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 identify the precise relationship between the five-parameter τ (2) -family in the N -state chiral Potts model and XXZ chains with U q ( sl 2 )-cyclic representation. By studying the Yang–Baxter relation of the six-vertex model, we discover a one-parameter family of L -operators in terms of the quantum group U q ( sl 2 ). When N is odd, the N -state τ (2) -model can be regarded as the XXZ chain of cyclic representations with . The symmetry algebra of the τ (2) -model is described by the quantum affine algebra via the canonical representation. In general, for an arbitrary N , we show that the XXZ chain with a U q ( sl 2 )-cyclic representation for q 2 N = 1 is equivalent to two copies of the same N -state τ (2) -model.
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 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.
Following Baxter's method of producing Q 72 -operator, we construct the Q -operator of the root-of-unity eight-vertex model for the crossing parameter with odd N where Q 72 does not exist. We use this new Q -operator to study the functional relations in the Fabricius–McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N -state chiral Potts model. By the compatibility of the constructed Q -operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q -operator and fusion weights of the SOS model.
We construct the fusion operators in the generalized τ (2) -model using the fused L -operators, and verify the fusion relations with the truncation identity. The algebraic Bethe-ansatz discussion is conducted on two special classes of τ (2) which include the superintegrable chiral Potts model. We then perform the parallel discussion on the XXZ spin chain at roots of unity, and demonstrate that the sl 2 -loop-algebra symmetry exists for the root-of-unity XXZ spin chain with a higher spin, where the evaluation parameters for the symmetry algebra are identified by the explicit Fabricius–McCoy current for the Bethe states. Parallels are also drawn to the comparison with the superintegrable chiral Potts model.