On ${\tau }^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(s{l}_2)$

Abstract

We identify the precise relationship between the five-parameter ${\tau }^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q(s{l}_2)$-cyclic representation. By studying the Yang-Baxter relation of the six-vertex model, we discover an one-parameter family of $L$-operators in terms of the quantum group $U_q(s{l}_2)$. When $N$ is odd, the $N$-state ${\tau }^{(2)}$-model can be regarded as the XXZ chain of $U_{\mathsf{q}}(s{l}_2)$ cyclic representations with ${\mathsf{q}}^N=1$. The symmetry algebra of the ${\tau }^{(2)}$-model is described by the quantum affine algebra $U_{\mathsf{q}}({\widehat{sl}}_2)$ via the canonical representation. In general for an arbitrary $N$, we show that the XXZ chain with a $U_q(s{l}_2)$-cyclic representation for $q^{2N}=1$ is equivalent to two copies of the same $N$-state ${\tau }^{(2)}$-model.

arXiv commentLatex 11 pages; Typos corrected, Minor changes for clearer presentation, References added and updated-Journal version

References in corpus (12)

• How Algebraic Bethe Ansatz works for integrable model
• Transfer matrix functional relations for the generalized tau_2(t_q) model
• Derivation of the order parameter of the chiral Potts model
• Baxter-Bazhanov-Stroganov model: Separation of Variables and Baxter Equation
• The six and eight-vertex models revisited
• The sl_2 loop algebra symmetry of the six-vertex model at roots of unity
• The L(sl_2) symmetry of the Bazhanov-Stroganov model associated with the superintegrable chiral Potts model
• Eigenvectors in the Superintegrable Model I: sl_2 Generators
• Fusion Operators in the Generalized ${\tau }^{(2)}$-model and Root-of-unity Symmetry of the XXZ Spin Chain of Higher Spin
• The Q-operator for Root-of-Unity Symmetry in Six Vertex Model
• The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(s{l}_2)$
• Bethe Equation of ${\tau }^{(2)}$-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities

Cited by in corpus (2)

• Factorized finite-size Ising model spin matrix elements from Separation of Variables
• Duality and Symmetry in Chiral Potts Model