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

Journal of Physics Conference Series

The Heine–Stieltjes correspondence and the polynomial approach to the Gaudin–Richardson models

Feng Pan, Kristina D. Launey, Xin Guan

Published Dec 18, 2012• 0 citations
Publisher

Abstract

The Heine–Stieltjes correspondence is extended and applied to solve Bethe ansatz equations of the Gaudin–Richardson models, from which the extended Heine–Stieltjes polynomial approach to these models is proposed. As examples for the application of this approach, exact solutions of the standard two-site Bose–Hubbard model and the standard pairing model for nuclei are formulated from the corresponding polynomials.

DOI
10.1088/1742-6596/403/1/012007

Related papers

Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem

Xin Guan, Kristina D. Launey, Mingxia Xie +3 more

A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For $k$ pairs of valence nucleons on $n$ different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one $(k+1)\ifmmode\times\else\texttimes\fi{}(k+1)$ and one $(n\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(k+1)$ matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem, which by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multiparticle systems.

arXiv:1106.5237•Physical Review C•Aug 28, 2012•79 citations

Gaudin models solver based on the correspondence between Bethe ansatz and ordinary differential equations

Alexandre Faribault, Omar El Araby, Christoph Sträter +1 more

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a different set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root-finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model, and the generalized Dicke model.

arXiv:1103.0472•Physical Review B•Jun 13, 2011•100 citations

Generalized Heine–Stieltjes and Van Vleck polynomials associated with two-level, integrable BCS models

Ian Marquette, Jon Links

We study the Bethe ansatz/ordinary differential equation (BA/ODE) correspondence for Bethe ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalized Heine–Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of integrable Bardeen–Cooper–Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p + ip-wave pairing model, the p + ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d + id-wave pairing model with additional interactions. The zeros of the generalized Heine–Stieltjes polynomials provide solutions of the corresponding Bethe ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterization of ground-state phases in these systems. Our techniques also permit the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.

arXiv:1206.2988•Journal of Statistical Mechanics Theory and Experiment•Aug 31, 2012•36 citations

An exact solution of the homogenous trimer Bose-Hubbard model

Feng Pan, Aoxue Li, Yingxin Wu +1 more

Abstract It is shown that the homogenous three-site (trimer) Bose–Hubbard model with a periodic boundary condition can be solved exactly by using an extended Bethe ansatz based on the solution of the dimer model and the site-permutation symmetry. A solution of the model within the S 3 symmetric or non-symmetric subspace is presented. Coupled differential equations of a series of extended Heine–Stieltjes polynomials, the zeros of which are related to the solution of the model, are derived. Numerical examples of the solution of the model with N ⩽ 4 bosons, including the related Heine–Stieltjes polynomials and the Van Vleck polynomials, are presented, which serve to validate the procedure and illustrate the completeness of the solutions they render.

Journal of Statistical Mechanics Theory and Experiment•Mar 1, 2023•2 citations

Richardson—Gaudin form of Bethe Ansatz solutions for an atomic-molecular Bose-Einstein condensate model

Yibing Shen, Jon Links

An integrable model describing the interconversion between Bose-Einstein condensates of atomic and molecular degrees of freedom is studied. An exact solution for this model has been previously derived via the algebraic Bethe Ansatz formalism. Here we derive an alternative solution of the Richardson-Gaudin form, and highlight some of the implications of the result.

Journal of Physics Conference Series•Apr 13, 2015•1 citation

Exact solution of the two-axis two-spin Hamiltonian

Feng Pan, Yao-Zhong Zhang, Xiaohan Qi +3 more

Abstract Bethe ansatz solution of the two-axis two-spin Hamiltonian is derived based on the Jordan–Schwinger boson realization of the SU(2) algebra. It is shown that the solution of the Bethe ansatz equations can be obtained as zeros of the related extended Heine–Stieltjes polynomials. Symmetry properties of excited levels of the system and zeros of the related extended Heine–Stieltjes polynomials are discussed. As an example of an application of the theory, the two equal spin case is studied in detail, which demonstrates that the levels in each band are symmetric with respect to the zero energy plane perpendicular to the level diagram and that excited states are always well entangled.

arXiv:2108.00604•Journal of Statistical Mechanics Theory and Experiment•Oct 1, 2021•1 citation
The Heine–Stieltjes correspondence is extended and applied to solve Bethe ansatz equations of the Gaudin–Richardson models, from which the extended Heine–Stieltjes polynomial approach to these models is proposed. As examples for the application of this approach, exact solutions of the standard two-site Bose–Hubbard model and the standard pairing model for nuclei are formulated from the corresponding polynomials.
A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For kkk pairs of valence nucleons on nnn different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one $(k+1)\ifmmode\times\else\texttimes\fi{}(k+1)$ and one $(n\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(k+1)$ matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem, which by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multiparticle systems.
We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a different set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root-finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model, and the generalized Dicke model.
We study the Bethe ansatz/ordinary differential equation (BA/ODE) correspondence for Bethe ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalized Heine–Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of integrable Bardeen–Cooper–Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p + ip-wave pairing model, the p + ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d + id-wave pairing model with additional interactions. The zeros of the generalized Heine–Stieltjes polynomials provide solutions of the corresponding Bethe ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterization of ground-state phases in these systems. Our techniques also permit the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.
Abstract It is shown that the homogenous three-site (trimer) Bose–Hubbard model with a periodic boundary condition can be solved exactly by using an extended Bethe ansatz based on the solution of the dimer model and the site-permutation symmetry. A solution of the model within the S 3 symmetric or non-symmetric subspace is presented. Coupled differential equations of a series of extended Heine–Stieltjes polynomials, the zeros of which are related to the solution of the model, are derived. Numerical examples of the solution of the model with N ⩽ 4 bosons, including the related Heine–Stieltjes polynomials and the Van Vleck polynomials, are presented, which serve to validate the procedure and illustrate the completeness of the solutions they render.
Abstract Bethe ansatz solution of the two-axis two-spin Hamiltonian is derived based on the Jordan–Schwinger boson realization of the SU(2) algebra. It is shown that the solution of the Bethe ansatz equations can be obtained as zeros of the related extended Heine–Stieltjes polynomials. Symmetry properties of excited levels of the system and zeros of the related extended Heine–Stieltjes polynomials are discussed. As an example of an application of the theory, the two equal spin case is studied in detail, which demonstrates that the levels in each band are symmetric with respect to the zero energy plane perpendicular to the level diagram and that excited states are always well entangled.