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arXiv:cond-mat/0505698 · Journal of Statistical Mechanics Theory and Experiment

The Onsager algebra symmetry of τ(j)-matrices in the superintegrable chiral Potts model

Shi-Shyr Roan

Published Sep 23, 2005• 11 citations
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Abstract

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.

DOI
10.1088/1742-5468/2005/09/p09007

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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.
We examine the Onsager algebra symmetry of τ(j)τ^{(j)}τ(j)-matrices in the superintegrable chiral Potts model. The comparison of Onsager algebra symmetry of the chiral Potts model with the sl2sl_2sl2​-loop algebra symmetry of six-vertex model at roots of unity is made from the aspect of functional relations using the QQQ-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.
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Eigenvectors of an Arbitrary Onsager Sector in Superintegrable τ(2)τ^{(2)}τ(2)-model and Chiral Potts Model
We study the eigenvector problem in homogeneous superintegrable NNN-state chiral Potts model (CPM) by the symmetry principal. Using duality symmetry and (spin-)inversion in CPM, together with Onsager-algebra symmetry and sl2sl_2sl2​-loop-algebra symmetry of the superintegrable τ(2)τ^{(2)}τ(2)-model, we construct the complete k′k'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)τ^{(2)}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)τ^{(2)}τ(2)-model through its equivalent spin-N−12\frac{N-1}{2}2N−1​-XXZ chain. The τ(2)τ^{(2)}τ(2)-eigenvectors in a sector are derived from the Bethe state and the sl2sl_2sl2​-product structure determined by the Fabricius-McCoy current of the sector. From those τ(2)τ^{(2)}τ(2)-eigenvectors, the k′k'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 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.
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 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.