arXiv:2503.24223

Jordanian deformation of the non-compact and $\mathfrak{sl}_2 $-invariant $XXX_{-1/2}$ spin-chain

Riccardo Borsato, Fernández, Miguel García

Published Mar 31, 2025• 1 citation

Abstract

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.

DOI: 10.48550/arxiv.2503.24223

Related papers

Jordanian 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.

Journal of High Energy PhysicsAug 12, 20254 citations

Groenewold-Moyal twists, integrable spin-chains and AdS/CFT

Riccardo Borsato, Miguel García Fernández

hep-thCross-listedarXiv:2604.07291Apr 9, 2026

s l 2 Gaudin model with jordanian twist

N. Cirilo António, N. Manojlović

arXiv:nlin/0506029Journal of Mathematical PhysicsOct 1, 200520 citations

The quantum group structure of long-range integrable deformations

Koen Schouten, Marius de Leeuw

arXiv:2604.27189Apr 29, 20260 citations

$$ T\overline{T} $$-deformation and long range spin chains

Balázs Pozsgay, Yunfeng Jiang, G. Takács

arXiv:1911.11118Journal of High Energy PhysicsMar 1, 202076 citations

Quantum-group-invariant $$ {D}_{n+1}^{(2)} $$ models: Bethe ansatz and finite-size spectrum

Holger Frahm, Sascha Gehrmann, Rafael I. Nepomechie +1 more

A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra $$ {D}_{n+1}^{(2)} $$ D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra $$ {D}_3^{(2)} $$ D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1 $$ {D}_2^{(2)} $$ D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the $$ {D}_2^{(2)} $$ D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.

arXiv:2509.00610Journal of High Energy PhysicsDec 16, 20250 citations

arXiv:2503.24223

Jordanian deformation of the non-compact and $\mathfrak{sl}_2 $-invariant $XXX_{-1/2}$ spin-chainJordanian deformation of the non-compact and $\mathfrak{sl}_2 $-invariant $XXX_{-1/2}$ spin-chain

Riccardo Borsato, Fernández, Miguel García

Published Mar 31, 2025• 1 citation

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2503.24223

Related papers

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

Groenewold-Moyal twists, integrable spin-chains and AdS/CFT

Riccardo Borsato, Miguel García Fernández

We take the first steps to address via integrability the spectral problem of AdS/CFT dual pairs deformed by Groenewold-Moyal twists. In particular, we start by considering a twisted spin-chain that couples, through a Groenewold-Moyal twist deformation, two $\mathfrak{sl}(2)$ -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.We take the first steps to address via integrability the spectral problem of AdS/CFT dual pairs deformed by Groenewold-Moyal twists. In particular, we start by considering a twisted spin-chain that couples, through a Groenewold-Moyal twist deformation, two $\mathfrak{sl}(2)$-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.

hep-thCross-listedarXiv:2604.07291Apr 9, 2026

s l 2 Gaudin model with jordanian twist

N. Cirilo António, N. Manojlović

s l 2 Gaudin model with jordanian twist is studied. This system can be obtained as the semiclassical limit of the XXX spin chain deformed by the jordanian twist. The appropriate creation operators that yield the Bethe states of the Gaudin model and consequently its spectrum are defined. Their commutation relations with the generators of the corresponding loop algebra as well as with the generating function of integrals of motion are given. The inner products and norms of Bethe states and the relation to the solutions of the Knizhnik-Zamolodchikov equations are discussed.s l 2 Gaudin model with jordanian twist is studied. This system can be obtained as the semiclassical limit of the XXX spin chain deformed by the jordanian twist. The appropriate creation operators that yield the Bethe states of the Gaudin model and consequently its spectrum are defined. Their commutation relations with the generators of the corresponding loop algebra as well as with the generating function of integrals of motion are given. The inner products and norms of Bethe states and the relation to the solutions of the Knizhnik-Zamolodchikov equations are discussed.

