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arXiv:1003.3621

Eigenvectors of an Arbitrary Onsager Sector in Superintegrable $τ^{(2)}$-model and Chiral Potts Model

Shi-shyr Roan

Published Mar 18, 2010• 5 citations
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Abstract

We study the eigenvector problem in homogeneous superintegrable $N$-state chiral Potts model (CPM) by the symmetry principal. Using duality symmetry and (spin-)inversion in CPM, together with Onsager-algebra symmetry and $sl_2$-loop-algebra symmetry of the superintegrable $τ^{(2)}$-model, we construct the complete $k'$-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 $τ^{(2)}$-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 $τ^{(2)}$-model through its equivalent spin-$\frac{N-1}{2}$-XXZ chain. The $τ^{(2)}$-eigenvectors in a sector are derived from the Bethe state and the $sl_2$-product structure determined by the Fabricius-McCoy current of the sector. From those $τ^{(2)}$-eigenvectors, the $k'$-dependence of CPM state vectors in local-spin-basis form is obtained by the Onsager-algebra symmetry of the superintegrable chiral Potts quantum chain.

DOI
10.48550/arxiv.1003.3621

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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 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.
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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.
In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic τ 2 ( t q ) model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of τ 2 ( t q ) in the Q = 0 sector is 2 r , with r = ( N − 1) L / N when the size of the transfer matrix L is a multiple of N . We introduce chiral Potts model operators, different from the more commonly used generators of quantum group . From these we can form the generators of a loop algebra . For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra .
It has been shown in earlier works that for Q = 0 and L a multiple of N , the ground state sector eigenspace of the superintegrable τ 2 ( t q ) model is highly degenerate and is generated by a quantum loop algebra . Furthermore, this loop algebra can be decomposed into r = ( N − 1) L / N simple algebras. For Q ≠ 0, we shall show here that the corresponding eigenspace of τ 2 ( t q ) is still highly degenerate, but splits into two spaces, each containing 2 r − 1 independent eigenvectors. The generators for the subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q = 0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q ≠ 0 ground state sectors for the transfer matrix of the superintegrable 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.