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Quantum Integrable Models in Discrete 2 + 1 Dimensional Space-Time: Auxiliary Linear Problem on a Lattice, Zero-Curvature Representation, Isospectral Deformation of the Zamolodchikov-Bazhanov-Baxter Model

С. М. Сергеев

Published Jan 1, 2004• 11 citations
OpenAlex

Abstract

An invariant approach to quantum integrable models in wholly discrete 2 + 1-dimensional spacetime is considered. An auxiliary linear problem on two-dimensional lattices generalizing quantum chains is formulated. A method of constructing a complete set of

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L. D. Faddeev, A. Yu. Volkov

NATO ASI series. Series B : Physics•Jan 1, 1997•3 citations
The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.
Exact Spectrum of the Critical Three-State Potts Chain (G Albertini) New Results Related to the Chiral Potts Model (H Au-Yang & J H H Perk) Chiral Potts Model: Corner Transfer Matrices and Parametrizations (R J Baxter) Solvable Lattice Models in Three Dimensions (VV Bazhanov) Dressing Symmetries (D Bernard) Knot Theory and Chem-Simons Lagrangian: Some Remarks about the corresponding Diagrams (P Cartier) Recent Progress in 2-D Quantum Gravity (V Dotsenko) On Coordinate Algebras of Quantum Groups at Root of Unity (B Enriquez) Integrability in Discrete Systems (B Grammaticos) New Construction of Solvable RSOS Models (S 0 Warnaar) Birational Automorphisms and Yang-Baxter Equations (J-M Maillard) Yang-Baxter Equations and Non-Commutative Differential Calculus (G Maltsiniotis) Physics Beyond Quasi-Particules: Spectrum and Completeness of the Three-State Superintegrable Chiral Potts Model (B M McCoy) Analytic Calulation of Conformal Weights for A-D-E Lattice Models (P Pearce) Quantum Tops (A Reiman) Remarks on Quasi-Hopf Algebras (M Rosso) Symmetries of the Yang-Baxter Equations: A Remarkable Family of Discrete Groups (C-M Viallet) The Yang-Baxter Equation in Knot Theory (F Y Wu).
A sort of two dimensional linear auxiliary problem for the complex of 3D RRR -- operators associated with the Zamolodchikov -- Bazhanov -- Baxter statistical model is proposed. This problem resembles the problem of the local Yang -- Baxter equation but does not coincide with it. The formulation of the auxiliary problem admits a notion of a ``fusion'', and usual local Yang -- Baxter equation appears among other results of this ``fusion''.
We apply a 3-dimensional approach to describe a new parametrization of the L-operators for the 2-dimensional Bazhanov-Stroganov (BS) integrable spin model related to the chiral Potts model. This parametrization is based on the solution of the associated classical discrete integrable system. Using a 3-dimensional vertex satisfying a modified tetrahedron equation, we construct an operator which generalizes the BS quantum intertwining matrix S. This operator describes the isospectral deformations of the integrable BS model.