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Journal of Physics G Nuclear and Particle Physics

Finite temperatureSU(3) LMG model: mean-field versus exact calculation

Constança Providência, João da Providência, Maisa O. Terra

Published Mar 1, 1996• 1 citation
Publisher

Abstract

A variational approach is applied to the description of temperature effects in a N-particle exactly soluble schematic model, with SU(3) symmetry. The ground-state energy for different couplings and the strength distribution for a weak coupling versus temperature are compared with the exact results. For a weak coupling, or for a large enough N, the mean-field approximation agrees well with the exact values. For , new RPA modes appear which did not exist at T = 0.

DOI
10.1088/0954-3899/22/3/008

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A variational approach is applied to the description of temperature effects in a N-particle exactly soluble schematic model, with SU(3) symmetry. The ground-state energy for different couplings and the strength distribution for a weak coupling versus temperature are compared with the exact results. For a weak coupling, or for a large enough N, the mean-field approximation agrees well with the exact values. For , new RPA modes appear which did not exist at T = 0.
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.
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.
An exactly soluble three-level model for a system of fermions with Hamiltonian formed from generators of the group SU(3) is studied. A basis for the representation of which the ground state is a member is constructed. It is demonstrated that in this representation, the generators can each be expanded in a series as functions of a pair of ``kinematical'' boson operators; the series are uniquely determined to satisfy the operator algebra and the invariants of the representation by a method of Marumori. It is seen that the lowest anharmonic approximation to the Hamiltonian and other operators yields excellent numerical agreement with exact results for all regimes of interaction strength considered. An alternative description of the system in terms of a dynamically more meaningful boson, called the ``physical'' boson, is shown to be appropriate for relatively weak coupling where one has a near harmonic spectrum.
The "phase transitions" predicted within finite microscopic systems by Hartree-Fock-Bogoliubov (HFB) and aspects of the use of the associated broken-syrometry bases in the random-phase approximation (RPA) are considered using soluble models-Evidence is presented in support of the conjecture that the success of these techniques lies in the fact that they mimic singularities in the dependence on interaction strengths of the exact solution.This conjecture provides a natural explanation for why such methods fail close to a point where a phase transition occurs and indicates possible directions for improvement.Phase diagrams at both zero and finite temperature are determined, and simple analytic expressions for the way in which critical scrcn;the scale with particle number are found.It is shown that the ''phase transitions" predicted at finite temperature are relevant.A connection between the singularities referred to above and real phase transitions found in the thermodynamic limit is discussed.It is found that only stable bases can be used in an RPA calculation.This is in particular true for those RPA modes which are not associated with the onset of instability of the basis; these modes do not describe any excited state when tht. basis is unstable.Outside transitional regions certain undesirable features of HFB are unearthed, notably that the HFB ground state energy is not necessarily an upper bound to the exact ground state energy.The effectiveness in this regime of the Hartree-Fock Seniority approximation as a substitute to projection methods is ■ -■■.ated.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.