The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method

TL;DR

This paper extends Sklyanin's SOV method to fully characterize the spectrum and eigenstates of the tau_2-model and the chiral Potts model, proving the completeness of Bethe equations for broad classes of representations.

Contribution

It provides a complete spectral characterization of the tau_2-model and chiral Potts model using an extended SOV approach, establishing the completeness of Bethe ansatz equations.

Findings

Spectrum is proven to be simple for general representations.

Complete eigenvalues and eigenstates characterization for the models.

Bethe ansatz equations are shown to be complete for self-adjoint representations.

Abstract

The most general cyclic representations of the quantum integrable tau_2-model are analyzed. The complete characterization of the tau_2-spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin's Separation of Variables (SOV) method by extending and adapting the ideas first introduced in [1, 2]: i) The determination of the tau_2-spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials. ii) The determination of the tau_2-eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proven to be polynomials for a quite general class of tau_2-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 tau_2-model on the chiral Potts algebraic curves.