Diagonalization of the \(XXZ\) Hamiltonian by vertex operators. (English) Zbl 0769.17020

The \(XXZ\) Hamiltonian is an operator \(H\) on the doubly infinite tensor product \(\bigotimes^ \infty_{k=-\infty}\mathbb{C}^ 2\) of copies of \(\mathbb{C}^ 2\). Viewing \(\mathbb{C}^ 2\) as the natural 2-dimensional representation of the quantum affine algebra \(U_ q(\widehat{sl}_ 2)\), it turns out that \(H\) commutes with the action of \(U_ q(\widehat{sl}_ 2)\). The authors analyze this representation of \(U_ q(\widehat{sl}_ 2)\) by crystal basis techniques, and use their results to diagonalize \(H\).


17B81 Applications of Lie (super)algebras to physics, etc.
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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[1] Frenkel, I.B., Reshetikhin, N.Yu.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys.146, 1–60 (1992) · Zbl 0760.17006 · doi:10.1007/BF02099206
[2] Kang, S.-J., Kashiwara, M., Misra, K., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. Int. J. Mod. Phys. A.7, Suppl. 1A, 449–484 (1992) · Zbl 0925.17005 · doi:10.1142/S0217751X92003896
[3] Foda, O., Miwa, T.: Corner transfer matrices and quantum affine algebras. ibid. 279–302 · Zbl 0925.17008 · doi:10.1142/S0217751X92003811
[4] Date, E., Jimbo, M., Okado, M.: Crystal base andq vertex operators. Osaka Univ. Math. Sci. preprint,1 (1991) Commun. Math. Phys., to appear · Zbl 0790.17007
[5] Smirnov, F.A.: Dynamical symmetries of massive integrable models. Int. J. Mod. Phys. A7, Suppl. 1B, 813–837, 839–858 (1992) · Zbl 0925.17027 · doi:10.1142/S0217751X92004063
[6] Bernard, D., LeClair, A.: Quantum Group Symmetries and Non-Local Currents in 2D QFT. Commun. Math. Phys.142, 99–138 (1991) · Zbl 0746.35037 · doi:10.1007/BF02099173
[7] Felder, G., LeClair, A.: Restricted Quantum Affine Symmetry of Restricted Minimal Conformal Models. Int. J. Mod. Phys. A7, Suppl. 1A, 239–278 (1992) · Zbl 0925.35140 · doi:10.1142/S0217751X9200380X
[8] Eßler, F.H.L., Korepin, V.E., Schoutens, K.: Fine structure of the Bethe Ansatz for the spin-1/2 HeisenbergXXX model. preprint (1992)
[9] Destri, C., Lowenstein, J.H.: Analysis of the Bethe-Ansatz equations of the chiral-invariant Gross-Neveu model. Nucl. Phys. B205 [FS5], 369–385 (1982) · doi:10.1016/0550-3213(82)90363-7
[10] Babelon, O., de Vega, H.J., Viallet, C.M.: Analysis of the Bethe Ansatz equations of theXXZ model. Nucl. Phys. B220 [FS8], 13–34 (1983) · doi:10.1016/0550-3213(83)90131-1
[11] Faddeev, L., Takhatajan, L.A.: Spectrum and scattering of excitations in the one-dimensional Heisenberg model. J. Soviet Math.24, 241–267 (1984) · Zbl 0532.47009 · doi:10.1007/BF01087245
[12] Pasquier, V., Saleur, H.: Common structures between finite size systems and conformal field theories through quantum groups. Nucl. Phys. B330, 523–556 (1990) · doi:10.1016/0550-3213(90)90122-T
[13] Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press 1982 · Zbl 0538.60093
[14] Wu T.-T., McCoy, B.M., Tracy, C.A., Barouch, E.: Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region. Phys. Rev. B13, 316–374 (1976) · doi:10.1103/PhysRevB.13.316
[15] Sato, M., Miwa, T., Jimbo, M.: Holonomic quantum fields I–V. Publ. RIMS,14–17 223–267 (1978); 201–278, 577–629, 871–972 (1979); 531–584 (1980) · Zbl 0383.35066 · doi:10.2977/prims/1195189284
[16] Smirnov, F.A.: Form factors in completely integrable models of quantum field theory. Advanced Series in Mathematical Physics vol.14. Singapore: World Scientific 1992 · Zbl 0788.46077
[17] Belavin, A.A., Polyako, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys.B241, 333–380 (1984) · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[18] Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory onP 1 and monodromy representations of braid group. Advanced Studies in Pure Mathematics, vol.16, Tokyo: Kinokuniya 1988 · Zbl 0661.17021
[19] Zamolodchikov, A.B.: Integrable field theory from conformal field theory. Advanced Studies in Pure Mathematics vol.19. Tokyo: Kinokuniya 1989 · Zbl 0703.17014
