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

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

Published Mar 10, 2017• 8 citations
OpenAlex

Abstract

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.

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

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.

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Bethe Vectors for Orthogonal Integrable Models

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

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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.

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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:2005.09249•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2020•12 citations
We find Bethe vectors for quantum integrable models associated with the
supersymmetric Yangians Y(gl(m∣n)Y(\mathfrak{gl}(m|n)Y(gl(m∣n) in terms of the current
generators of the Yangian double DY(gl(m∣n))DY(\mathfrak{gl}(m|n))DY(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(gl(m∣n))DY(\mathfrak{gl}(m|n))DY(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.
Bethe vectors in integrable models based on the super-Yangian Y(gl(m∣n))Y(\mathfrak{gl}(m|n))Y(gl(m∣n))
We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians Y(gl(m∣n)Y(\mathfrak{gl}(m|n)Y(gl(m∣n) in terms of the current generators of the Yangian double DY(gl(m∣n))DY(\mathfrak{gl}(m|n))DY(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(gl(m∣n))DY(\mathfrak{gl}(m|n))DY(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 consider quantum integrable models associated with the so3\mathfrak{so}_3so3​ algebra and describe Bethe vectors of these models in terms of the current generators of the DY(so3)\mathcal{D}Y(\mathfrak{so}_3)DY(so3​) 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 so3\mathfrak{so}_3so3​ algebra.
We study Bethe vectors of integrable models based on the super-YangianY(gl(m∣n))Y(\mathfrak{gl}(m|n))Y(gl(m∣n)). Starting from the super-trace formula, we exhibitrecursion relations for these vectors in the case of Y(gl(2∣1))Y(\mathfrak{gl}(2|1))Y(gl(2∣1))and Y(gl(1∣2))Y(\mathfrak{gl}(1|2))Y(gl(1∣2)). These recursion relations allow to get explicitexpressions for the Bethe vectors. Using an antimorphism of the super-YangianY(gl(m∣n))Y(\mathfrak{gl}(m|n))Y(gl(m∣n)), we also construct a super-trace formula for dual Bethevectors, and, for Y(gl(2∣1))Y(\mathfrak{gl}(2|1))Y(gl(2∣1)) and Y(gl(1∣2))Y(\mathfrak{gl}(1|2))Y(gl(1∣2))super-Yangians, show recursion relations for them. Again, the latter allow usto get explicit expressions for dual Bethe vectors.