arXiv:1802.07576 · Journal of High Energy Physics

A note on $$ \mathfrak{g}{\mathfrak{l}}_2 $$-invariant Bethe vectors

S. Belliard, N. A. Slavnov

Published Apr 1, 2018• 12 citations

Abstract

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.

DOI: 10.1007/jhep04(2018)031

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arXiv:1802.07576 · Journal of High Energy Physics

A note on $\mathfrak{g}{\mathfrak{l}}_2$ -invariant Bethe vectorsA note on $$ \mathfrak{g}{\mathfrak{l}}_2 $$-invariant Bethe vectors

S. Belliard, N. A. Slavnov

Published Apr 1, 2018• 12 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1007/jhep04(2018)031

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Abstract We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{g}{\mathfrak{l}}_3$ g l 3 -invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.Abstract We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $$ \mathfrak{g}{\mathfrak{l}}_3 $$ g l 3 -invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.

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Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak

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Abstract We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing -invariant R -matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors. The q -deformed case, as well as the superalgebra case, are also evoked in the conclusion.Abstract We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing -invariant R -matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors. The q -deformed case, as well as the superalgebra case, are also evoked in the conclusion.

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Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak, E. Ragoucy +1 more

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arXiv:1906.03202Theoretical and Mathematical PhysicsNov 1, 201912 citations

A note on

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

-invariant Bethe vectors

Abstract We consider

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

On Bethe vectors in

\mathfrak{g}{\mathfrak{l}}_3

-invariant integrable models

Abstract We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing

\mathfrak{g}{\mathfrak{l}}_3

g l 3 -invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.

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.