arXiv:2109.07528 · Journal of Physics A Mathematical and Theoretical

Recurrence relations for off-shell Bethe vectors in trigonometric integrable models

Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak

Published Jan 7, 2022• 5 citations

Abstract

Abstract The zero modes method is applied in order to get the action of the monodromy matrix entries on off-shell Bethe vectors in quantum integrable models associated with U q ( g l N ) -invariant R -matrices. The action formulas allow to get recurrence relations for off-shell Bethe vectors and for highest coefficients of the Bethe vectors scalar product.

DOI: 10.1088/1751-8121/ac491b

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arXiv:2109.07528 · Journal of Physics A Mathematical and Theoretical

Recurrence relations for off-shell Bethe vectors in trigonometric integrable models

Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak

Published Jan 7, 2022• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8121/ac491b

Related papers

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arXiv:2005.09249Journal of Statistical Mechanics Theory and ExperimentSep 1, 202012 citations

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Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak, Laboratoire d'Annecy-le-Vieux de Physique Théorique (LAPTh), France +2 more

A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type $\mathfrak{gl}_n$ or type $\mathfrak{o}_{2n+1}$ symmetries. These recurrence relations describe how to add a single parameter $z$ to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on $z$ . We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type $\mathfrak{gl}_n$ or type $\mathfrak{o}_{2n+1}$ symmetries. These recurrence relations describe how to add a single parameter $z$ to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on $z$. We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.

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Rectangular Recurrence Relations in

\mathfrak{gl}_{n}

and

\mathfrak{o}_{2n+1}

Invariant Integrable Models

A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type

\mathfrak{gl}_n

or type

\mathfrak{o}_{2n+1}

symmetries. These recurrence relations describe how to add a single parameter

z

to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on

z

. We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.

We consider quantum integrable models associated with the

\mathfrak{so}_3

algebra and describe Bethe vectors of these models in terms of the current generators of the

\mathcal{D}Y(\mathfrak{so}_3)

algebra. To implement this program, we use an isomorphism between the R-matrix and the Drinfeld current realizations of the Yangians and their doubles for classical type B-, C-, and D-series algebras. Using these results, we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We obtain recurrence relations for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The formulas for the action of the monodromy matrix elements can also be used to calculate scalar products in the models associated with the

\mathfrak{so}_3

algebra.

A note on

\mathfrak{g}{\mathfrak{l}}_2

-invariant Bethe vectors

Abstract We consider

\mathfrak{g}{\mathfrak{l}}_2

g l 2 -invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the actions of the twisted monodromy matrix entries on the twisted off-shell Bethe vectors.