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arXiv:1102.1694 · Journal of High Energy Physics

Completeness of Bethe ansatz by Sklyanin SOV for cyclic representations of integrable quantum models

Giuliano Niccoli

Published Mar 1, 2011• 46 citations
arXivHTMLPDF
DOI
10.1007/jhep03(2011)123

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The most general cyclic representations of the quantum integrable τ 2 -model are analyzed. The complete characterization of the τ 2 -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 τ 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 τ 2 -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 τ 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 the τ 2 -model on the chiral Potts algebraic curves.
The antiperiodic transfer matrices associated to higher spin representations of the rational 6-vertex Yang-Baxter algebra are analyzed by generalizing the approach introduced recently in the framework of Sklyanin's quantum separation of variables (SOV) for cyclic representations, spin-1/2 highest weight representations, and also for spin-1/2 representations of the 6-vertex reflection algebra. Such SOV approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on Bethe ansatz and Baxter Q-operator. In particular, we both prove the completeness of the SOV characterization of the transfer matrix spectrum and its simplicity. Then, the derived characterization of local operators by Sklyanin's quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products.
We present a microscopic approach in the framework of Sklyanin's quantum separation of variables (SOV) for the exact solution of 1+1-dimensional quantum field theories by integrable lattice regularizations. Sklyanin's SOV is the natural quantum analogue of the classical method of separation of variables and it allows a more symmetric description of classical and quantum integrability w.r.t. traditional Bethe ansatz methods. Moreover, it has the advantage to be applicable to a more general class of models for which its implementation gives a characterization of the spectrum complete by construction. Our aim is to introduce a method in this framework which allows at once to derive the spectrum (eigenvalues and eigenvectors) and the dynamics (time dependent correlation functions) of integrable quantum field theories (IQFTs). This approach is presented for a paradigmatic example of relativistic IQFT, the sine-Gordon model.