On the Equivalent Theory of the Generalized τ^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities

Abstract

By the Baxter's $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized $τ^{(2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $ς$) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $ς^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $ς^N \neq 1$) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{T}$ of the chiral Potts model (including the degenerate ones) serve as the $Q_R, Q_L$-operators of the corresponding $τ^{(2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.