain closed operator identities for constructing the functional T − Q relations and the Bethe ansatz equations are derived. It is found that a variety of inhomogeneous T − Q relations obeying the operator product identities can be constructed. Numerical results for two-site s = 1 case indicate that an arbitrary choice of the derived T − Q relations is enough to give the complete spectrum of the transfer matrix.
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References
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
S. Novikov, The Hamiltonian formalism and a many-valued analogue of the Morse theory, Usp. Math. Nauk. 37 (1982) 3.
E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
R. Thomale, S. Rachel, P. Schmitteckert and M. Greiter, A Family of spin-S chain representations of SU(2) k Wess-Zumino-Witten models, Phys. Rev. B 85 (2012) 195149 [arXiv:1110.5956] [INSPIRE].
R. Shankar and E. Witten, The S Matrix of the Supersymmetric Nonlinear σ-model, Phys. Rev. D 17 (1978) 2134 [INSPIRE].
C. Ahn, D. Bernard and A. LeClair, Fractional Supersymmetries in Perturbed Coset CFTs and Integrable Soliton Theory, Nucl. Phys. B 346 (1990) 409 [INSPIRE].
T. Inami, S. Odake and Y.-Z. Zhang, Supersymmetric extension of the sine-Gordon theory with integrable boundary interactions, Phys. Lett. B 359 (1995) 118 [hep-th/9506157] [INSPIRE].
R.I. Nepomechie, The Boundary supersymmetric sine-Gordon model revisited, Phys. Lett. B 509 (2001) 183 [hep-th/0103029] [INSPIRE].
Z. Bajnok, L. Palla and G. Takács, Spectrum of boundary states in N = 1 SUSY sine-Gordon theory, Nucl. Phys. B 644 (2002) 509 [hep-th/0207099] [INSPIRE].
H. Frahm and M. Stahlsmeier, Spinon statistics in integrable spin- S Heisenberg chains, Phys. Lett. A 250 (1998) 293 [cond-mat/9803381].
N. Andrei and C. Destri, Solution of the multichannel Kondo problem, Phys. Rev. Lett. 52 (1984) 364.
A. Tsvelik and P. Wiegmann, Solution of two-channel Kondo problem (Scaling and Integrability), Z. Phys. 54 (1984) 201.
J. Dai, Y. Wang and U. Eckern, Ghost spins and quantum critical behavior in a spin chain with local bond deformation, Phys. Rev. B 60 (1999) 6594.
A.B. Zamolodchikov and V.A. Fateev, Model factorized s matrix and an integrable heisenberg chain with spin 1., in russian, Sov. J. Nucl. Phys. 32 (1980) 298 [INSPIRE].
P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method, recent developments, Lect. Notes Phys. 151 (1982) 61.
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 N.Y. Reshetikhin, Quantum linear problem for the sine-Gordon equation and higher representation, J. Sov. Math. 23 (1983) 2435 [INSPIRE].
A.N. Kirillov and N.Yu. Reshetikhin, Exact solution of the Heisenberg XXZ model of spin s, 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].
C.-N. Yang, Some exact results for the many body problems in one dimension with repulsive delta function interaction, Phys. Rev. Lett. 19 (1967) 1312 [INSPIRE].
R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.
H.M. Babujian, Exact solution of the isotropic Heisenberg chain with arbitrary spins: thermodynamics of the model, Nucl. Phys. B 215 (1983) 317 [INSPIRE].
L.A. Takhtajan, The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins, Phys. Lett. A 87 (1982) 479 [INSPIRE].
H.M. Babujian, Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spin S, Phys. Lett. A 90 (1982) 479 [INSPIRE].
I.V. Cherednik, Factorizing Particles on a Half Line and Root Systems, Theor. Math. Phys. 61 (1984) 977 [INSPIRE].
E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A 21 (1988) 2375 [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].
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] [INSPIRE].
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.
R.I. Nepomechie, Bethe ansatz solution of the open XX spin chain with nondiagonal boundary terms, J. Phys. A 34 (2001) 9993.
R.I. Nepomechie, Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity, Nucl. Phys. B 622 (2002) 615.
R.I. Nepomechie, Functional relations and Bethe Ansatz for the XXZ chain, J. Statist. 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].
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, Refined Razumov-Stroganov conjectures for open boundaries, J. Stat. Mech. 9 (2004) 9 [math-ph/0408042].
A. Nichols, V. Rittenberg and J. de Gier, One-boundary Temperley-Lieb algebras in the XXZ and loop models, J. Stat. Mech. 0503 (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 Delta = 0 and Delta = -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 [INSPIRE].
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) 004 [hep-th/0503019] [INSPIRE].
A. Doikou, The Open XXZ and associated models at q root of unity, J. Stat. Mech. 0605 (2006) P05010 [hep-th/0603112] [INSPIRE].
