The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(sl_2)$

TL;DR

This paper establishes a connection between the transfer matrix of the superintegrable chiral Potts model and the Q-operator of the root-of-unity XXZ chain with cyclic $U_q(sl_2)$ representation, unifying different models through Baxter's method.

Contribution

It introduces a novel approach to construct the superintegrable chiral Potts transfer matrix from the $ au^{(2)}$-model using the Q-operator, linking it to the XXZ chain at roots of unity.

Findings

Unified the chiral Potts model and XXZ chain at roots of unity as a single physical theory.

Constructed Q-operators for a superintegrable $ au^{(2)}$-model using Baxter's method.

Provided a new method to derive the chiral Potts transfer matrix from the $ au^{(2)}$-model.

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 $U_{\sf q}(sl_2)$ with $N$th root-of-unity ${\sf q}$ and representation-parameter for odd $N$. The symmetry problem of XXZ chain with a general cyclic $U_{\sf q}(sl_2)$-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 ${\sf q}^N=1$ 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.