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arXiv:2412.05224 · Symmetry Integrability and Geometry Methods and Applications

Rectangular Recurrence Relations in $\mathfrak{gl}_{n}$ and $\mathfrak{o}_{2n+1}$ Invariant Integrable Models

Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak, Laboratoire d'Annecy-le-Vieux de Physique Théorique (LAPTh), France, E. Ragoucy, Laboratoire d'Annecy-le-Vieux de Physique Théorique (LAPTh), France

Published Sep 21, 2025• 1 citation

Abstract

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.

DOI: 10.3842/sigma.2025.078

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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.

Algebraic Bethe ansatz for

\mathfrak o_{2n+1}

-invariant integrable models

We study the class of

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

-invariant quantum integrable models in the framework of the algebraic Bethe ansatz and propose a construction of the

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

-invariant Bethe vector in terms of the Drinfeld currents for the Yangian double

\mathcal{D}Y ( \mathfrak{o} _{2n+1})

. We calculate the action of the monodromy matrix elements on the off-shell Bethe vectors for these models and obtain recurrence relations for these vectors. The action formulas can be used to investigate scalar products of Bethe vectors in

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

-invariant models.

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.

We study integrable models solvable by the nested algebraic Bethe ansatz and possessing GL(3)-invariant R-matrix. Assuming that the monodromy matrix of the model can be expanded into series with respect to the inverse spectral parameter, we define zero modes of the monodromy matrix entries as the first nontrivial coefficients of this series. Using these zero modes we establish new relations between form factors of the elements of the monodromy matrix. We prove that all of them can be obtained from the form factor of a diagonal matrix element in special limits of Bethe parameters. As a result we obtain determinant representations for form factors of all the entries of the monodromy matrix.

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.