Bethe vectors for models based on the super-Yangian Y(gl(m|n))

Pakuliak, S. Z., E. Ragoucy, N. A. Slavnov

Published Jan 1, 2017• 1 citation

Abstract

We study Bethe vectors of integrable models based on the super-Yangian$Y(\mathfrak{gl}(m|n))$. Starting from the super-trace formula, we exhibitrecursion relations for these vectors in the case of $Y(\mathfrak{gl}(2|1))$and $Y(\mathfrak{gl}(1|2))$. These recursion relations allow to get explicitexpressions for the Bethe vectors. Using an antimorphism of the super-Yangian$Y(\mathfrak{gl}(m|n))$, we also construct a super-trace formula for dual Bethevectors, and, for $Y(\mathfrak{gl}(2|1))$ and $Y(\mathfrak{gl}(1|2))$super-Yangians, show recursion relations for them. Again, the latter allow usto get explicit expressions for dual Bethe vectors.

Related papers

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

Current presentation for the double super-Yangian DY(gl(m|n)) and Bethe vectors

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

Mar 10, 20178 citations

Bethe Vectors for Composite Models with gl(2|1) and gl(1|2) Supersymmetry

Jan Fuksa

Supersymmetric composite generalized quantum integrable models solvable by the algebraic Bethe ansatz are studied.Using a coproduct in the bialgebra of monodromy matrix elements and their action on Bethe vectors, formulas for Bethe vectors in the composite models with supersymmetry based on the super-Yangians Y [gl(2|1)] and Y [gl(1|2)] are derived.

arXiv:1611.00943Symmetry Integrability and Geometry Methods and ApplicationsMar 13, 20175 citations

Bethe vectors and recurrence relations for twisted Yangian based models

Vidas Regelskis

arXiv:2302.11842SciPost PhysicsNov 5, 20242 citations

Bethe vectors in integrable models based on the super-Yangian Y(gl(m|n)) Y(gl(m|n))

Lyashik Andrey, Arthur Hutsalyuk, Stanislav Zdislavovich Pakuliak +2 more

Jan 1, 20160 citations

Scalar products of Bethe vectors in the models with gl(m|n) symmetry

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

We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by gl(m|n) superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for the Bethe vectors. This allows us to find recursions for the highest coefficient of the scalar product.

arXiv:1704.08173Nuclear Physics BAug 3, 201735 citations

Bethe vectors for models based on the super-Yangian Y(gl(m|n))

Pakuliak, S. Z., E. Ragoucy, N. A. Slavnov

Published Jan 1, 2017• 1 citation

OpenAlex

Abstract

We study Bethe vectors of integrable models based on the super-Yangian

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

. Starting from the super-trace formula, we exhibitrecursion relations for these vectors in the case of

Y(\mathfrak{gl}(2|1))

and

Y(\mathfrak{gl}(1|2))

. These recursion relations allow to get explicitexpressions for the Bethe vectors. Using an antimorphism of the super-Yangian

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

, we also construct a super-trace formula for dual Bethevectors, and, for

Y(\mathfrak{gl}(2|1))

and

Y(\mathfrak{gl}(1|2))

super-Yangians, show recursion relations for them. Again, the latter allow usto get explicit expressions for dual Bethe vectors.

Related papers

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

Current presentation for the double super-Yangian DY(gl(m|n)) and Bethe vectors

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

Mar 10, 20178 citations

Bethe Vectors for Composite Models with gl(2|1) and gl(1|2) Supersymmetry

Jan Fuksa

arXiv:1611.00943Symmetry Integrability and Geometry Methods and ApplicationsMar 13, 20175 citations

Bethe vectors and recurrence relations for twisted Yangian based models

Vidas Regelskis

We study Olshanski twisted Yangian based models, known as one-dimensional “soliton non-preserving” open spin chains, by means of algebraic Bethe Ansatz. The even case, when the bulk symmetry is \mathfrak{gl}_{2n} 𝔤 𝔩 2 n and the boundary symmetry is \mathfrak{sp}_{2n} 𝔰 𝔭 2 n or \mathfrak{so}_{2n} 𝔰 𝔬 2 n , was studied in [Ann. Henri Poincaré 20, 339 (2018)]. In the present work, we focus on the odd case, when the bulk symmetry is \mathfrak{gl}_{2n+1} 𝔤 𝔩 2 n + 1 and the boundary symmetry is \mathfrak{so}_{2n+1} 𝔰 𝔬 2 n + 1 . We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and Y(\mathfrak{gl}_n) Y ( 𝔤 𝔩 n ) -type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.We study Olshanski twisted Yangian based models, known as one-dimensional “soliton non-preserving” open spin chains, by means of algebraic Bethe Ansatz. The even case, when the bulk symmetry is \mathfrak{gl}_{2n} 𝔤 𝔩 2 n and the boundary symmetry is \mathfrak{sp}_{2n} 𝔰 𝔭 2 n or \mathfrak{so}_{2n} 𝔰 𝔬 2 n , was studied in [Ann. Henri Poincaré 20, 339 (2018)]. In the present work, we focus on the odd case, when the bulk symmetry is \mathfrak{gl}_{2n+1} 𝔤 𝔩 2 n + 1 and the boundary symmetry is \mathfrak{so}_{2n+1} 𝔰 𝔬 2 n + 1 . We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and Y(\mathfrak{gl}_n) Y ( 𝔤 𝔩 n ) -type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.

arXiv:2302.11842SciPost PhysicsNov 5, 20242 citations

Bethe vectors in integrable models based on the super-Yangian Y(gl(m|n)) Y(gl(m|n))

Lyashik Andrey, Arthur Hutsalyuk, Stanislav Zdislavovich Pakuliak +2 more

Jan 1, 20160 citations

Scalar products of Bethe vectors in the models with gl(m|n) symmetry

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

arXiv:1704.08173Nuclear Physics BAug 3, 201735 citations

We study Bethe vectors of integrable models based on the super-Yangian

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

. Starting from the super-trace formula, we exhibitrecursion relations for these vectors in the case of

Y(\mathfrak{gl}(2|1))

and

Y(\mathfrak{gl}(1|2))

. These recursion relations allow to get explicitexpressions for the Bethe vectors. Using an antimorphism of the super-Yangian

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

, we also construct a super-trace formula for dual Bethevectors, and, for

Y(\mathfrak{gl}(2|1))

and

Y(\mathfrak{gl}(1|2))

super-Yangians, show recursion relations for them. Again, the latter allow usto get explicit expressions for dual Bethe vectors.

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.

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))

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.