39 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

    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

    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

    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

    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

    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

    Density and current profiles in Uq(A2(1))U_q(A^{(1)}_2) zero range process

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    The stochastic RR matrix for Uq(An(1))U_q(A^{(1)}_n) introduced recently gives rise to an integrable zero range process of nn classes of particles in one dimension. For n=2n=2 we investigate how finitely many first class particles fixed as defects influence the grand canonical ensemble of the second class particles. By using the matrix product stationary probabilities involving infinite products of qq-bosons, exact formulas are derived for the local density and current of the second class particles in the large volume limit.Comment: 31 pages, 12 figures. Minor revision from version

    Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation

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    We study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.Comment: Plenary talk at the XVI International Congress on Mathematical Physics, 3-8 August 2009, Prague, Czech Republi
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