arXiv:2005.09249 · Journal of Statistical Mechanics Theory and Experiment

Actions of the monodromy matrix elements onto g l ( m | n ) -invariant Bethe vectors

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak, E. Ragoucy, N. A. Slavnov

Published Sep 1, 2020• 12 citations

Abstract

Abstract Multiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the g l ( m | n ) -invariant quantum integrable models are calculated. These actions are used to describe recursions for the highest coefficients in the sum formula for the scalar product. For simplicity, detailed proofs are given for the g l ( m ) case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs.

DOI: 10.1088/1742-5468/abacb2

Related papers

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

Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak

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.

arXiv:2109.07528Journal of Physics A Mathematical and TheoreticalJan 7, 20225 citations

Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak +4 more

We study gl(2|1) symmetric integrable models solvable by the nested algebraic Bethe ansatz.Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors.We show that the result of these actions is a finite linear combination of Bethe vectors.The obtained formulas open a way for studying scalar products of Bethe vectors.

arXiv:1605.06419Symmetry Integrability and Geometry Methods and ApplicationsOct 8, 201626 citations

Scalar products of Bethe vectors in models with gl ( 2 ∣ 1 ) symmetry 1. Super-analog of Reshetikhin formula *

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak +2 more

Journal of Physics A Mathematical and TheoreticalOct 19, 201619 citations

Bethe Vectors for Orthogonal Integrable Models

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

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.

arXiv:1906.03202Theoretical and Mathematical PhysicsNov 1, 201912 citations

Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 1. Super-analog of Reshetikhin formula

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak +2 more

arXiv:1605.09189May 30, 20162 citations

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

S. Belliard, N. A. Slavnov

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.

arXiv:1802.07576Journal of High Energy PhysicsApr 1, 201812 citations

arXiv:2005.09249 · Journal of Statistical Mechanics Theory and Experiment

Actions of the monodromy matrix elements onto g l ( m | n ) -invariant Bethe vectors

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak, E. Ragoucy, N. A. Slavnov

Published Sep 1, 2020• 12 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1742-5468/abacb2

Related papers

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

Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak

arXiv:2109.07528Journal of Physics A Mathematical and TheoreticalJan 7, 20225 citations

Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak +4 more

arXiv:1605.06419Symmetry Integrability and Geometry Methods and ApplicationsOct 8, 201626 citations

Scalar products of Bethe vectors in models with gl ( 2 ∣ 1 ) symmetry 1. Super-analog of Reshetikhin formula *

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak +2 more

Abstract We study the scalar products of Bethe vectors in integrable models solvable by the nested algebraic Bethe ansatz and possessing gl ( 2 ∣ 1 ) symmetry. Using explicit formulas of the monodromy matrix entries’ multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with gl ( 1 ∣ 1 ) symmetry.Abstract We study the scalar products of Bethe vectors in integrable models solvable by the nested algebraic Bethe ansatz and possessing gl ( 2 ∣ 1 ) symmetry. Using explicit formulas of the monodromy matrix entries’ multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with gl ( 1 ∣ 1 ) symmetry.

Journal of Physics A Mathematical and TheoreticalOct 19, 201619 citations

Bethe Vectors for Orthogonal Integrable Models

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

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 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.

arXiv:1906.03202Theoretical and Mathematical PhysicsNov 1, 201912 citations

Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 1. Super-analog of Reshetikhin formulaScalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 1. Super-analog of Reshetikhin formula

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak +2 more

We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(2|1)$ symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with $\mathfrak{gl}(1|1)$ symmetry.We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(2|1)$ symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with $\mathfrak{gl}(1|1)$ symmetry.

arXiv:1605.09189May 30, 20162 citations

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

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.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.

arXiv:1802.07576Journal of High Energy PhysicsApr 1, 201812 citations

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.

Scalar products of Bethe vectors in models with

\mathfrak{gl}(2|1)

symmetry 1. Super-analog of Reshetikhin formula

We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing

\mathfrak{gl}(2|1)

symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with

\mathfrak{gl}(1|1)

symmetry.

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.