Geodesic
Functional Equations and Quantum Separation of Variables for 3d Spin Models
Teoretičeskaâ i matematičeskaâ fizika, Tome 138 (2004) no. 2, pp. 269-282 Cet article a éte moissonné depuis la source Math-Net.Ru

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The recently proposed invariant formulation of the auxiliary linear problem for 3d integrable models provides several new ideas for solving the spectral problem of 3d spin models, e. g., the Zamolodchikov–Bazhanov–Baxter model in its vertex formulation. This paper announces results following from the invariant formulation. We formulate the class of 3d spin models that are essentially appropriately parameterized inhomogeneous Zamolodchikov–Bazhanov–Baxter models, present an expression for the generating function of the complete set of matrices commuting with the transfer matrix of this model (integrals of motion), give the functional equations defining the eigenvalues of the integrals of motion and the transfer matrices, explicitly describe the groupoid of isospectral transformations of the initial system of integrals of motion, and finally give an explicit parameterization of a projection operator onto the separated states in the sense of the quantum separation of variables (functional Bethe ansatz).
Keywords: 3d integrable models, Zamolodchikov–Bazhanov–Baxter model. 
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     title = {Functional {Equations} and {Quantum} {Separation} of {Variables} for 3d {Spin} {Models}},
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S. M. Sergeev. Functional Equations and Quantum Separation of Variables for 3d Spin Models. Teoretičeskaâ i matematičeskaâ fizika, Tome 138 (2004) no. 2, pp. 269-282. http://geodesic.mathdoc.fr/item/TMF_2004_138_2_a5/

[1] S. Sergeev, J. Phys. A, 32 (1999), 5693–5714 ; Phys. Lett. A, 265 (2000), 364–368 ; J. Nonlinear Math. Phys., 7 (2000), 57–72 ; С. М. Сергеев, ТМФ, 124 (2000), 391–409 ; Зап. научн. семин. ПОМИ, 269, 2000, 292–307 ; Вполне интегрируемые дискретные системы в трех измерениях, Автореферат дисс. ...докт. физ.-матем. наук, ПОМИ, С.-Петербург, 2001 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | Zbl

[2] A. B. Zamolodchikov, ZhETF, 79 (1980), 641–664 ; A. B. Zamolodchikov, Commun. Math. Phys., 79 (1981), 489–505 ; R. J. Baxter, Commun. Math. Phys., 88 (1983), 185–205 ; Phys. Rev. Lett., 53 (1984), 1795–1798 ; Physica D, 18 (1986), 321–347 ; V. V. Bazhanov, R. J. Baxter, J. Stat. Phys., 69 (1992), 453–485 | MR | DOI | MR | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[3] S. M. Sergeev, V. V. Mangazeev, Yu. G. Stroganov, J. Stat. Phys., 82 (1996), 31–50 | DOI | MR

[4] L. D. Faddeev, Lett. Math. Phys., 34 (1995), 249–254 ; “Modular double of quantum group”, Quantization, Deformation, and Symmetries, Conférence Moshé Flato. V. 1 (Dijon, France, September 5–8, 1999), eds. G. Dito et al., Kluwer Acad. Publ., Dordrecht, 2000, 149–156 ; ; V. V. Bazhanov, N. Yu. Reshetikhin, J. Phys. A, 28 (1995), 2217–2226 ; V. Bazhanov, A. Bobenko, N. Reshetikhin, Commun. Math. Phys., 175 (1996), 377–400 ; R. Kashaev, N. Reshetikhin, Invariants of links with flat connections in their complements. II: Holonomy $R$-matrices related to quantized universal enveloping algebras at roots of 1, E-print math.qn/9912078E-print math.AT/0202212 | DOI | MR | Zbl | MR | DOI | MR | Zbl | DOI | MR | Zbl

[5] E. K. Sklyanin, “Functional Bethe ansatz”, Integrable and Superintegrable Systems, ed. B. A. Kupershmidt, World Scientific, Teanect, NJ, 1990, 8–33 ; “Quantum Inverse Scattering Method. Selected Topics”, Quantum Group and Quantum Integrable Systems, Nankai Lectures in Mathematical Physics, ed. Mo-Lin Ge, World Scientific, Singapore, 1992, 63–97 ; Progr. Theor. Phys. (suppl.), 118 (1995), 35–60 ; F. A. Smirnov, Separation of variables for quantum integrable models related to $U_q(\widehat{sl}_N)$, E-print math-ph/0109013 | DOI | MR | MR | DOI | MR | Zbl | MR

[6] S. Pakuliak, S. Sergeev, Relativistic Toda chain at root of unity, E-print nlin.SI/0101024 | MR