Abstract
Exact solution of the quantum integrable \( {D}_2^{(2)} \) spin chain with generic integrable boundary fields is constructed. It is found that the transfer matrix of this model can be factorized as the product of those of two open staggered anisotropic XXZ spin chains. Based on this identity, the eigenvalues and Bethe ansatz equations of the \( {D}_2^{(2)} \) model are derived via off-diagonal Bethe ansatz.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N.F. Robertson, J.L. Jacobsen and H. Saleur, Lattice regularisation of a non-compact boundary conformal field theory, JHEP 02 (2021) 180 [arXiv:2012.07757] [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3 and SL(2, ℝ) WZW model 1.: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
J.M. Maldacena, H. Ooguri and J. Son, Strings in AdS3 and the SL(2, ℝ) WZW model. Part 2. Euclidean black hole, J. Math. Phys. 42 (2001) 2961 [hep-th/0005183] [INSPIRE].
A. Hanany, N. Prezas and J. Troost, The Partition function of the two-dimensional black hole conformal field theory, JHEP 04 (2002) 014 [hep-th/0202129] [INSPIRE].
N.Y. Reshetikhin, The spectrum of the transfer matrices connected with Kac-Moody algebras, Lett. Math. Phys. 14 (1987) 235.
M.J. Martins, Unified algebraic Bethe ansatz for two-dimensional lattice models, Phys. Rev. E 59 (1999) 7220.
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].
L. Mezincescue and R.I. Nepomechie, Integrable open spin chains with nonsymmetrical R-matrices, J. Phys. A 24 (1991) L17.
M.J. Martins and X.W. Guan, Integrability of the \( {D}_n^2 \) vertex models with open boundary, Nucl. Phys. B 583 (2000) 721 [nlin/0002050] [INSPIRE].
R.I. Nepomechie, R.A. Pimenta and A.L. Retore, The integrable quantum group invariant \( {A}_{2n-1}^{(2)} \) and \( {D}_{n+1}^{(2)} \) open spin chains, Nucl. Phys. B 924 (2017) 86 [arXiv:1707.09260] [INSPIRE].
R.I. Nepomechie and A.L. Retore, The spectrum of quantum-group-invariant transfer matrices, Nucl. Phys. B 938 (2019) 266 [arXiv:1810.09048] [INSPIRE].
N.F. Robertson, M. Pawelkiewicz, J.L. Jacobsen and H. Saleur, Integrable boundary conditions in the antiferromagnetic Potts model, JHEP 05 (2020) 144 [arXiv:2003.03261] [INSPIRE].
V.V. Bazhanov, Trigonometric Solution of Triangle Equations and Classical Lie Algebras, Phys. Lett. B 159 (1985) 321 [INSPIRE].
M. Jimbo, Quantum r Matrix for the Generalized Toda System, Commun. Math. Phys. 102 (1986) 537 [INSPIRE].
V.V. Bazhanov, Integrable Quantum Systems and Classical Lie Algebras (in Russian), Commun. Math. Phys. 113 (1987) 471 [INSPIRE].
R.B. Potts, Some generalized order-disorder transformations, Proc. Camb. Phil. Soc. 48 (1952) 106.
H. Saleur, The Antiferromagnetic Potts model in two-dimensions: Berker-Kadanoff phases, antiferromagnetic transition, and the role of Beraha numbers, Nucl. Phys. B 360 (1991) 219 [INSPIRE].
J.L. Jacobsen and H. Saleur, The Antiferromagnetic transition for the square-lattice Potts model, Nucl. Phys. B 743 (2006) 207 [cond-mat/0512058] [INSPIRE].
Y. Ikhlef, J. Jacobsen and H. Saleur, A staggered six-vertex model with non-compact continuum limit, Nucl. Phys. B 789 (2008) 483 [INSPIRE].
Y. Ikhlef, J.L. Jacobsen and H. Saleur, An Integrable spin chain for the SL(2, ℝ)/U(1) black hole sigma model, Phys. Rev. Lett. 108 (2012) 081601 [arXiv:1109.1119] [INSPIRE].
C. Candu and Y. Ikhlef, Nonlinear integral equations for the SL(2, ℝ)/U(1) black hole sigma model, J. Phys. A 46 (2013) 415401 [arXiv:1306.2646] [INSPIRE].
R.I. Nepomechie and A.L. Retore, Factorization identities and algebraic Bethe ansatz for \( {D}_2^{(2)} \) models, JHEP 03 (2021) 089 [arXiv:2012.08367] [INSPIRE].
R.I. Nepomechie and R.A. Pimenta, New \( {D}_{n+1}^{(2)} \) K-matrices with quantum group symmetry, J. Phys. A 51 (2018) 39LT02.
R.I. Nepomechie, R.A. Pimenta and A.L. Retore, Towards the solution of an integrable \( {D}_2^{(2)} \)(2) spin chain, J. Phys. A 52 (2019) 434004 [arXiv:1905.11144] [INSPIRE].
A. Lima-Santos and R. Malara, \( {C}_n^{(1)} \), \( {D}_n^{(1)} \) and \( {A}_{2n-1}^{(2)} \) reflection K-matrices, Nucl. Phys. B 675 (2003) 661 [nlin/0307046] [INSPIRE].
R. Malara and A. Lima-Santos, On \( {A}_{n-1}^{(1)} \), \( {B}_n^{(1)} \), \( {C}_n^{(1)} \), \( {D}_n^{(1)} \), \( {A}_{2n}^{(2)} \), \( {A}_{2n-1}^{(2)} \) and \( {D}_{n+1}^{(2)} \) reflection K-matrices, J. Stat. Mech. 0609 (2006) P09013 [nlin/0412058] [INSPIRE].
Y. Wang, W.-L. Yang, J. Cao and K. Shi, Off-Diagonal Bethe Ansatz for Exactly Solvable Models, Springer, Berlin, Heidelberg, Germany (2015) [DOI].
H. Frahm and M.J. Martins, Phase Diagram of an Integrable Alternating Uq[sl(2|1)] Superspin Chain, Nucl. Phys. B 862 (2012) 504 [arXiv:1202.4676] [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 N.Y. Reshetikhin, Quantum linear problem for the sine-Gordon equation and higher representation, Zap. Nauchn. Semin. 101 (1981) 101 [INSPIRE].
A.N. Kirillov and N.Y. 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].
L. Mezincescu and R.I. Nepomechie, Fusion procedure for open chains, J. Phys. A 25 (1992) 2533 [INSPIRE].
L. Mezincescu and R.I. Nepomechie, Analytical Bethe Ansatz for quantum algebra invariant spin chains, Nucl. Phys. B 372 (1992) 597 [hep-th/9110050] [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].
J. Cao, W.-L. Yang, K. Shi and Y. Wang, On the complete-spectrum characterization of quantum integrable spin chains via inhomogeneous T − Q relation, J. Phys. A 48 (2015) 444001 [arXiv:1409.5303] [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].
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].
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].
X. Zhang, Y.-Y. Li, J. Cao, W.-L. Yang, K. Shi and Y. Wang, Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields, Nucl. Phys. B 893 (2015) 70 [arXiv:1412.6905] [INSPIRE].
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2202.06531
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Li, GL., Cao, J., Yang, WL. et al. Spectrum of the quantum integrable \( {D}_2^{(2)} \) spin chain with generic boundary fields. J. High Energ. Phys. 2022, 101 (2022). https://doi.org/10.1007/JHEP04(2022)101
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2022)101