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

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.