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Physical Review A

Three-level Lipkin model in the context of the suq (3) algebra

L. Brito, Constança Providência, João da Providência, Sidney S. Avancini, F. Cruz, Débora Peres Menezes, Mello Moraes

Published Jul 1, 1995• 7 citations
Publisher

Abstract

In this work we study the quantum deformation of the three-level Lipkin model. The exact spectra are obtained. The reliability of the mean-field approximation is discussed in the framework of the model. Using adequate coherent states, the mean-field ground-state and the random-phase-approximation modes are calculated. The Dyson realization of ${\mathrm{su}}_{\mathit{q}}$(3) in the Bargmann representation is derived. The effect of the quantum deformation on the ground-state energy and on the phase transitions of the model is discussed. The simulation of temperature effects by the quantum deformation is investigated.

DOI
10.1103/physreva.52.92

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Journal of Physics A Mathematical and General•Aug 7, 1996•1 citation
In this work we study the quantum deformation of the three-level Lipkin model. The exact spectra are obtained. The reliability of the mean-field approximation is discussed in the framework of the model. Using adequate coherent states, the mean-field ground-state and the random-phase-approximation modes are calculated. The Dyson realization of suq{\mathrm{su}}_{\mathit{q}}suq​(3) in the Bargmann representation is derived. The effect of the quantum deformation on the ground-state energy and on the phase transitions of the model is discussed. The simulation of temperature effects by the quantum deformation is investigated.
The behaviour of the Moszkowski model within the context of quantum algebras is studied. The Moszkowski Hamiltonian is exactly diagonalized for various values of the deformation parameter of the involved quantum algebra. The phase transition from a vibrational regime to a rotational regime is discussed in terms of the q-deformation. A coherent state for the ground state is introduced, and the Hartree-Fock energy and the RPA frequencies are compared with the exact values. The meaning of the q-deformation in this model is discussed.
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.
Assuming q-deformed commutation relations for the fermions, an extension of the standard Lipkin Hamiltonian is presented. The usual quasi-spin representation of the standard Lipkin model is also obtained in this q-deformed framework. A variationally obtained energy functional is used to analyse the phase transition associated with the spherical symmetry breaking. The only phase transitions in this q-deformed model are of second order. As an outcome of this analysis a critical parameter is obtained which is dependent on the deformation of the algebra and on the number of particles.
We compute the energy functional of a three-level Lipkin model via a Legrendre transform and compare exact numerical results with analytical solutions obtained from the random phase approximation (RPA). Except for the region of the phase transition, the RPA solutions perform very well. We also study the case of three nondegenerate levels and again find that the RPA solution agrees well with the exact numerical result. For this case, the analytical results give us insight into the form of the energy functional in the presence of symmetry-breaking one-body potentials.