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A convenient basis for the Izergin-Korepin model. (English) Zbl 1404.82023
Summary: We propose a convenient orthogonal basis of the Hilbert space for the quantum spin chain associated with the $$A_2^{(2)}$$ algebra (or the Izergin-Korepin model). It is shown that compared with the original basis the monodromy-matrix elements acting on this basis take relatively simple forms, which is quite similar as that for the quantum spin chain associated with $$A_n$$ algebra in the so-called F-basis. As an application of our general results, we present the explicit recursive expressions of the Bethe states in this basis for the Izergin-Korepin model.
##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 17B81 Applications of Lie (super)algebras to physics, etc. 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 82B23 Exactly solvable models; Bethe ansatz
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##### References:
 [1] Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G., Quantum Inverse Scattering Method and Correlation Functions, (1993), Cambridge University Press [2] Wang, Y.; Yang, W.-L.; Cao, J.; Shi, K., Off-Diagonal Bethe Ansatz for Exactly Solvable Models, (2015), Springer Press [3] Maillet, J. M.; Sanchez de Santos, J., Drinfel’d twists and algebraic Bethe ansatz, Transl. Am. Math. Soc., Ser. 2, 201, 137-178, (1996) [4] Drinfeld, V. G., Constant quasiclassical solutions of the Yang-Baxter quantum equation, Sov. Math. Dokl., 28, 667-671, (1983) [5] Kitanine, N.; Maillet, J. M.; Terras, V., Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B, 554, 647-678, (1999) [6] Terras, V., Drinfel’d twists and functional Bethe ansatz, Lett. Math. Phys., 48, 263-276, (1999) [7] Albert, T. D.; Boos, H.; Flume, R.; Ruhlig, K., Resolution of the nested hierarchy for rational $$s l(n)$$ models, J. Phys. A, 33, 4963, (2000) [8] Albert, T. D.; Boos, H.; Flume, R.; Poghossian, R. H.; Ruhlig, K., An F-twisted XYZ model, Lett. Math. Phys., 53, 201-214, (2000) [9] Albert, T. D.; Ruhlig, K., Polarization-free generators for the Belavin model, J. Phys. A, 34, 1569, (2001) [10] Yang, W.-L.; Zhang, Y.-Z.; Zhao, S.-Y., Drinfeld twists and algebraic Bethe ansatz of the supersymmetric t-J model, J. High Energy Phys., 12, (2004); Drinfeld twists and algebraic Bethe ansatz of the supersymmetric model associated with $$U_q(g l(m | n))$$, Commun. Math. Phys., 264, 87-114, (2006) [11] Zhao, S.-Y.; Yang, W.-L.; Zhang, Y.-Z., Drinfeld twists and symmetric Bethe vectors of supersymmetric Fermion models, J. Stat. Mech., 04005, (2005); Determinant representations of correlation functions for the supersymmetric t-J model, Commun. Math. Phys., 268, 505-541, (2006); On the construction of correlation functions for the integrable supersymmetric fermion models, Int. J. Mod. Phys. B, 20, 505-549, (2006) [12] Izergin, A. G.; Korepin, V. E., The inverse scattering method approach to the quantum Shabat-Mikhailov model, Commun. Math. Phys., 79, 303-316, (1981) [13] Dodd, R. K.; Bullough, R. K., Polynomial conserved densities for the sine-Gordon equations, Proc. R. Soc. Lond. A, 352, 481-503, (1977) [14] Zhiber, A. V.; Shabat, A. B., Klein-Gordon equations with a nontrivial group, Dokl. Akad. Nauk SSSR, 247, 1103, (1979) [15] Yung, C. M.; Batchelor, M. T., Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices, Nucl. Phys. B, 435, 430-462, (1995) [16] Batchelor, M. T.; Yung, C. M., Exact results for the adsorption of a flexible self-avoiding polymer chain in two dimensions, Phys. Rev. Lett., 74, 2026, (1995) [17] Reshetikhin, N. Y., The functional equation method in the theory of exactly soluble quantum systems, Sov. Phys. JETP, 57, 691, (1983) [18] Tarasov, V. O., Algebraic Bethe ansatz for the Izergin-Korepin R matrix, Theor. Math. Phys., 56, 793-803, (1988) [19] Martins, M. J., The exact solution and the finite-size behaviour of the $$O s p(1 | 2)$$-invariant spin chain, Nucl. Phys. B, 450, 768-788, (1995) [20] Fan, H., Bethe ansatz for the Izergin-Korepin model, Nucl. Phys. B, 488, 409-425, (1997) [21] Lima-Santos, A., Reflection K-matrices for 19-vertex models, Nucl. Phys. B, 558, 637-667, (1999) [22] Hou, B.-Y.; Yang, W.-L.; Zhang, Y.-Z., The twisted quantum affine algebra $$U_q(A_2^{(2)})$$ and correlation functions of the Izergin-Korepin model, Nucl. Phys. B, 556, 485-504, (1999) [23] Li, G.-L.; Shi, K.-J.; Yue, R.-H., The algebraic Bethe ansatz for the Izergin-Korepin model with open boundary conditions, Nucl. Phys. B, 670, 401-438, (2003) [24] Sklyanin, E. K., The quantum Toda chain, Lect. Notes Phys., 226, 196-233, (1985); Goryachev-Chaplygin top and the inverse scattering method, J. Sov. Math., 31, 3417-3431, (1985); Separation of variables new trends, Prog. Theor. Phys. Suppl., 118, 35-60, (1995) [25] Niccoli, G., 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., (2012) [26] Hao, K.; Cao, J.-P.; Li, G.-L.; Yang, W.-L.; Shi, K.; Wang, Y., Exact solution of an $$s u(n)$$ spin torus, J. Stat. Mech., (2016); A representation basis for the quantum integrable spin chain associated with the $$s u(3)$$ algebra, J. High Energy Phys., 05, (2016) [27] Göhmann, F.; Korepin, V. E., Solution of the quantum inverse problem, J. Phys. A, 33, 1199, (2000) [28] Maillet, J. M.; Terras, V., On the quantum inverse scattering problem, Nucl. Phys. B, 575, 627-644, (2000)
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