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

N. Debergh, Floarea Stancu

Published Jan 1, 2001• 1 citation
OpenAlex

Abstract

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.

Related papers

On the exact solutions of the Lipkin-Meshkov-Glick model

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We present the many-particle Hamiltonian model of Lipkin, Meshkov and Glick in the context of deformed polynomial algebras and show that its exact solutions can be easily and naturally obtained within this formalism. The Hamiltonian matrix of each j multiplet can be split into two submatrices associated with two distinct irreps of the deformed algebra. Their invariant subspaces correspond to even and odd numbers of particle-hole excitations.

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Equations of motion for a spectrum-generating algebra: Lipkin–Meshkov–Glick model

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In this paper, we have used the q-deformed version of the Lipkin-Meshkov-Glick (LMG) model to summarize the utilization of quantum algebras already applied to soluble pseudospin models. Within the suq(2) context, we have checked the validity of the co herent states variational method, then used the Time Dependent Hartree-Fock (TDHF) approximation to compare with the exact evolution in time of the system, and finally observed the temperature effects undergone by the system, through the study of the Helmholtz free energy. We have also investigated the LMG model within the contexts of the supq(2) and suq(3) algebras and some of the features mentioned above have been rediscussed.

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In this paper we investigate the dynamics of the Lipkin-Meshkov-Glick model within the context of quantum algebras. For this purpose we obtain the equations of motion with the help of the time-dependent variational principle. To analyse the results, graphs for equi-energies, potentials and mean-square deviation of the J z operator have been plotted. Modifications in the usual libration and rotational motions due to the introduction of quantum deformation are discussed.

Journal of Physics A Mathematical and General•Feb 7, 1994•8 citations
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.
For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin–Meshkov–Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 106, computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point.
In this paper, we have used the q-deformed version of the Lipkin-Meshkov-Glick (LMG) model to summarize the utilization of quantum algebras already applied to soluble pseudospin models. Within the suq(2) context, we have checked the validity of the co herent states variational method, then used the Time Dependent Hartree-Fock (TDHF) approximation to compare with the exact evolution in time of the system, and finally observed the temperature effects undergone by the system, through the study of the Helmholtz free energy. We have also investigated the LMG model within the contexts of the supq(2) and suq(3) algebras and some of the features mentioned above have been rediscussed.
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.