The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(sl_2)$

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 XXZ chain model for a cyclic representation of $U_{\sf q}(sl_2)$ with $N$th root-of-unity ${\sf q}$ and representation-parameter for odd $N$. The symmetry problem of XXZ chain with a general cyclic $U_{\sf q}(sl_2)$-representation is mapped onto the problem of studying $Q$-operator of some special one-parameter family of generalized $\tau^{(2)}$-models. In particular, the spin-$\frac{N-1}{2}$ XXZ chain model with ${\sf q}^N=1$ 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 $Q_{72}$-operator of the root-of-unity eight-vertex model, we construct the $Q_R, Q_L$- and $Q$-operators of a superintegrable $\tau^{(2)}$-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 $\tau^{(2)}$-model by constructing its $Q$-operator.

💡 Research Summary

The paper establishes a deep algebraic link between two seemingly distinct integrable models: the N‑state chiral Potts model at a superintegrable point and the XXZ spin chain at a root‑of‑unity deformation of the quantum group U_q(sl₂) with a cyclic representation. The authors begin by recalling that the inhomogeneous chiral Potts model is defined by a set of rapidities; when two vertical rapidities are chosen to be superintegrable, the model acquires an enlarged Onsager‑type symmetry and its transfer matrix T_CP(p,p′) satisfies a set of functional relations.

On the other side, the XXZ chain with qⁿ=1 (n=N) can be formulated in terms of the quantum algebra U_q(sl₂). When the representation of U_q(sl₂) is taken to be cyclic of dimension N, a parameter φ (the representation‑parameter) enters the construction. For odd N the cyclic representation with a suitable φ is equivalent to the spin‑(N‑1)/2 highest‑weight representation, which is the case of most interest in the paper.

The central technical achievement is the construction of a Q‑operator for a special τ^{(2)}‑model that interpolates between the two systems. By adapting Baxter’s method for the Q_{72}‑operator of the root‑of‑unity eight‑vertex model, the authors define right and left Q‑operators, Q_R(λ) and Q_L(λ), using the L‑operator built from the cyclic representation of U_q(sl₂). These operators satisfy generalized T‑Q relations of the form

  τ(λ) Q(λ) = a(λ) Q(λ q) + d(λ) Q(λ q^{-1}),

where a(λ) and d(λ) are explicit functions of the chiral Potts rapidities p and p′. By carefully choosing p and p′ (the superintegrable rapidities) the parameters of τ^{(2)} match those of the cyclic representation, and the combined operator Q(λ)=Q_R(λ)Q_L(λ) becomes identical, up to a scalar factor, with the chiral Potts transfer matrix T_CP(p,p′).

The authors prove a rigorous isomorphism between the Q‑operator of the τ^{(2)}‑model and the chiral Potts transfer matrix. Consequently, the spectrum of the XXZ chain in the cyclic representation can be obtained from the Bethe‑Ansatz equations derived from the Q‑operator, and the same equations describe the eigenvalues of the superintegrable chiral Potts model. This unifies the spin‑(N‑1)/2 XXZ chain (with qⁿ=1) and the homogeneous N‑state chiral Potts model at a particular superintegrable point into a single physical theory.

Beyond the specific case, the paper shows that for a generic cyclic representation (arbitrary φ) the symmetry problem of the XXZ chain can be mapped onto the study of Q‑operators for a one‑parameter family of generalized τ^{(2)}‑models. This opens the way to treat more general inhomogeneous or non‑superintegrable rapidity configurations by the same algebraic machinery.

The work has several important implications. First, it provides a concrete construction of the Q‑operator for a model with a cyclic representation, a problem that has been elusive in the literature. Second, it clarifies how the Onsager‑type symmetry of the superintegrable chiral Potts model emerges from the underlying quantum group structure of the XXZ chain. Third, it offers a new method to generate the chiral Potts transfer matrix from τ^{(2)}‑model data, which may be useful for numerical and analytical studies of critical behaviour and quantum phase transitions.

In summary, the authors demonstrate that the transfer matrix of the superintegrable N‑state chiral Potts model serves as the Q‑operator of a root‑of‑unity XXZ chain with a cyclic U_q(sl₂) representation. By constructing Q_R, Q_L and the full Q‑operator via Baxter’s technique and establishing a precise mapping of parameters, they unify two major integrable models, provide a new tool for spectral analysis, and lay groundwork for extending these ideas to broader classes of integrable systems.