The tau_2-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 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.

40 pages. Minor modifications in the text and some notations

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• Separation of variables for higher rank integrable models: review of recent progress
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint