New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz

Details New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz

1997-07-17

arxiv.org/abs/hep-th/9707151v1

Annals Phys. 266 (1998) 81-134

Physics

High Energy Physics

High Energy Physics - Theory

49 pages, LaTeX

Scientific paper

10.1006/aphy.1998.5797

We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence ``Yangian --- quasi-Yangian'' and ``$XXX$ spin models --- quasi-$XXX$ spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang-Baxter algebras and show that they naturally lead to new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable in the sense of classical inverse scattering method, i.e. admit only partial construction of action-angle variables.

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