Z. Bajnok, Equivalences between spin models induced by defects, J. Stat. Mech. 0606 (2006) P06010 [hep-th/0601107] [INSPIRE].
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].
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] [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, 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, 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. 1210 (2012) P10025 [arXiv:1206.0646] [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.
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].
S. Faldella, N. Kitanine and G. Niccoli, The complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms, J. Stat. Mech. 1401 (2014) P01011 [arXiv:1307.3960] [INSPIRE].
S. Belliard, N. Crampé, E. Ragoucy, Algebraic Bethe Ansatz for Open XXX Model with Triangular Boundary Matrices, Lett. Math. Phys. 103 (2013) 493 [arXiv:1209.4269].
L. Frappat, R. Nepomechie and É. Ragoucy, Complete Bethe Ansatz solution of the open spin-s XXZ chain with general integrable boundary terms, J. Stat. Mech. 0709 (2007) P09009 [arXiv:0707.0653] [INSPIRE].
R. Murgan, Bethe ansatz of the open spin-s XXZ chain with nondiagonal boundary terms, JHEP 04 (2009) 076 [arXiv:0901.3558] [INSPIRE].
R. Baiyasi and R. Murgan, Generalized T-Q relations and the open spin-s XXZ chain with nondiagonal boundary terms, J. Stat. Mech. 1210 (2012) P10003 [arXiv:1206.0814] [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.
J. Cao, W.-L. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz and exact solution of a topological spin ring, Phys. Rev. Lett. 111 (2013) 137201.
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. Shi and Y. Wang, Spin-1/2 XYZ model revisit: general solutions via off-diagonal Bethe ansatz, Nucl. Phys. B 866 (2014) 185.
J. Cao, W.-L. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz solutions of the anisotropic spin- \( \frac{1}{2} \) chains with arbitrary boundary fields, Nucl. Phys. B 877 (2013) 152 [arXiv:1307.2023] [INSPIRE].
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, J. Stat. Mech. 2014 (2014) P04031 [arXiv:1312.0376] [INSPIRE].
J. Cao, W.-L. Yang, K. Shi and Y. Wang, Nested off-diagonal Bethe ansatz and exact solutions of the SU(N ) spin chain with generic integrable boundaries, JHEP 04 (2014) 143 [arXiv:1312.4770] [INSPIRE].
K. Hao, J. Cao, G.-L. Li, W.-L. Yang, K. Shi et al., Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms, JHEP 06 (2014) 128 [arXiv:1403.7915] [INSPIRE].
Y.-Y. Li, J. Cao, W.-L. Yang, K. Shi and Y. Wang, Thermodynamic limit and surface energy of the XXZ spin chain with arbitrary boundary fields, Nucl. Phys. B 884 (2014) 17 [arXiv:1401.3045] [INSPIRE].
P. Baseilhac and K. Koizumi, Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory, J. Stat. Mech. 0709 (2007) P09006 [hep-th/0703106] [INSPIRE].
N. Kitanine, J.-M. Maillet and G. Niccoli, Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables, J. Stat. Mech. 1405 (2014) P05015 [arXiv:1401.4901] [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].
X. Zhang, Y.-Y. Li, J. Cao, W.-L. Yang, K. Shi et al., Retrieve the Bethe states of quantum integrable models solved via off-diagonal Bethe Ansatz, arXiv:1407.5294 [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].
E.C. Fireman, A. Lima-Santos and W. Utiel, Bethe ansatz solution for quantum spin 1 chains with boundary terms, Nucl. Phys. B 626 (2002) 435 [nlin/0110048] [INSPIRE].
A. Doikou, Fused integrable lattice models with quantum impurities and open boundaries, Nucl. Phys. B 668 (2003) 447 [hep-th/0303205] [INSPIRE].
T. Inami, S. Odake and Y.-Z. Zhang, Reflection K matrices of the 19 vertex model and XXZ spin 1 chain with general boundary terms, Nucl. Phys. B 470 (1996) 419 [hep-th/9601049] [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].
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].
C. Korff, PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields, J. Phys. A 41 (2008) 295206 [arXiv:0803.4500] [INSPIRE].
Y. Jiang, S. Cui, J. Cao, W.-L. Yang and Y. Wang, Completeness and Bethe root distribution of the spin-1/2 Heisenberg chain with arbitrary boundary fields, arXiv:1309.6456 [INSPIRE].
G. Niccoli, Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables, J. Math. Phys. 54 (2013) 053516 [arXiv:1206.2418] [INSPIRE].
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Cao, J., Cui, S., Yang, WL. et al. Exact spectrum of the spin-s Heisenberg chain with generic non-diagonal boundaries. J. High Energ. Phys. 2015, 36 (2015). https://doi.org/10.1007/JHEP02(2015)036
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DOI: https://doi.org/10.1007/JHEP02(2015)036