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:1603.03542 · SciPost Physics

Completeness of the Bethe states for the rational, spin-1/2 Richardson-Gaudin system

Jon Links

Published Jul 28, 2017• 30 citations
arXivHTMLPDF

Abstract

Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson–Gaudin system it will be argued that, for generic values of the system’s coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.

DOI
10.21468/scipostphys.3.1.007

Related papers

‘Bethe-ansatz-free’ eigenstates for spin-1/2 Richardson–Gaudin integrable models

Alexandre Faribault, Claude Dimo

Abstract In this work we construct the eigenstates of the most general spin-1/2 Richardson–Gaudin model integrable in an external magnetic field. This includes the possibility for fully anisotropic XYZ coupling such that the S i x S j x , S i y S j y and S i z S j z terms all have distinct coupling strengths. While insuring that integrability is maintained in the presence of an external field excludes the elliptic XYZ model which is only integrable at zero field, this work still covers a wide class of fully anisotropic (XYZ) models associated with non skew-symmetric r -matrices. The eigenstates, as constructed here, do not require any usable Bethe ansatz and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin raising (Gaudin) operators have to be defined. Indeed, the eigenstates are generically built only through the conserved charges which define the model of interest and the specification of the set of eigenvalues defining the particular eigenstate. Since these eigenvalues are, in general, solutions to a simple set of quadratic equations, the proposed approach is simpler to implement than any Bethe ansatz and, moreover, it remains completely identical independently of the symmetries of the model. Indeed, the construction removes any distinction between XYZ models and XXZ/XXX models and, generically, that between models with or without U (1) symmetry so that any difficulties associated with the use of a Bethe ansatz in any of these cases are avoided.

arXiv:1812.06428•Journal of Physics A Mathematical and Theoretical•Sep 16, 2022•1 citation

On completeness of Bethe Ansatz solutions for sl(2) Richardson–Gaudin systems

Jon Links

Springer eBooks•Jan 1, 2017•7 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

Solutions of nuclear pairing

A. B. Balantekin, Yamaç Pehlivan

We give the exact solution of orbit dependent nuclear pairing problem between two nondegenerate energy levels using the Bethe ansatz technique. Our solution reduces to previously solved cases in the appropriate limits including Richardson's treatment of reduced pairing in terms of rational Gaudin algebra operators.

arXiv:0710.3941•Physical Review C•Nov 14, 2007•18 citations

Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

Rafael I. Nepomechie, Francesco Ravanini

A Bethe Ansatz solution of the open spin- XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al , we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.

arXiv:hep-th/0307095•Journal of Physics A Mathematical and General•Oct 28, 2003•116 citations

Addendum to Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

Rafael I. Nepomechie, Francesco Ravanini

Addendum to the paper J. Phys. A36 (2003) 1391-1401: \nA Bethe Ansatz solution of the open spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al, we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.

Journal of Physics A Mathematical and General•Jan 19, 2004•41 citations
Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson–Gaudin system it will be argued that, for generic values of the system’s coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.
Abstract In this work we construct the eigenstates of the most general spin-1/2 Richardson–Gaudin model integrable in an external magnetic field. This includes the possibility for fully anisotropic XYZ coupling such that the S i x S j x , S i y S j y and S i z S j z terms all have distinct coupling strengths. While insuring that integrability is maintained in the presence of an external field excludes the elliptic XYZ model which is only integrable at zero field, this work still covers a wide class of fully anisotropic (XYZ) models associated with non skew-symmetric r -matrices. The eigenstates, as constructed here, do not require any usable Bethe ansatz and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin raising (Gaudin) operators have to be defined. Indeed, the eigenstates are generically built only through the conserved charges which define the model of interest and the specification of the set of eigenvalues defining the particular eigenstate. Since these eigenvalues are, in general, solutions to a simple set of quadratic equations, the proposed approach is simpler to implement than any Bethe ansatz and, moreover, it remains completely identical independently of the symmetries of the model. Indeed, the construction removes any distinction between XYZ models and XXZ/XXX models and, generically, that between models with or without U (1) symmetry so that any difficulties associated with the use of a Bethe ansatz in any of these cases are avoided.
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.
Addendum to the paper J. Phys. A36 (2003) 1391-1401: \nA Bethe Ansatz solution of the open spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al, we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.