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arXiv:2604.27189

The quantum group structure of long-range integrable deformations

Koen Schouten, Marius de Leeuw

Published Apr 29, 2026• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2604.27189

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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.

We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary short-range (e.g. nearest-neighbor) integrable spin chains, it allows to construct an infinite set of conserved long-range charges. We explain the moduli space of deformation parameters by different classes of generating operators. The rapidity map and dressing phase in the long-range Bethe equations are a result of these deformations. The closed chain asymptotic Bethe equations for long-range spin chains transforming under a generic symmetry algebra are derived. Notably, our construction applies to generalizations of standard nearest-neighbor chains such as alternating spin chains. We also discuss relevant properties for its application to planar D=4, N=4 and D=3, N=6 supersymmetric gauge theories. Finally, we present a map between long-range and inhomogeneous spin chains delivering more insight into the structures of these models as well as their limitations at wrapping order.

We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary short-range (e.g. nearest neighbor) integrable spin chains, it allows us to construct an infinite set of conserved long-range charges. We explain the moduli space of deformation parameters by different classes of generating operators. The rapidity map and dressing phase in the long-range Bethe equations are a result of these deformations. The closed chain asymptotic Bethe equations for long-range spin chains transforming under a generic symmetry algebra are derived. Notably, our construction applies to generalizations of standard nearest neighbor chains such as alternating spin chains. We also discuss relevant properties for its application to planar and supersymmetric gauge theories. Finally, we present a map between long-range and inhomogeneous spin chains delivering more insight into the structures of these models, as well as their limitations at wrapping order.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of ﬁnite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a ﬁnite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-ﬁeld six-vertex model.

Matrix representations of

{\rm BC}_n

associative Yang--Baxter equations and related long-range spin chains

We construct a matrix representation of the algebra defined by the associative Yang--Baxter equations of

{\rm BC}_n

type. In the elliptic case, two sets of generators are represented in terms of the Baxter--Belavin quantum

R

-matrix in the fundamental representation of

{\rm GL}_N

, while the remaining generators are represented by the elliptic

K

-matrix found by Komori and Hikami. We revisit its derivation using the Felder--Pasquier procedure and correct the unitarity relation for odd

N

. We then use these matrix representations to construct examples of long-range quantum spin chains from

R

-matrix-valued Lax pairs for the classical Calogero--Inozemtsev system associated with the

{\rm BC}_n

root system. We also construct

R

-matrix-valued Lax pairs for Calogero--Moser models in an external field, leading to Haldane--Shastry and Polychronakos--Frahm chains with modified spin positions.

Abstract In this note, we study the eigenvectors and the scalar products the integrable long-range deformation of the XXX spin chain defined in [1]. The model is solved exactly by algebraic Bethe ansatz, and it coincides in the bulk with the Inozemtsev spin chain. At the closing point it contains a defect which effectively removes the wrapping interactions. Here we concentrate on determining the defect term for the first non-trivial order in perturbation in the deformation parameter and how it affects the Bethe ansatz equations. Our study is motivated by the relation with the dilatation operator of the $ \mathcal{N} $ = 4 gauge theory in the su(2) sector.

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.