100 research outputs found

    On the Yang-Baxter equation for the six-vertex model

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    In this paper we review the theory of the Yang-Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin RR-matrix associated with the affine quantum algebra Uq(sl(2)^)U_q(\widehat{sl(2)}). In the simplest case of the spin s=1/2s=1/2 this RR-matrix naturally reduces to the RR-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the QQ-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin ss. We also give all functional equations satisfied by the transfer-matrices and QQ-operators.Comment: 25 pages, 1 figur

    An Analytic Formula for the A_2 Jack Polynomials

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    In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003), 451-482, math.CA/0306242] on separation of variables for the AnA_n Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995), 27-34, solv-int/9508002] where the integral representations for the A2A_2 Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the A2A_2 Jack polynomials in terms of generalised hypergeometric functions.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

    Q-operators in the six-vertex model

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    In this paper we continue the study of QQ-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin RR-matrix associated with the affine quantum algebra Uq(sl(2)^)U_q(\widehat{sl(2)}). Taking a special limit in this RR-matrix we obtained new formulas for the QQ-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct QQ-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-) integer spins.Comment: 18 pages, no figure

    The eight-vertex model and Painleve VI

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    In this letter we establish a connection of Picard-type elliptic solutions of Painleve VI equation with the special solutions of the non-stationary Lame equation. The latter appeared in the study of the ground state properties of Baxter's solvable eight-vertex lattice model at a particular point, η=π/3\eta=\pi/3, of the disordered regime.Comment: 9 pages, LaTeX, submitted to the special issue on Painleve VI, Journal of Physics

    Elliptic solution for modified tetrahedron equations

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    As is known, tetrahedron equations lead to the commuting family of transfer-matrices and provide the integrability of corresponding three-dimensional lattice models. We present the modified version of these equations which give the commuting family of more complicated two-layer transfer-matrices. In the static limit we have succeeded in constructing the solution of these equations in terms of elliptic functions.Comment: 11 page

    Electromagnetic Structure Functions of Nucleons in the Region of Very Small X

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    A two component model describing the electromagnetic nucleon structure functions in the low-x region, based on generalized vector dominance and color dipole approaches is briefly described.Comment: 3 pages, 1 figure, Talk given at the 14th Lomonosov Conference, Moscow, August 200

    Eight-vertex model and Painlev\'e VI equation. II. Eigenvector results

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    We study a special anisotropic XYZ-model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with Picard's elliptic solutions of the Painlev\'e VI equation. Connections with other problems related to the eight-vertex model are briefly discussed.Comment: 18 page

    Eight-vertex model and non-stationary Lame equation

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    We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the Δ=−1/2\Delta=-1/2 six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrodinger equation with the time-dependent potential given by the Weierstrass elliptic P-function where the modular parameter τ\tau plays the role of (imaginary) time. In the scaling limit the equation transforms into a ``non-stationary Mathieu equation'' for the vacuum eigenvalues of the Q-operators in the finite-volume massive sine-Gordon model at the super-symmetric point, which is closely related to the theory of dilute polymers on a cylinder and the Painleve III equation.Comment: 11 pages, LaTeX, minor misprints corrected, references adde
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