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:2502.15636 · SciPost Physics Core

Spin-$s$ $Q$-systems: Twist and open boundaries

Yi-Jun He, Jue Hou, Yichao Liu, Zi-Xi Tan

Published Aug 27, 2025• 0 citations
arXivHTMLPDF

Abstract

In integrable spin chains, the spectral problem can be solved by the method of Bethe Ansatz, which transforms the problem of diagonalization of the Hamiltonian into the problem of solving a set of algebraic equations named Bethe equations. In this work, we systematically investigate the spin- s s XXX chain with twisted and open boundary conditions using the rational Q Q -system, which is a powerful tool to solve Bethe equations. We establish basic frameworks of the rational Q Q -system and confirm its completeness numerically in both cases. For twisted boundaries, we investigate the polynomiality conditions of the rational Q Q -system and derive physical conditions for singular solutions of Bethe equations. For open boundaries, we uncover novel phenomena such as hidden symmetries and boundary induced strings under specific boundary parameters. Hidden symmetries lead to the appearance of extra degeneracies in the Hilbert space, while the boundary induced string is a novel type of exact string configuration, whose length depends on the boundary magnetic fields. These findings, supported by both analytical and numerical evidences, offer new insights into the interplay between symmetries and boundary conditions.

DOI
10.21468/scipostphyscore.8.3.057

Related papers

Rational $Q$-systems for integrable spin chains without $U(1)$ symmetry

Yunfeng Jiang, Yichao Liu, Yuan Miao +1 more

The Q Q -system is an efficient method for finding complete physical solutions of Bethe Ansatz equations, but so far its application has been confined to systems possessing U(1) U ( 1 ) symmetry. We extend the rational Q Q -system framework to integrable spin chains without U(1) U ( 1 ) symmetry, exemplified by the closed XXZ model with anti-diagonal twists and the open XXZ model with non-diagonal boundary fields. We demonstrate that the Q Q -system can be derived by combining TQ T Q -relation with fusion relations of higher-spin transfer matrices. This yields QQ Q Q -relations analogous to the U(1) U ( 1 ) symmetric case but incorporating additional inhomogeneous terms. We present numerical solutions that are validated against exact diagonalization, confirming that it generates all and exclusively physical solutions.

arXiv:2512.01551•SciPost Physics•Jun 22, 2026•0 citations

Bethe-root configurations and spectral degeneracy in the open XXZ chain with degenerate boundaries

Shizhang Wu, Shu Chen, Junpeng Cao +1 more

We investigate the structure of Bethe-root configurations in the spin-1/2 XXZ chain with degenerate open boundaries. The physical solutions of the Bethe Ansatz equations are classified into three types in the fully degenerate case and two types in the partially degenerate case, for which we propose counting formulas for the number of physical solution sets of each type. Furthermore, we observe spectral degeneracies in the fully degenerate case and show that they can be naturally explained by the presence of specific phantom strings in the Bethe roots. The classification and resulting spectral degeneracies in the diagonal limit are also discussed.

arXiv:2608.20815•Aug 21, 2026•0 citations

Spin-$s$ rational $Q$-system

Jue Hou, Yunfeng Jiang, Rui-Dong Zhu

Bethe Ansatz equations for spin-s Heisenberg spin chain with s≥1 are significantly more difficult to analyze than the spin-1/2 case, due to the presence of repeated roots. As a result, it is challenging to derive extra conditions for the Bethe roots to be physical and study the related completeness problem. In this paper, we propose the rational Q-system for the XXX _s s spin chain. Solutions of the proposed Q-system give all and only physical solutions of the Bethe Ansatz equations required by completeness. This is checked numerically and proved rigorously. The rational Q-system is equivalent to the requirement that the solution and the corresponding dual solution of the TQ-relation are both polynomials, which we prove rigorously. Based on this analysis, we propose the extra conditions for solutions of the XXX _s s Bethe Ansatz equations to be physical.

arXiv:2303.07640•SciPost Physics•Apr 26, 2024•10 citations

Completeness of Wronskian Bethe equations for rational $\mathfrak{gl}_{m|n}$ spin chain

Dmitry Chernyak, Sébastien Leurent, Dmytro Volin

We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities $\theta_\ell$, $\ell=1,\ldots,L$ satisfy $\theta_\ell+\hbar\neq\theta_{\ell'}$ for $\ell<\ell'$. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of $\theta_\ell$. In particular, we consider the regime $\theta_{\ell+1}/\theta_{\ell}\gg 1$ for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.

•Apr 22, 2020•0 citations

Completeness of Wronskian Bethe equations for rational gl(m|n) spin chain

Dmitry Chernyak, Sébastien Leurent, Dmytro Volin

We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities $\theta_\ell$, $\ell=1,\ldots,L$ satisfy $\theta_\ell+\hbar\neq\theta_{\ell'}$ for $\ell<\ell'$. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of $\theta_\ell$. In particular, we consider the regime $\theta_{\ell+1}/\theta_{\ell}\gg 1$ for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.

