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

arXiv:1610.08032 · Journal of High Energy Physics

New construction of eigenstates and separation of variables for SU(N) quantum spin chains

Nikolay Gromov, Fedor Levkovich-Maslyuk, Grigory Sizov

Published Sep 1, 2017• 108 citations
arXivHTMLPDF

Abstract

We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.

DOI
10.1007/jhep09(2017)111

Related papers

New compact construction of eigenstates for supersymmetric spin chains

Nikolay Gromov, Fedor Levkovich-Maslyuk

Abstract The problem of separation of variables (SoV) in supersymmetric spin chains is closely related to the calculation of correlation functions in $$ \mathcal{N}=4 $$ N = 4 SYM theory which is integrable in the planar limit. To address this question we find a compact formula for the spin chain eigenstates, which does not have any sums over auxiliary roots one usually gets in the widely adopted nested Bethe ansatz. Our construction only involves one application of a simple B g (u k ) operator to the reference state for each of the magnons, in complete analogy with the $$ \mathfrak{s}\mathfrak{u}(2) $$ s u 2 algebraic Bethe ansatz. This generalizes our SoV based construction for $$ \mathfrak{s}\mathfrak{u}(n) $$ s u n to the supersymmetric $$ \mathfrak{s}\mathfrak{u}\left(1\Big|2\right) $$ s u 1 | 2 case.

arXiv:1805.03927•Journal of High Energy Physics•Sep 1, 2018•22 citations

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

Kun Hao, Junpeng Cao, Guang-Liang Li +3 more

An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.

arXiv:1601.04771•Journal of High Energy Physics•May 1, 2016•4 citations

The nested Bethe ansatz for ‘all’ open spin chains with diagonal boundary conditions

S. Belliard, E. Ragoucy

We present in a unified and detailed way the nested Bethe ansatz for open spin chains based on(super)algebras, with arbitrary representations (i.e., 'spins') on each site of the chain and diagonal boundary matrices (K + (u), K -(u)).The nested Bethe ansatz applies for a general K -(u), but a particular form of the K + (u) matrix.The construction extends and unifies the results already obtained for open spin chains based on the fundamental representation and for some particular superspin chains.We give the eigenvalues, Bethe equations and the explicit form of the Bethe vectors for the corresponding models.The Bethe vectors are also expressed using a trace formula.

arXiv:0902.0321•Journal of Physics A Mathematical and Theoretical•Apr 29, 2009•28 citations

Separation of variables for higher rank integrable models: review of recent progress

Fedor Levkovich-Maslyuk

Abstract Separation of variables (SoVs) is a powerful method expected to be applicable for a wide range of quantum integrable systems, from models in condensed matter physics to gauge and string theories. Yet its full implementation for many higher rank examples, such as SU(N) spin chains with N > 2, has remained elusive for a long time. In this pedagogical review we discuss the major progress achieved recently in understanding SoV for models of this type. In particular, for rational SU(N) spin chains we describe different constructions of the SoV basis, novel compact forms for spin chain eigenstates, the functional SoV approach, and explicit computation of the SoV measure. We also discuss key first applications of these results, namely the new compact determinant representations for many observables such as scalar products and correlators.

arXiv:2503.15398•Journal of Physics A Mathematical and Theoretical•Jun 4, 2025•3 citations

On factorized overlaps: Algebraic Bethe Ansatz, twists, and separation of variables

Tamás Gombor, Balázs Pozsgay

We investigate the exact overlaps between eigenstates of integrable spin chains and a special class of states called “integrable initial/final states”. These states satisfy a special integrability constraint, and they are closely related to integrable boundary conditions. We derive new algebraic relations for the integrable states, which lead to a set of recursion relations for the exact overlaps. We solve these recursion relations and thus we derive new overlap formulas, valid in the XXX Heisenberg chain and its integrable higher spin generalizations. Afterwards we generalize the integrability condition to twisted boundary conditions, and derive the corresponding exact overlaps. Finally, we embed the integrable states into the “Separation of Variables” framework, and derive an alternative representation for the exact overlaps of the XXX chain. Our derivations and proofs are rigorous, and they can form the basis of future investigations involving more complicated models such as nested or long-range deformed systems.

