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

TL;DR

This paper constructs and classifies eigenvectors of the superintegrable chiral Potts model using algebraic Bethe ansatz, duality, and symmetry principles, revealing detailed sector relationships and explicit eigenvector structures.

Contribution

It provides a complete classification of quantum numbers and explicit construction of eigenvectors for all Onsager sectors in the superintegrable $ au^{(2)}$-model, utilizing symmetry and algebraic techniques.

Findings

Complete classification of quantum numbers for superintegrable $ au^{(2)}$-model.

Explicit construction of CPM eigenvectors in all sectors.

Relationships among sectors under duality and inversion are established.

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 $\tau^{(2)}$-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 $\tau^{(2)}$-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 $\tau^{(2)}$-model through its equivalent spin-$\frac{N-1}{2}$-XXZ chain. The $\tau^{(2)}$-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 $\tau^{(2)}$-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.