The transfer matrix of a superintegrable chiral Potts model as theQ operator of root-of-unity XXZ chain with cyclic representation of

Abstract

We demonstrate that the transfer matrix of the inhomogeneousN-state chiral Potts model with two vertical superintegrable rapidities serves as theQ operator of the XXZ chain model for a cyclic representation of with Nth root-of-unity and representation parameter for oddN. The symmetry problem of the XXZ chain with a general cyclic representation is mapped onto the problem of studying theQ operator of some special one-parameter family of generalizedτ(2) models. Inparticular, the spin-(N−1)/2 XXZ chain model with and the homogeneous N-state chiral Potts model at a specific superintegrable point are unifiedas one physical theory. By Baxter’s method, developed for producing aQ72 operator of the root-of-unity eight-vertex model, we construct theQR,QL andQ operators of asuperintegrable τ(2) model, then identify them with transfer matrices of theN-state chiral Potts model for a positive integerN. We thus obtain a new method of producing the superintegrableN-state chiral Potts transfer matrix from theτ(2) model byconstructing its Q operator.