Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On reflection algebras and twisted Yangians

Anastasia Doikou

Published Jan 1, 2004• 16 citations
OpenAlex

Abstract

It is known that integrable models associated to rational R matrices give rise to certain non-abelian symmetries known as Yangians. Analogously ‘boundary ’ symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton non-preserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary non-local charges directly at the quantum level. The non-local charges are simply coproduct realizations of particular boundary quantum algebras called ‘boundary ’ or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the symmetry of the open spin chain, namely we show that a number of the boundary non-local charges are in fact conserved quantities.

Related papers

On reflection algebras and twisted Yangians

Anastasia Doikou

It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.

arXiv:hep-th/0403277•Journal of Mathematical Physics•Apr 6, 2005•22 citations

General boundary conditions for the and open spin chains

D. Arnaudon, Jean Avan, Nicolas Crampé +3 more

Two types of boundary conditions (‘soliton preserving’ and ‘soliton non-preserving’) are investigated for the and open spin chains. The appropriate reflection equations are formulated and the corresponding solutions are classified. The symmetry and the Bethe ansatz equations are derived for each case. The general treatment for non-diagonal reflection matrices associated with the ‘soliton preserving’ case is worked out. The connection between the ‘soliton non-preserving’ boundary conditions and the twisted (super-) Yangians is also discussed.

arXiv:math-ph/0406021•Journal of Statistical Mechanics Theory and Experiment•Aug 24, 2004•79 citations

Boundary non-local charges from the open spin chain

Anastasia Doikou

The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine–Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non-local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary one of the non-local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian.

arXiv:math-ph/0402067•Journal of Statistical Mechanics Theory and Experiment•Dec 7, 2005•22 citations

Boundary non-local charges from the open spin chain

Anastasia Doikou

The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non–local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary, one of the non–local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian. LAPTH-1029/04 Quantum groups have attracted much interest recently because of their intrinsic mathematical interest [1]–[3], but also because they provide a powerful scheme for deriving solutions of the Yang-Baxter equation [4]–[6] R12(λ1 − λ2) R13(λ1) R23(λ2) = R23(λ2) R13(λ1) R12(λ1 − λ2). (1.1) R acts on V ⊗2, whereas the Yang–Baxter equation on V ⊗3, and as customary R12 = R ⊗ I, R23 = I ⊗ R. The

•Sep 14, 2013•18 citations

Scattering matrices in the $\mathfrak{sl}({\mbox{\sffamily\bfseries 3}})$ twisted Yangian

Jean Avan, Anastasia Doikou, Nikos Karaiskos

A quantum spin chain with non-conventional boundary conditions is studied. The distinct nature of these boundary conditions arises from the conversion of a soliton to an anti-soliton after being reflected by the boundary, hence the appellation soliton non-preserving boundary conditions. We focus on the simplest non-trivial case of this class of models based on the twisted Yangian quadratic algebra. Our computations are performed through the Bethe ansatz equations in the thermodynamic limit. We formulate a suitable quantization condition describing the scattering process and proceed with explicitly determining the bulk and boundary scattering amplitudes. The energy and quantum numbers of the low lying excitations are also derived.

arXiv:1410.5991•Journal of Statistical Mechanics Theory and Experiment•Feb 3, 2015•2 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.

arXiv:2302.11842•SciPost Physics•Nov 5, 2024•2 citations
It is known that integrable models associated to rational R matrices give rise to certain non-abelian symmetries known as Yangians. Analogously ‘boundary ’ symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton non-preserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary non-local charges directly at the quantum level. The non-local charges are simply coproduct realizations of particular boundary quantum algebras called ‘boundary ’ or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the symmetry of the open spin chain, namely we show that a number of the boundary non-local charges are in fact conserved quantities.
It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.
Two types of boundary conditions (‘soliton preserving’ and ‘soliton non-preserving’) are investigated for the and open spin chains. The appropriate reflection equations are formulated and the corresponding solutions are classified. The symmetry and the Bethe ansatz equations are derived for each case. The general treatment for non-diagonal reflection matrices associated with the ‘soliton preserving’ case is worked out. The connection between the ‘soliton non-preserving’ boundary conditions and the twisted (super-) Yangians is also discussed.
The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine–Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non-local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary one of the non-local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian.
The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non–local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary, one of the non–local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian. LAPTH-1029/04 Quantum groups have attracted much interest recently because of their intrinsic mathematical interest [1]–[3], but also because they provide a powerful scheme for deriving solutions of the Yang-Baxter equation [4]–[6] R12(λ1 − λ2) R13(λ1) R23(λ2) = R23(λ2) R13(λ1) R12(λ1 − λ2). (1.1) R acts on V ⊗2, whereas the Yang–Baxter equation on V ⊗3, and as customary R12 = R ⊗ I, R23 = I ⊗ R. The
A quantum spin chain with non-conventional boundary conditions is studied. The distinct nature of these boundary conditions arises from the conversion of a soliton to an anti-soliton after being reflected by the boundary, hence the appellation soliton non-preserving boundary conditions. We focus on the simplest non-trivial case of this class of models based on the twisted Yangian quadratic algebra. Our computations are performed through the Bethe ansatz equations in the thermodynamic limit. We formulate a suitable quantization condition describing the scattering process and proceed with explicitly determining the bulk and boundary scattering amplitudes. The energy and quantum numbers of the low lying excitations are also derived.
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.