The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(s{l}_2)$

Abstract

We demonstrate that the transfer matrix of the inhomogeneous $N$-state chiral Potts model with two vertical superintegrable rapidities serves as the $Q$-operator of XXZ chain model for a cyclic representation of $U_{\mathsf{q}}(s{l}_2)$ with $N$th root-of-unity $\mathsf{q}$ and representation-parameter for odd $N$. The symmetry problem of XXZ chain with a general cyclic $U_{\mathsf{q}}(s{l}_2)$-representation is mapped onto the problem of studying $Q$-operator of some special one-parameter family of generalized ${\tau }^{(2)}$-models. In particular, the spin-$\frac{N-1}{2}$ XXZ chain model with ${\mathsf{q}}^N=1$ and the homogeneous $N$-state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method developed for producing $Q_{72}$-operator of the root-of-unity eight-vertex model, we construct the $Q_R,Q_L$- and $Q$-operators of a superintegrable ${\tau }^{(2)}$-model, then identify them with transfer matrices of the $N$-state chiral Potts model for a positive integer $N$. We thus obtain a new method of producing the superintegrable $N$-state chiral Potts transfer matrix from the ${\tau }^{(2)}$-model by constructing its $Q$-operator.

Latex 27 Pages; Typos and errors corrected, Improved version with clearer explanations for better presentation. Terminology and notations refined. References added and updated-Journal version

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Cited by in corpus (5)

• Onsager symmetries in $U(1)$-invariant clock models
• An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model
• Conjectures on Hidden Onsager Algebra Symmetries in Interacting Quantum Lattice Models
• On ${\tau }^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(s{l}_2)$
• Bethe Equation of ${\tau }^{(2)}$-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities