Quantum Group Theory in $τ^{(2)}$-model, Duality of $τ^{(2)}$-model and XXZ-model with Cyclic ${\bf U_q(sl_2)}$-representation for ${\bf q^n =1}$, and Chiral Potts Model
We identify the quantum group ${\Large\textsl{U}}_\textsl{w}(sl_2)$ in the $L$-operator of $τ^{(2)}$-model for a generic $\textsl{w}$ as a subalgebra of $U_{\sf q} (sl_2)$ with $\textsl{w} = {\sf q}^{-2}$. In the roots of unity case, ${\sf q}=q, \textsl{w} = ω$ with $q^{\bf n} = ω^N = 1$, the eigenvalues and eigenvectors of XXZ-model with the $U_q (sl_2)$-cyclic representation are determined by the $τ^{(2)}$-model with the induced ${\Large\textsl{U}}_ω(sl_2)$-cyclic representation, which is decomposed as a finite sum of $τ^{(2)}$-models in non-superintegrable inhomogeneous $N$-state chiral Potts model. Through the theory of chiral Potts model, the $Q$-operator of XXZ-model can be identified with the related chiral Potts transfer matrices, with special features appeared in the ${\bf n}=2N$, e.g. $N$ even, case. We also establish the duality of $τ^{(2)}$-models related to cyclic representations of $U_q (sl_2)$, analogous to the $τ^{(2)}$-duality in chiral Potts model; and identify the model dual to the XXZ model with $U_q (sl_2)$-cyclic representation.
Movable seams in root-of-unity XXZ chains: Relative flux classes and antiunitary spectral pairing
School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；Shanghai Key Laboratory of Particle Physics and Cosmology（上海粒子物理与宇宙学重点实验室）；School of Physics, East China University of Science and Technology（华东理工大学物理系）
