A bstract Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and $$ \mathfrak{s}{\mathfrak{l}}_2 $$ s l 2 -invariant XXX −1/2 spin-chain. Before the deformation, the seed model can be understood as a sector of the $$ \mathfrak{psu} $$ psu (2, 2|4)-invariant spin-chain encoding the spectral problem of $$ \mathcal{N} $$ N = 4 super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact U q ( $$ \mathfrak{s}{\mathfrak{l}}_2 $$ s l 2 )-invariant XXZ −1/2 spin-chain.

Journal of High Energy PhysicsAug 12, 20254 citations

Integrable Quantum Spin Chains and Some Problems Related to Integrable Systems

A. Kundu, B. Basu-Mallick

Research reports in physicsJan 1, 19900 citations

Lecture notes in physics

Twist Deformations of Quantum Integrable Spin Chains

P. P. Kulish

Published Jan 1, 2009• 11 citations

Publisher

DOI: 10.1007/978-3-540-89793-4_9

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Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and $\mathfrak{sl}_2$ -invariant $XXX_{-1/2}$ spin-chain. Before the deformation, the seed model can be understood as a sector of the $\mathfrak{psu}(2,2|4)$ -invariant spin-chain encoding the spectral problem of $\mathcal{N}=4$ super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact $U_q(\mathfrak{sl}_2)$ -invariant $XXZ_{-1/2} $ spin-chain.Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and $\mathfrak{sl}_2$-invariant $XXX_{-1/2}$ spin-chain. Before the deformation, the seed model can be understood as a sector of the $\mathfrak{psu}(2,2|4)$-invariant spin-chain encoding the spectral problem of $\mathcal{N}=4$ super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact $U_q(\mathfrak{sl}_2)$-invariant $XXZ_{-1/2} $ spin-chain.

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Jordanian deformation of the non-compact and $\mathfrak{s}{\mathfrak{l}}_2$ -invariant XXX−1/2 spin-chainJordanian deformation of the non-compact and $$ \mathfrak{s}{\mathfrak{l}}_2 $$-invariant XXX−1/2 spin-chain

Riccardo Borsato, M. Fernández

A bstract Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and $\mathfrak{s}{\mathfrak{l}}_2$ s l 2 -invariant XXX −1/2 spin-chain. Before the deformation, the seed model can be understood as a sector of the $\mathfrak{psu}$ psu (2, 2|4)-invariant spin-chain encoding the spectral problem of $\mathcal{N}$ N = 4 super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact U q ( $\mathfrak{s}{\mathfrak{l}}_2$ s l 2 )-invariant XXZ −1/2 spin-chain.A bstract Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and $$ \mathfrak{s}{\mathfrak{l}}_2 $$ s l 2 -invariant XXX −1/2 spin-chain. Before the deformation, the seed model can be understood as a sector of the $$ \mathfrak{psu} $$ psu (2, 2|4)-invariant spin-chain encoding the spectral problem of $$ \mathcal{N} $$ N = 4 super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact U q ( $$ \mathfrak{s}{\mathfrak{l}}_2 $$ s l 2 )-invariant XXZ −1/2 spin-chain.

Journal of High Energy PhysicsAug 12, 20254 citations

Integrable Quantum Spin Chains and Some Problems Related to Integrable Systems

A. Kundu, B. Basu-Mallick

Research reports in physicsJan 1, 19900 citations

\mathfrak{sl}(2)

-invariant spin-chains. We interpret this deformed spin-chain as a deformation of a subsector of the

AdS_3/CFT_2

spin-chain, but the construction shares qualitative features also with the corresponding deformation of the

AdS_5/CFT_4

spin-chain, for example. As in similar types of deformations, we show that there exists a certain basis in which the spin-chain Hamiltonian takes a Jordan-block form. At the same time, by working in the basis of eigenstates of the generators used to construct the Groenewold-Moyal twist, the Hamiltonian appears to be diagonalisable and with a deformed spectrum. Employing the method of the Baxter equation, we write down the energy of the ground state and of excited states in a perturbation of the deformation parameter. We then consider the string-theory side of the duality, where the twist is realised as a deformation of AdS of the type of Maldacena-Russo-Hashimoto-Itzhaki. We construct a deformation of the usual BMN classical solution, and in the large-

J

limit we match the leading

\mathcal O(J^{-3})

term of the energy of the spin-chain groundstate with a conserved charge of the string classical solution. Differently from the undeformed setup as well as similar kinds of deformations, we find that the general expression of this charge of the string sigma-model is non-local, and that it does not correspond to a standard isometry. Nevertheless, it can be computed from the monodromy matrix and it is part of the tower of conserved charges provided by integrability.

Jordanian deformation of the non-compact and $\mathfrak{sl}_2 $-invariant

XXX_{-1/2}

spin-chain

Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and

\mathfrak{sl}_2

-invariant

XXX_{-1/2}

spin-chain. Before the deformation, the seed model can be understood as a sector of the

\mathfrak{psu}(2,2|4)

-invariant spin-chain encoding the spectral problem of

\mathcal{N}=4

super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact

U_q(\mathfrak{sl}_2)

-invariant $XXZ_{-1/2} $ spin-chain.

Jordanian deformation of the non-compact and

\mathfrak{s}{\mathfrak{l}}_2

-invariant XXX−1/2 spin-chain

A bstract Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and

\mathfrak{s}{\mathfrak{l}}_2

s l 2 -invariant XXX −1/2 spin-chain. Before the deformation, the seed model can be understood as a sector of the

\mathfrak{psu}

psu (2, 2|4)-invariant spin-chain encoding the spectral problem of

\mathcal{N}

N = 4 super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact U q (

\mathfrak{s}{\mathfrak{l}}_2

s l 2 )-invariant XXZ −1/2 spin-chain.