We discover a Ising-type duality in the general N-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ′) to those of another chiral Potts model at a high temperature (of k ′−1). The τ (2)-model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl2-loop-algebra symmetry of its associated spin- N−1 2 XXZ chain through the identification of their eigenstates.
We examine the Onsager algebra symmetry of
τ(j)-matrices in the superintegrable chiral Potts model. The comparison of Onsager algebra symmetry of the chiral Potts model with the
sl2-loop algebra symmetry of six-vertex model at roots of unity is made from the aspect of functional relations using the
Q-operator and fusion matrices. The discussion of Bethe ansatz for both models is conducted in a uniform manner through the evaluation parameters of their symmetry algebras.
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
Uq(sl2) with $\textsl{w} = {\sf q}^{-2}$. In the roots of unity case, ${\sf q}=q, \textsl{w} = ω$ with
qn=ωN=1, the eigenvalues and eigenvectors of XXZ-model with the
Uq(sl2)-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
n=2N, e.g.
N even, case. We also establish the duality of
τ(2)-models related to cyclic representations of
Uq(sl2), analogous to the
τ(2)-duality in chiral Potts model; and identify the model dual to the XXZ model with
Uq(sl2)-cyclic representation.
We study the eigenvector problem in homogeneous superintegrable
N-state chiral Potts model (CPM) by the symmetry principal. Using duality symmetry and (spin-)inversion in CPM, together with Onsager-algebra symmetry and
sl2-loop-algebra symmetry of the superintegrable
τ(2)-model, we construct the complete
k′-dependent CPM-eigenvectors in the local spin basis for an arbitrary Onsager sector. In this paper, we present the complete classification of quantum numbers of superintegrable
τ(2)-model. Accordingly, there are four types of sectors. The relationships among Onsager sectors under duality and inversion, together with their Bethe roots and CPM-eigenvectors, are explicitly found. Using algebraic-Bethe-ansatz techniques and duality of CPM, we construct the Bethe states and the Fabricius-McCoy currents of the superintegrable
τ(2)-model through its equivalent spin-
2N−1-XXZ chain. The
τ(2)-eigenvectors in a sector are derived from the Bethe state and the
sl2-product structure determined by the Fabricius-McCoy current of the sector. From those
τ(2)-eigenvectors, the
k′-dependence of CPM state vectors in local-spin-basis form is obtained by the Onsager-algebra symmetry of the superintegrable chiral Potts quantum chain.
We demonstrate that the τ ( j ) -matrices in the superintegrable chiral Potts model possess the Onsager algebra symmetry for their degenerate eigenvalues. The Fabricius–McCoy comparison of functional relations of the eight-vertex model for roots of unity and the superintegrable chiral Potts model has been carefully analysed by identifying equivalent terms in the corresponding equations, by which we extract the conjectured relation of Q -operators and all fusion matrices in the eight-vertex model corresponding to the -relation in the chiral Potts model.