<!-- *** Custom HTML *** --> The discrete quantum Sine-Gordon model at roots of unity remarkably combines a classical integrable system with an integrable quantum spin system, whose parameters obey classical equations of motion. We show that the fundamental $R$-matrix of the model (which satisfies a difference property Yang–Baxter equation) naturally splits into a product of a singular "classical" part and a finite dimensional quantum part. The classical part of the $R$-matrix itself satisfies the quantum Yang–Baxter equation, and therefore can be factored out producing, however, a certain "twist" of the quantum part. We show that the resulting equation exactly coincides with the star-triangle relation of the $N$-state chiral Potts model. The associated spin model on the whole lattice is, in fact, more general than the chiral Potts and reduces to the latter only for the simplest (constant) classical background. In a general case the model is inhomogeneous: its Boltzmann weights are determined by non-trivial background solutions of the equations of motion of the classical discrete sine-Gordon model.
A new type of Yang-Baxter equation (YBE) for
R-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the
ZN symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for
N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated
R-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter
R-matrix---realized as an intertwiner of cyclic representations of
Uq(sl2) at a root of unity---and Shastry's two-parameter
R-operator obtained from the decorated YBE.
We obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known.This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters.If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on
We extend the notion of space shifts introduced in [FV3] for certain quantum light cone lattice equations of sine-Gordon type at root of unity (e.g. [FV1][FV2][BKP][BBR]). As a result we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation which is obtained by exponentiating these roots is exactly the evolution equation for the "classical background" as described in [BBR]. As an application for the introduced constructions, a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice as constructed in [DV] is derived. Supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 288 e-mail: nadja@math.tu-berlin.de 1 Introduction Let us consider a class of integrable lattice systems defined by an evolution equation of the following type: g u = V 0 (g l \\Gamma g r ) + g d = 0; (1) where...
An operator defoemed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its q → 1 limit seem to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. PACS numbers 02.30.Ik, 02.20.Uw, 05.50+q, 03.65.Fd 1
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).