arXiv:2101.10354•Nuclear Physics B•Apr 6, 2021•28 citations

Bethe ansatz without nesting

Gleb Eduardovich Arutyunov, Hrachya M. Babujian, Minghao Gao

A bstract We develop a non-nested Bethe ansatz description of rational 𝔤𝔩 ℓ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the 𝔤𝔩 3 and 𝔤𝔩 4 spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank- ℓ equation ℛ ℓ = 0. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial Q 1 ( u ). We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational 𝔤𝔩 ℓ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated Q -system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the 𝔤𝔩 3 spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order 𝔤𝔩 3 oper.

arXiv:2607.11617•Journal of High Energy Physics•Oct 8, 2026•0 citations
We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.
Abstract The problem of separation of variables (SoV) in supersymmetric spin chains is closely related to the calculation of correlation functions in N=4 \mathcal{N}=4 N=4 N = 4 SYM theory which is integrable in the planar limit. To address this question we find a compact formula for the spin chain eigenstates, which does not have any sums over auxiliary roots one usually gets in the widely adopted nested Bethe ansatz. Our construction only involves one application of a simple B g (u k ) operator to the reference state for each of the magnons, in complete analogy with the su(2) \mathfrak{s}\mathfrak{u}(2) su(2) s u 2 algebraic Bethe ansatz. This generalizes our SoV based construction for su(n) \mathfrak{s}\mathfrak{u}(n) su(n) s u n to the supersymmetric su(1∣2) \mathfrak{s}\mathfrak{u}\left(1\Big|2\right) su(1​2) s u 1 | 2 case.
An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.
We present in a unified and detailed way the nested Bethe ansatz for open spin chains based on(super)algebras, with arbitrary representations (i.e., 'spins') on each site of the chain and diagonal boundary matrices (K + (u), K -(u)).The nested Bethe ansatz applies for a general K -(u), but a particular form of the K + (u) matrix.The construction extends and unifies the results already obtained for open spin chains based on the fundamental representation and for some particular superspin chains.We give the eigenvalues, Bethe equations and the explicit form of the Bethe vectors for the corresponding models.The Bethe vectors are also expressed using a trace formula.
Abstract Separation of variables (SoVs) is a powerful method expected to be applicable for a wide range of quantum integrable systems, from models in condensed matter physics to gauge and string theories. Yet its full implementation for many higher rank examples, such as SU(N) spin chains with N > 2, has remained elusive for a long time. In this pedagogical review we discuss the major progress achieved recently in understanding SoV for models of this type. In particular, for rational SU(N) spin chains we describe different constructions of the SoV basis, novel compact forms for spin chain eigenstates, the functional SoV approach, and explicit computation of the SoV measure. We also discuss key first applications of these results, namely the new compact determinant representations for many observables such as scalar products and correlators.
We investigate the exact overlaps between eigenstates of integrable spin chains and a special class of states called “integrable initial/final states”. These states satisfy a special integrability constraint, and they are closely related to integrable boundary conditions. We derive new algebraic relations for the integrable states, which lead to a set of recursion relations for the exact overlaps. We solve these recursion relations and thus we derive new overlap formulas, valid in the XXX Heisenberg chain and its integrable higher spin generalizations. Afterwards we generalize the integrability condition to twisted boundary conditions, and derive the corresponding exact overlaps. Finally, we embed the integrable states into the “Separation of Variables” framework, and derive an alternative representation for the exact overlaps of the XXX chain. Our derivations and proofs are rigorous, and they can form the basis of future investigations involving more complicated models such as nested or long-range deformed systems.
A bstract We develop a non-nested Bethe ansatz description of rational 𝔤𝔩 ℓ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the 𝔤𝔩 3 and 𝔤𝔩 4 spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank- ℓ equation ℛ ℓ = 0. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial Q 1 ( u ). We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational 𝔤𝔩 ℓ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated Q -system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the 𝔤𝔩 3 spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order 𝔤𝔩 3 oper.