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

New polynomials associated with a special LMG model

Feng Pan, Lina Bao, Liyuan Zhai, Xiaoyue Cui, Jerry Paul Draayer

Published Jan 5, 2011• 0 citations
OpenAlex

Abstract

Abstract: New polynomials associated with a special Lipkin-Meshkov-Glick (LMG) model corresponding to the standard two-site Bose-Hubbard model are derived based on the Stieltjes correspondence. It is shown that there is a one-to-one correspondence between zeros of this new polynomial and solutions of the Bethe ansatz equations for the LMG model. A one-dimensional classical electrostatic analogue corresponding to the special LMG model is established according to Stieltjes early work. It shows that any possible configuration of equilibrium positions of the charges in the electrostatic problem corresponds uniquely to one set of roots of the Bethe ansatz equations for the LMG model, and the number of possible configurations of equilibrium positions of the charges equals exactly to the number of energy levels in the LMG model. Some relations of sums of powers and inverse powers of zeros of the new polynomials related to the eigenenergies of the LMG model are derived.

Related papers

The Lipkin-Meshkov-Glick model from the perspective of theSU(1,1) Richardson-Gaudin models

Sergio Lerma-Hernández, J. Dukelsky

Originally introduced in nuclear physics as a numerical laboratory to test different many-body approximation methods, the Lipkin-Meshkov-Glick (LMG) model has received much attention as a simple enough but non-trivial model with many interesting features for areas of physics beyond the nuclear one. In this contribution we look at the LMG model as a particular example of an SU (1,1) Richardson-Gaudin model. The characteristics of the model are analyzed in terms of the behavior of the spectral-parameters or pairons which determine both eigenvalues and eigenfunctions of the model Hamiltonian. The problem of finding these pairons is mathematically equivalent to obtain the equilibrium positions of a set of electric charges moving in a two dimensional space. The electrostatic problems for the different regions of the model parameter space are discussed and linked to the different energy density of states already identified in the LMG spectrum.

Journal of Physics Conference Series•Mar 18, 2014•6 citations

SOLUTION FOR THE LMG MODEL HAMILTONIAN BASED ON THE RECURRENCE FORMULA AND JACOBI MATRIX

Z. ALIPOUR, M. A. Jafarizadeh, N. FOOLADI +1 more

Based on the recursion relation formulas and the Jacobi matrix from the theory of orthogonal polynomials, a numeric solution for the Lipkin–Meshkov–Glick (LMG) model Hamiltonian is given, without considering any restriction on the parameters specifying the strengths of the interactions included in the LMG model. Moreover, via this method, we have studied the shape phase transitions for one example of nuclei. At the end we compare the results thus obtained with those of Bethe ansatz and Hartree–Fock methods.

International Journal of Modern Physics E•Apr 5, 2011•2 citations

The SU(N) Fermi-Hubbard Model on two sites: Bethe Ansatz solution and Quantum Phase Transition of the Lipkin-Meshkov-Glick Model in the large-N limit

Pierre Nataf

We show that the SU(N) Fermi-Hubbard model (FHM) on two sites, where N is the number of flavors of each fermion, corresponds to an exactly solvable two-level many-boson model that Richardson [J. Math. Phys. 9, 1327 (1968)] analytically solved long ago. We express the Bethe ansatz solutions as a function of the physical parameters of the SU(N) FHM, and recast its eigenvalues and eigenstates in terms of the Richardson pair energies and creation operators. In this context, the connection with the well-studied Lipkin-Meshkov-Glick (LMG) model, known as equivalent to the Richardson model, is established and serves as a guideline to the prediction of some N-body physics phenomena in the two-site SU(N) FHM with N particles. In particular, the LMG second-order quantum phase transition (QPT) is shown to occur in the SU(N) FHM for an attractive density-density interaction U equal to U_c=-1/(2N), in units of the (absolute value of the) tunneling amplitude between the two sites. We show the finite-size energies, the gap, and the kinetic energy, which all reveal the transition, as a function of U for values of N from N = 3 to N = 36, suggesting that the QPT could be experimentally achieved with current technologies involving SU(N) ultracold atoms or molecules. Finally, we show the entanglement entropy of the first site with respect to the second, and it scales like N at the transition, in contrast with several two-mode models.

arXiv:2412.13327•Dec 17, 2024•0 citations

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

Feng Pan, Kristina D. Launey, Xin Guan

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.

