In this thesis the nested algebraic Bethe ansatz technique is applied to various orthogonal and symplectic closed and open spin chain models. Each spin chain considered is regarded as a representation of an underlying quantum group algebra, and expressions for eigenvectors of transfer matrices associated to these models are constructed using the algebra relations, reducing the problem to a set of Bethe equations. The specific models considered are the Ol'shanskii twisted Yangian spin chain, where gl_n bulk symmetry is broken to orthogonal or symplectic symmetry; the MacKay twisted Yangian spin chain, an open spin chain with bulk orthogonal or symplectic symmetry and various boundary types; and the q-deformed orthogonal or symplectic closed spin chain. For the first and third cases, a closed 'trace formula' expression for the eigenvector is also provided.In this thesis the nested algebraic Bethe ansatz technique is applied to various orthogonal and symplectic closed and open spin chain models. Each spin chain considered is regarded as a representation of an underlying quantum group algebra, and expressions for eigenvectors of transfer matrices associated to these models are constructed using the algebra relations, reducing the problem to a set of Bethe equations. The specific models considered are the Ol'shanskii twisted Yangian spin chain, where gl_n bulk symmetry is broken to orthogonal or symplectic symmetry; the MacKay twisted Yangian spin chain, an open spin chain with bulk orthogonal or symplectic symmetry and various boundary types; and the q-deformed orthogonal or symplectic closed spin chain. For the first and third cases, a closed 'trace formula' expression for the eigenvector is also provided.

Sep 1, 20200 citations

Nested algebraic Bethe ansatz for orthogonal and symplectic open spin chains

Allan John Gerrard, Vidas Regelskis

We present a nested algebraic Bethe ansatz for one-dimensional so2n- and sp2n-symmetric open spin chains with diagonal boundary conditions. The monodromy matrix of these spin chains satisfies the defining relations on the extended twisted Yangians Xρ(so2n,so2nρ)tw and Xρ(sp2n,sp2nρ)tw, respectively. We use a generalisation of the De Vega and Karowski approach allowing us to relate the spectral problem of so2n- or sp2n-symmetric open spin chain to that of gln-symmetric open spin chain studied by Belliard and Ragoucy. We explicitly derive the structure of Bethe vectors, their eigenvalues and the nested Bethe equations. We also provide a proof of Belliard and Ragoucy's trace formula for Bethe vectors of gln-symmetric open spin chains.We present a nested algebraic Bethe ansatz for one-dimensional so2n- and sp2n-symmetric open spin chains with diagonal boundary conditions. The monodromy matrix of these spin chains satisfies the defining relations on the extended twisted Yangians Xρ(so2n,so2nρ)tw and Xρ(sp2n,sp2nρ)tw, respectively. We use a generalisation of the De Vega and Karowski approach allowing us to relate the spectral problem of so2n- or sp2n-symmetric open spin chain to that of gln-symmetric open spin chain studied by Belliard and Ragoucy. We explicitly derive the structure of Bethe vectors, their eigenvalues and the nested Bethe equations. We also provide a proof of Belliard and Ragoucy's trace formula for Bethe vectors of gln-symmetric open spin chains.

arXiv:1909.12123Nuclear Physics BJan 2, 20207 citations

Bethe ansatz solution of the invariant spin chain

Marcio Jose Martins

arXiv:hep-th/9502135Physics Letters BOct 1, 199517 citations

On the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$ -symmetryOn the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$-symmetry

Allan John Gerrard

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with $\mathfrak{g}$ -symmetry for any simple Lie algebra $\mathfrak{g}$ . Starting the nesting process by removing a single simple root from $\mathfrak{g}$ , we use the residual $U(1)$ charge and the block Gauss decomposition of the $R$ -matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with $\mathfrak{g}$-symmetry for any simple Lie algebra $\mathfrak{g}$. Starting the nesting process by removing a single simple root from $\mathfrak{g}$, we use the residual $U(1)$ charge and the block Gauss decomposition of the $R$-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.

arXiv:2405.20177May 30, 20241 citation

Bethe ansatz of the open spin-sXXZ chain with nondiagonal boundary terms

Rajan Murgan

arXiv:0901.3558Journal of High Energy PhysicsApr 20, 200928 citations

On the nested algebraic Bethe ansatz for spin chains with simple

\mathfrak{g}

-symmetry

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with

\mathfrak{g}

-symmetry for any simple Lie algebra

\mathfrak{g}

. Starting the nesting process by removing a single simple root from

\mathfrak{g}

, we use the residual

U(1)

charge and the block Gauss decomposition of the

R

-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.