We demonstrate that the transfer matrix of the inhomogeneous N -state chiral Potts model with two vertical superintegrable rapidities serves as the Q operator of the XXZ chain model for a cyclic representation of with N th root-of-unity and representation parameter for odd N . The symmetry problem of the XXZ chain with a general cyclic representation is mapped onto the problem of studying the Q operator of some special one-parameter family of generalized τ (2) models. In particular, the spin-( N −1)/2 XXZ chain model with and the homogeneous N -state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method, developed for producing a Q 72 operator of the root-of-unity eight-vertex model, we construct the Q R , Q L and Q operators of a superintegrable τ (2) model, then identify them with transfer matrices of the N -state chiral Potts model for a positive integer N . We thus obtain a new method of producing the superintegrable N -state chiral Potts transfer matrix from the τ (2) model by constructing its Q operator.
By the Baxter's
Q72-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
Uq(sl2) with
qN=1 for odd
N is identified with either (for
ςN=1) the chiral Potts model with two superintegrable vertical rapidities, or (for
ςN=1) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices
T,T^ of the chiral Potts model (including the degenerate ones) serve as the
QR,QL-operators of the corresponding
τ(2)-model, so that the functional relations hold as in the solvable
N-state chiral Potts model.
We discover an Ising-type duality in the general N -state chiral Potts model, which is the Kramers–Wannier duality of a 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 sl 2 -loop-algebra symmetry of its associated spin- ( N −1)/2 XXZ chain through the identification of their eigenstates.
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 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.
A new type of Yang-Baxter equation (YBE) for
R-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the
ZN symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for
N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated
R-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter
R-matrix---realized as an intertwiner of cyclic representations of
Uq(sl2) at a root of unity---and Shastry's two-parameter
R-operator obtained from the decorated YBE.