Journal of Physics Conference Series•Dec 18, 2012•0 citations

Extended Lipkin–Meshkov–Glick Hamiltonian

R Romano, X. Roca-Maza, G. Colò +1 more

Abstract The Lipkin–Meshkov–Glick (LMG) model was devised to test the validity of different approximate formalisms to treat many-particle systems. The model was constructed to be exactly solvable and yet non-trivial, in order to capture some of the main features of real physical systems. In the present contribution, we explicitly review the fact that different many-body approximations commonly used in different fields in physics clearly fail to describe the exact LMG solution. With similar assumptions as those adopted for the LMG model, we propose a new Hamiltonian based on a general two-body interaction. The new model (extended LMG) is not only more general than the original LMG model and, therefore, has a potentially larger spectrum of applicability, but also the physics behind its exact solution can be much better captured by common many-body approximations. At the basis of this improvement lies a new term in the Hamiltonian that depends on the number of constituents and polarizes the system; the associated symmetry breaking is discussed, together with some implications for the study of more realistic systems.

arXiv:2009.03593•Journal of Physics G Nuclear and Particle Physics•Jan 5, 2021•3 citations

The Lipkin-Meshkov-Glick Model and its Deformations through Polynomial Algebras

N. Debergh, Floarea Stancu

We consider the many-particle Hamiltonian of Lipkin, Meshkov and Glick in the context of deformed algebras. The reducibility of the original model is proved according to the representations of these polynomial algebras. The LMG spectrum is recovered in such a way as well as supplementary eigenvalues associated to deformed LMG models.

•Jan 1, 2001•1 citation
Abstract: New polynomials associated with a special Lipkin-Meshkov-Glick (LMG) model corresponding to the standard two-site Bose-Hubbard model are derived based on the Stieltjes correspondence. It is shown that there is a one-to-one correspondence between zeros of this new polynomial and solutions of the Bethe ansatz equations for the LMG model. A one-dimensional classical electrostatic analogue corresponding to the special LMG model is established according to Stieltjes early work. It shows that any possible configuration of equilibrium positions of the charges in the electrostatic problem corresponds uniquely to one set of roots of the Bethe ansatz equations for the LMG model, and the number of possible configurations of equilibrium positions of the charges equals exactly to the number of energy levels in the LMG model. Some relations of sums of powers and inverse powers of zeros of the new polynomials related to the eigenenergies of the LMG model are derived.
Originally introduced in nuclear physics as a numerical laboratory to test different many-body approximation methods, the Lipkin-Meshkov-Glick (LMG) model has received much attention as a simple enough but non-trivial model with many interesting features for areas of physics beyond the nuclear one. In this contribution we look at the LMG model as a particular example of an SU (1,1) Richardson-Gaudin model. The characteristics of the model are analyzed in terms of the behavior of the spectral-parameters or pairons which determine both eigenvalues and eigenfunctions of the model Hamiltonian. The problem of finding these pairons is mathematically equivalent to obtain the equilibrium positions of a set of electric charges moving in a two dimensional space. The electrostatic problems for the different regions of the model parameter space are discussed and linked to the different energy density of states already identified in the LMG spectrum.
Based on the recursion relation formulas and the Jacobi matrix from the theory of orthogonal polynomials, a numeric solution for the Lipkin–Meshkov–Glick (LMG) model Hamiltonian is given, without considering any restriction on the parameters specifying the strengths of the interactions included in the LMG model. Moreover, via this method, we have studied the shape phase transitions for one example of nuclei. At the end we compare the results thus obtained with those of Bethe ansatz and Hartree–Fock methods.
We show that the SU(N) Fermi-Hubbard model (FHM) on two sites, where N is the number of flavors of each fermion, corresponds to an exactly solvable two-level many-boson model that Richardson [J. Math. Phys. 9, 1327 (1968)] analytically solved long ago. We express the Bethe ansatz solutions as a function of the physical parameters of the SU(N) FHM, and recast its eigenvalues and eigenstates in terms of the Richardson pair energies and creation operators. In this context, the connection with the well-studied Lipkin-Meshkov-Glick (LMG) model, known as equivalent to the Richardson model, is established and serves as a guideline to the prediction of some N-body physics phenomena in the two-site SU(N) FHM with N particles. In particular, the LMG second-order quantum phase transition (QPT) is shown to occur in the SU(N) FHM for an attractive density-density interaction U equal to U_c=-1/(2N), in units of the (absolute value of the) tunneling amplitude between the two sites. We show the finite-size energies, the gap, and the kinetic energy, which all reveal the transition, as a function of U for values of N from N = 3 to N = 36, suggesting that the QPT could be experimentally achieved with current technologies involving SU(N) ultracold atoms or molecules. Finally, we show the entanglement entropy of the first site with respect to the second, and it scales like N at the transition, in contrast with several two-mode models.
Abstract The Lipkin–Meshkov–Glick (LMG) model was devised to test the validity of different approximate formalisms to treat many-particle systems. The model was constructed to be exactly solvable and yet non-trivial, in order to capture some of the main features of real physical systems. In the present contribution, we explicitly review the fact that different many-body approximations commonly used in different fields in physics clearly fail to describe the exact LMG solution. With similar assumptions as those adopted for the LMG model, we propose a new Hamiltonian based on a general two-body interaction. The new model (extended LMG) is not only more general than the original LMG model and, therefore, has a potentially larger spectrum of applicability, but also the physics behind its exact solution can be much better captured by common many-body approximations. At the basis of this improvement lies a new term in the Hamiltonian that depends on the number of constituents and polarizes the system; the associated symmetry breaking is discussed, together with some implications for the study of more realistic systems.