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

TL;DR

This paper establishes an equivalence between the generalized ^{(2)}-model and the N-state chiral Potts model with two alternating vertical rapidities, including degenerate cases, using Baxter's Q-operator method.

Contribution

It demonstrates the equivalence between the generalized ^{(2)}-model and the chiral Potts model with two alternating rapidities, extending to degenerate models and linking to XXZ chains with cyclic representations.

Findings

Transfer matrices serve as Q-operators in the ^{(2)}-model.

Functional relations mirror those in the solvable N-state chiral Potts model.

Degenerate models are included in the equivalence.

Abstract

By the Baxter's $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized $\tau^{(2)}$-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 $\varsigma$) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $\varsigma^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $\varsigma^N \neq 1$) 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 $\tau^{(2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.