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arXiv:nucl-th/9909023 · Physical Review C

Random-phase approximation approach to rotational symmetry restoration in a three-level Lipkin model

K. Hagino, G. F. Bertsch

Published Jan 14, 2000• 41 citations
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Abstract

We study an extended Lipkin-Meshkov-Glick model that permits a transition to a deformed phase with a broken continuous symmetry. Unlike simpler models, one sees a persistent zero-frequency Goldstone mode past the transition point into the deformed phase. We found that the RPA formula for the correlation energy provides a useful correction to the Hartree-Fock energy when the number of particle N satisfies $N>3,$ and becomes accurate for large N. We conclude that the RPA correlation energy formula offers a promising way to improve the Hartree-Fock energy in a systematic theory of nuclear binding energies.

DOI
10.1103/physrevc.61.024307

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We study an extended Lipkin-Meshkov-Glick model that permits a transition to a deformed phase with a broken continuous symmetry. Unlike simpler models, one sees a persistent zero-frequency Goldstone mode past the transition point into the deformed phase. We found that the RPA formula for the correlation energy provides a useful correction to the Hartree-Fock energy when the number of particle N satisfies N>3,N>3,N>3, and becomes accurate for large N. We conclude that the RPA correlation energy formula offers a promising way to improve the Hartree-Fock energy in a systematic theory of nuclear binding energies.
We show that it is possible to restore broken symmetries associated with the Goldstone mode within the self-consistent random phase approximation (SCRPA) applied to the three-level Lipkin model. To do this, it is necessary to include the so-called scattering terms in the RPA operator. We determine one- and two-body densities as very convergent expansions in terms of the generators of the RPA basis. We show that SCRPA excitations correspond to the heads of some rotational bands in the exact spectrum.
The random phase approximation (RPA) builds in correlations left out by mean-field theory. In full $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces we calculate the Hartree-Fock (HF)+RPA(\mathrm{HF})+\mathrm{RPA}(HF)+RPA binding energy, and compare it to exact diagonalization. We find that often, but not always, the HF+RPA\mathrm{HF}+\mathrm{RPA}HF+RPA gives a good approximation to the ``exact'' ground state energy. In those cases where the RPA is less satisfactory, however, there is no obvious correlation with properties of the HF state, such as deformation or overlap with the exact ground state wave function. This weakens the reliability of the RPA for computing binding energies.
Proton-neutron random phase approximation studied by the Lipkin-Meshkov-Glick model in the SU(2) ×\times× SU(2)
We study the proton-neutron RPA with an extended Lipikin-Meshkov-Glick model. We pay attention to the effect of correlated ground state and the case in which neutron and proton numbers are different. The effect of the correlated ground state are tested on the basis of quasi-boson approximation. We obtain the result that RPA excitation energies and transition strengths are in a good agreement with the exact solution up to a certain strength of the particle-particle interaction. However, the transition strength becomes worse if we consider the case in which neutron and proton numbers are different even at a weak particle-particle interaction.
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.
The Hartree-Fock approximation to the many-fermion problem can break exact symmetries, and in some cases by changing a parameter in the interaction one can drive the Hartree-Fock minimum from a symmetry-breaking state to a symmetry-conserving state (also referred to as a ``phase transition'' in the literature). The order of the transition is important when one applies the random-phase approximation (RPA) to the of the Hartree-Fock wave function: if first order, RPA is stable through the transition, but if second-order, then the RPA amplitudes become large and lead to unphysical results. The latter is known as ``collapse'' of the RPA. While the difference between first- and second-order transitions in the RPA was first pointed out by Thouless, we present for the first time nontrivial examples of both first- and second-order transitions in a uniform model, the interacting shell-model, where we can compare to exact numerical results.
We calculate the ground-state expectation value of scalar observables in the matrix formulation of the random phase approximation (RPA). Our expression, derived using the quasiboson approximation, is a straightforward generalization of the RPA correlation energy. We test the reliability of our expression by comparing against full diagonalization in $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces. In general the RPA values are an improvement over mean-field (Hartree-Fock) results, but are not always consistent with shell-model results. We also consider exact symmetries broken in the mean-field state and whether or not they are restored in RPA.