Quantum inverse scattering method and Yang-Baxter relation for integrable spin systems

Robin Steinigeweg, Heinz–Jürgen Schmidt

Kiyoshi Sogo, Miki Wadati

The Lax pair for spin 1/2 Heisenberg XYZ model is presented. This gives a basis for the quantum inverse scattering method. A generalization of the Yang-Baxter relation to arbitrary spin is presented through the factorized S-matrix theory. An equivalent vertex model is solved exactly by the quantum inverse scattering method.

A. G. Ushveridze

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.

A. Kundu, B. Basu-Mallick

千尋 松井

Pete Rigas

We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.