The τ2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin’s SOV method

Abstract

The most general cyclic representations of the quantum integrable τ2-model are analyzed. The complete characterization of the τ2-spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin’s separation of variables (SOV) method by extending and adapting previous results of one of the authors: (i) the determination of the τ2-spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials; (ii) the determination of the τ2-eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proved to be polynomials for a quite general class of τ2-self-adjoint representations and the completeness of the associated Bethe ansatz type equations is derived. Finally, the following results are derived for the inhomogeneous chiral Potts model: (i) simplicity of the spectrum, for general representations; (ii) complete characterization of the chiral Potts spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type equations, for the self-adjoint representations of the τ2-model on the chiral Potts algebraic curves.