•Apr 6, 2020•4 citations

On reflection algebras and twisted Yangians

Anastasia Doikou

It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.

arXiv:hep-th/0403277•Journal of Mathematical Physics•Apr 6, 2005•22 citations
Spin-sss QQQ-systems: Twist and open boundaries
In integrable spin chains, the spectral problem can be solved by the method of Bethe Ansatz, which transforms the problem of diagonalization of the Hamiltonian into the problem of solving a set of algebraic equations named Bethe equations. In this work, we systematically investigate the spin- s s XXX chain with twisted and open boundary conditions using the rational Q Q -system, which is a powerful tool to solve Bethe equations. We establish basic frameworks of the rational Q Q -system and confirm its completeness numerically in both cases. For twisted boundaries, we investigate the polynomiality conditions of the rational Q Q -system and derive physical conditions for singular solutions of Bethe equations. For open boundaries, we uncover novel phenomena such as hidden symmetries and boundary induced strings under specific boundary parameters. Hidden symmetries lead to the appearance of extra degeneracies in the Hilbert space, while the boundary induced string is a novel type of exact string configuration, whose length depends on the boundary magnetic fields. These findings, supported by both analytical and numerical evidences, offer new insights into the interplay between symmetries and boundary conditions.
Rational QQQ-systems for integrable spin chains without U(1)U(1)U(1) symmetry
The Q Q -system is an efficient method for finding complete physical solutions of Bethe Ansatz equations, but so far its application has been confined to systems possessing U(1) U ( 1 ) symmetry. We extend the rational Q Q -system framework to integrable spin chains without U(1) U ( 1 ) symmetry, exemplified by the closed XXZ model with anti-diagonal twists and the open XXZ model with non-diagonal boundary fields. We demonstrate that the Q Q -system can be derived by combining TQ T Q -relation with fusion relations of higher-spin transfer matrices. This yields QQ Q Q -relations analogous to the U(1) U ( 1 ) symmetric case but incorporating additional inhomogeneous terms. We present numerical solutions that are validated against exact diagonalization, confirming that it generates all and exclusively physical solutions.
We investigate the structure of Bethe-root configurations in the spin-1/2 XXZ chain with degenerate open boundaries. The physical solutions of the Bethe Ansatz equations are classified into three types in the fully degenerate case and two types in the partially degenerate case, for which we propose counting formulas for the number of physical solution sets of each type. Furthermore, we observe spectral degeneracies in the fully degenerate case and show that they can be naturally explained by the presence of specific phantom strings in the Bethe roots. The classification and resulting spectral degeneracies in the diagonal limit are also discussed.
Spin-sss rational QQQ-system
Bethe Ansatz equations for spin-s Heisenberg spin chain with s≥1 are significantly more difficult to analyze than the spin-1/2 case, due to the presence of repeated roots. As a result, it is challenging to derive extra conditions for the Bethe roots to be physical and study the related completeness problem. In this paper, we propose the rational Q-system for the XXX _s s spin chain. Solutions of the proposed Q-system give all and only physical solutions of the Bethe Ansatz equations required by completeness. This is checked numerically and proved rigorously. The rational Q-system is equivalent to the requirement that the solution and the corresponding dual solution of the TQ-relation are both polynomials, which we prove rigorously. Based on this analysis, we propose the extra conditions for solutions of the XXX _s s Bethe Ansatz equations to be physical.
Completeness of Wronskian Bethe equations for rational glm∣n\mathfrak{gl}_{m|n}glm∣n​ spin chain
We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities θℓ\theta_\ellθℓ​, ℓ=1,…,L\ell=1,\ldots,Lℓ=1,…,L satisfy θℓ+ℏ≠θℓ′\theta_\ell+\hbar\neq\theta_{\ell'}θℓ​+ℏ=θℓ′​ for ℓ<ℓ′\ell<\ell'ℓ<ℓ′. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of θℓ\theta_\ellθℓ​. In particular, we consider the regime θℓ+1/θℓ≫1\theta_{\ell+1}/\theta_{\ell}\gg 1θℓ+1​/θℓ​≫1 for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.
We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities θℓ\theta_\ellθℓ​, ℓ=1,…,L\ell=1,\ldots,Lℓ=1,…,L satisfy θℓ+ℏ≠θℓ′\theta_\ell+\hbar\neq\theta_{\ell'}θℓ​+ℏ=θℓ′​ for ℓ<ℓ′\ell<\ell'ℓ<ℓ′. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of θℓ\theta_\ellθℓ​. In particular, we consider the regime θℓ+1/θℓ≫1\theta_{\ell+1}/\theta_{\ell}\gg 1θℓ+1​/θℓ​≫1 for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.
It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.