The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(sl_2)$
Shi-shyr Roan
备注：
Latex 27 Pages; Typos and errors corrected, Improved version with clearer explanations for better presentation. Terminology and notations refined. References added and updated-Journal version
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 $τ^{(2)}$-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 $τ^{(2)}$-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.
Quantum Group Theory in $τ^{(2)}$-model, Duality of $τ^{(2)}$-model and XXZ-model with Cyclic ${\bf U_q(sl_2)}$-representation for ${\bf q^n =1}$, and Chiral Potts Model
The hamiltonian of the $N$-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by $N-1$ types of Temperley-Lieb generators. This generalises a previous result for $N=3$ obtained by J. F. Fjelstad and T. Månsson [J. Phys. A {\bf 45} (2012) 155208]. A pictorial representation of a related coupled algebra is given for the $N=3$ case which involves a generalisation of the pictorial presentation of the Temperley-Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the $N=3$ SICP chain …
We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on me…
The ground states of the spin-$ S $ antiferromagnetic chain $H_\textrm{AF}$ with a projection-based interaction and the spin-$ 1/2$ XXZ-chain $ H_\textrm{XXZ} $ at anisotropy parameter $Δ=\cosh(λ) $ share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar $Q$-state Potts model at $ \sqrt Q= 2S+1 =2\cosh(λ)$. The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground states of these two models: dimerization for $H_\textrm{AF}$ at all $S> 1/2$…