[20] Feigin, B., Frenkel, E.: Free field resolutions in affine Toda field theories. Phys. Lett.B276, 79–86 (1992) · doi:10.1016/0370-2693(92)90544-E
[21] Corrigan, E., Dorey, P.: A representation of the exchange relation for affine Toda field theory. Phys. Lett.,B273, 237–245 (1991) · doi:10.1016/0370-2693(91)91677-N
[22] Baxter, R.J.: Corner transfer matrices of the eight vertex model. Low-temperature expansions and conjectured properties. J. Stat. Phys.15, 485–503 (1976) · doi:10.1007/BF01020802
[23] Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: One dimensional configuration sums in vertex models and affine Lie algebra characters. Lett. Math. Phys.17, 69–77 (1989) · Zbl 0681.17016 · doi:10.1007/BF00420017
[24] Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models: Local height probabilities and theta function identities. Nucl. Phys. B290. [FS20], 231–273 (1987) · Zbl 0679.17010 · doi:10.1016/0550-3213(87)90187-8
[25] Kashiwara, M.: On crystal bases of theq-analogue of universal enveloping algebras. Duke Math. J.63, 465–516 (1991) · Zbl 0739.17005 · doi:10.1215/S0012-7094-91-06321-0
[26] Kashiwara, M.: Global crystal bases of quantum groups. RIMS preprint,756 (1991) · Zbl 0739.17005
[27] Baxter, R.J.: Spontaneous staggered polarization of theF model. J. Stat. Phys.9, 145–182 (1973) · doi:10.1007/BF01016845
[28] Tetel’man, M.G.: Lorentz group for two-dimensional integrable lattice systems. Zh. Eksp. Teor. Fiz.82, 528–535 (1982)
[29] Misra, K.lC., Miwa, T.: Crystal base for the basic representation of \(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) . Commun. Math. Phys.134, 79–88 (1990) · Zbl 0724.17010 · doi:10.1007/BF02102090
[30] Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of representations of \(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) atq=0. Commun. Math. Phys.136, 543–566 (1991) · Zbl 0749.17015 · doi:10.1007/BF02099073
[31] Frahm, H., Thacker, H.B.: Corner transfer matrix eigenstates for the six vertex model. J. Phys. A: Math. Gen.24, 5587–5603 (1991) · doi:10.1088/0305-4470/24/23/023
[32] Birkhoff, G.D.: The generalized Riemann problem for linear differential equations and the allied problems for linear difference andq-difference equations. Proc. Am. Acad. Arts and Sci.49, 521–568 (1914) · doi:10.2307/20025482
[33] Aomoto, K.: A note on holonomicq-difference systems, Algebraic Analysis, vol. I, Eds. Kashiwara, M., Kawai, T., pp. 25–28. San Diego: Academic Press 1988
[34] Lusztig, G.: EquivariantK-theory and representations of Hecke algebras. Proc. Am. Math. Soc.94, 337–342 (1985) · Zbl 0571.22014
[35] James, G.D., Kerber, A.: The representation theory of the symmetric group. Reading, MA: Addison-Wesley 1981 · Zbl 0491.20010
[36] Kirillov, A.N., Smirnov, F.A.: Form factors in theSU (2)-invariant Thirring model. J. Soviet Math.47, 2423–2450 (1989) · Zbl 0692.35106 · doi:10.1007/BF01840425
[37] des Cloiseau, Gaudin, M.: Anisotropic linear magnetic chain. J. Math. Phys.7, 1384–1400 (1966) · doi:10.1063/1.1705048
[38] Whittaker, E.T., Watson, G.N.: Modern Analysis. 4th ed., London: Cambridge University Press 1962 · Zbl 0105.26901
[39] Reshetikhin, N.Yu.:S-matrices, in integrable models of isotropic magnetic chains: I. J. Phys. A: Math. Gen.24, 3299–3309 (1991) · Zbl 0738.58024 · doi:10.1088/0305-4470/24/14/017
[40] Kashiwara, M., Miwa, T.: A class of elliptic solutions to the star-triangle relation. Nucl. Phys. B275 [FS17], 121–134 (1986) · doi:10.1016/0550-3213(86)90591-2
[41] Hasegawa, K., Yamada, Y.: Algebraic derivation of the brokenZ N symmetric model. Phys. Lett.146A, 387–396 (1990)
[42] Au-Yang, H., McCoy, B.M., Perk, J.H.H., Tang, S., Yan, M.-L.: Commuting transfer matrices in the chiral Potts models: Solutions of star-triangle equations with genus, >1. Phys. Lett.123A, 219–223 (1987)
[43] Baxter, R.J., Perk, J.H.H., Au-Yang, H.: New solutions of the star-triangle relations for the chiral Potts model. Phys. Lett.128A, 138–142 (1988)
[44] Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. of A.M.S.4, 365–421 (1991) · Zbl 0738.17011
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