arXiv:nlin/0506029Journal of Mathematical PhysicsOct 1, 200520 citations

The quantum group structure of long-range integrable deformations

Koen Schouten, Marius de Leeuw

Quantum integrable spin chains are known to possess a large family of long-range deformations generated by the local, boost and bilocal operators. Although these deformations are well-understood on the level of the pairwise commuting charges, the underlying quantum group structures had not yet been recognised. In this paper, we provide a quantum group-theoretical description for the family of long-range deformations of arbitrary homogeneous Yang-Baxter integrable spin chains up to first order in the deformation parameter. In particular, we show that the deformations are obtained via a twist of the algebraic structure of the underlying quantum group. This twisting results in a generally non-associative algebra that has a non-trivial Drinfeld associator. The Drinfeld associator is then shown to encode the information about the long-range interaction terms for the integrable spin chain. Moreover, the deformed quantum group is shown to contain a large perturbatively associative substructure, thus ensuring the perturbative integrability of the long-range model. The deformed quantum group provides explicit expressions for the Lax operators and R-matrices of the long-range deformed models, which manifestly satisfy the RLL relation and the Yang-Baxter equation up to first order in the deformation parameter. In order to derive the results, we introduce algebra elements that we call the algebraic charge densities. As a side result, we provide a conjecture for the explicit expressions of the undeformed charge densities in terms of these algebra elements.Quantum integrable spin chains are known to possess a large family of long-range deformations generated by the local, boost and bilocal operators. Although these deformations are well-understood on the level of the pairwise commuting charges, the underlying quantum group structures had not yet been recognised. In this paper, we provide a quantum group-theoretical description for the family of long-range deformations of arbitrary homogeneous Yang-Baxter integrable spin chains up to first order in the deformation parameter. In particular, we show that the deformations are obtained via a twist of the algebraic structure of the underlying quantum group. This twisting results in a generally non-associative algebra that has a non-trivial Drinfeld associator. The Drinfeld associator is then shown to encode the information about the long-range interaction terms for the integrable spin chain. Moreover, the deformed quantum group is shown to contain a large perturbatively associative substructure, thus ensuring the perturbative integrability of the long-range model. The deformed quantum group provides explicit expressions for the Lax operators and R-matrices of the long-range deformed models, which manifestly satisfy the RLL relation and the Yang-Baxter equation up to first order in the deformation parameter. In order to derive the results, we introduce algebra elements that we call the algebraic charge densities. As a side result, we provide a conjecture for the explicit expressions of the undeformed charge densities in terms of these algebra elements.

arXiv:2604.27189Apr 29, 20260 citations

$T\overline{T}$ -deformation and long range spin chains$$ T\overline{T} $$-deformation and long range spin chains

Balázs Pozsgay, Yunfeng Jiang, G. Takács

Abstract We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the $T\overline{T}$ T T ¯ -deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of $\mathcal{N}$ N = 4 super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the $T\overline{T}$ T T ¯ -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.Abstract We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the $$ T\overline{T} $$ T T ¯ -deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the $$ T\overline{T} $$ T T ¯ -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.

arXiv:1911.11118Journal of High Energy PhysicsMar 1, 202076 citations

Quantum-group-invariant ${D}_{n+1}^{(2)}$ models: Bethe ansatz and finite-size spectrumQuantum-group-invariant $$ {D}_{n+1}^{(2)} $$ models: Bethe ansatz and finite-size spectrum

Holger Frahm, Sascha Gehrmann, Rafael I. Nepomechie +1 more

A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra ${D}_{n+1}^{(2)}$ D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra ${D}_3^{(2)}$ D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1 ${D}_2^{(2)}$ D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the ${D}_2^{(2)}$ D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra $$ {D}_{n+1}^{(2)} $$ D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra $$ {D}_3^{(2)} $$ D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1 $$ {D}_2^{(2)} $$ D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the $$ {D}_2^{(2)} $$ D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.

arXiv:2509.00610Journal of High Energy PhysicsDec 16, 20250 citations

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

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.

\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.

T\overline{T}

-deformation and long range spin chains

Abstract We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the

T\overline{T}

T T ¯ -deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of

\mathcal{N}

N = 4 super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the

T\overline{T}

T T ¯ -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.

Quantum-group-invariant

{D}_{n+1}^{(2)}

models: Bethe ansatz and finite-size spectrum

A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra

{D}_{n+1}^{(2)}

D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra

{D}_3^{(2)}

D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1

{D}_2^{(2)}

D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the

{D}_2^{(2)}

D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.