On $\tau^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(sl_2)$

TL;DR

This paper establishes a detailed connection between the $ au^{(2)}$-model in the chiral Potts model and XXZ chains with cyclic quantum group representations, revealing symmetry structures and equivalences for arbitrary N.

Contribution

It identifies the precise relationship between the $ au^{(2)}$-model and $U_q(sl_2)$ cyclic representations, and demonstrates their equivalence to XXZ chains with specific quantum group parameters.

Findings

For odd N, $ au^{(2)}$-model corresponds to XXZ chains with $U_q(sl_2)$ cyclic representations.

The symmetry algebra of the $ au^{(2)}$-model is described by $U_q(\hat{sl}_2)$.

For general N, the XXZ chain with $U_q(sl_2)$ cyclic representation is equivalent to two copies of the $ au^{(2)}$-model.

Abstract

We identify the precise relationship between the five-parameter $\tau^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q (sl_2)$-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 $U_q (sl_2)$. When $N$ is odd, the $N$-state $\tau^{(2)}$-model can be regarded as the XXZ chain of $U_{\sf q} (sl_2)$ cyclic representations with ${\sf q}^N=1$. The symmetry algebra of the $\tau^{(2)}$-model is described by the quantum affine algebra $U_{\sf q} (\hat{sl}_2)$ via the canonical representation. In general for an arbitrary $N$, we show that the XXZ chain with a $U_q (sl_2)$-cyclic representation for $q^{2N}=1$ is equivalent to two copies of the same $N$-state $\tau^{(2)}$-model.