arXiv:2008.03664 · Theoretical and Mathematical Physics

Algebraic Bethe ansatz for $$\mathfrak o_{2n+1}$$-invariant integrable models

Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak

Published Jan 1, 2021• 8 citations

Abstract

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.

DOI: 10.1134/s0040577921010025

Related papers

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

Bethe vectors in integrable models based on the super-Yangian $Y(\mathfrak{gl}(m|n))$

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

Nov 29, 20160 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

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

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

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

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.

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

arXiv:2008.03664 · Theoretical and Mathematical Physics

Algebraic Bethe ansatz for $\mathfrak o_{2n+1}$ -invariant integrable modelsAlgebraic Bethe ansatz for $$\mathfrak o_{2n+1}$$-invariant integrable models

Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak

Published Jan 1, 2021• 8 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1134/s0040577921010025

Related papers

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

Bethe vectors in integrable models based on the super-Yangian $Y(\mathfrak{gl}(m|n))$ Bethe vectors in integrable models based on the super-Yangian $Y(\mathfrak{gl}(m|n))$

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

We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the current generators of the Yangian double $DY(\mathfrak{gl}(m|n))$ . More specifically, we use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double $DY(\mathfrak{gl}(m|n))$ we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the current generators of the Yangian double $DY(\mathfrak{gl}(m|n))$. More specifically, we use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double $DY(\mathfrak{gl}(m|n))$ we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.

Nov 29, 20160 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

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

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

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

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

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})

\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.

Bethe vectors in integrable models based on the super-Yangian

Y(\mathfrak{gl}(m|n))

We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians

Y(\mathfrak{gl}(m|n)

in terms of the current generators of the Yangian double

DY(\mathfrak{gl}(m|n))

. More specifically, we use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double

DY(\mathfrak{gl}(m|n))

we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.

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.

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.