New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz

TL;DR

This paper introduces quasi-Yang-Baxter algebras, revealing new quantum dynamical symmetries and leading to the development of quasi-exactly solvable integrable models with partial Bethe ansatz solutions.

Contribution

It proposes a new class of quasi-Yang-Baxter algebras, explores their applications in quantum inverse scattering, and develops a theory of deformations of integrable models.

Findings

Introduction of quasi-Yang-Baxter algebras as deformations of monodromy matrix algebras

Construction of new quasi-exactly solvable quantum models

Development of classical counterparts leading to new classical integrable models

Abstract

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.