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arXiv:nucl-th/0208025 · Physical Review C

Scalar ground-state observables in the random phase approximation

Calvin W. Johnson, Ionel Stetcu

Published Dec 4, 2002• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevc.66.064304

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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\ensuremathω0\ensuremath{\Elzxh}\ensuremath{\omega}0\ensuremath\Elzxh\ensuremathω 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.
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.
We reconsider the long-standing problem that the ground-state correlations predicted by the standard quasiboson (random phase) approximation are too large by a factor of two in the limit of weak residual interaction, and are thus inconsistent with the Pauli exclusion principle. We solve this problem by noting that for the derivation of the equations of the random phase approximation (RPA), which determine a set of matrix elements and excitation energies, it is unnecessary to assume that the ground state is the vacuum of a set of boson excitations, whereas in the faculty calculation of the ground-state correlation energy, this assumption is made. For the study of excitation energies and the restoration of symmetries broken by an initial mean-field solution, we apply a symmetry-preserving form of the equations of motion for fermion particle-hole excitation operators rather than for bosons. We illustrate this for the case of translations. We then describe a separate calculation of the ground-state energy that is consistent with the Pauli principle, but can nevertheless be evaluated in terms of the solutions of the RPA equations of motion. In contrast to some earlier work, we treat direct and exchange terms on an equal footing throughout.
We present a method which allows us to treat correlations in finite Fermi systems in a more consistent way than the random-phase approximation (RPA). The quasiboson approximation (QBA), where expectation values in the ground state of the system are approximated by their values in the uncorrelated reference state, is avoided in our approach where the correlated ground state is always used. We derive a closed, nonlinear set of equations which determines the energies and wave functions of the excited states as well as the single-particle occupation numbers in the ground state. As an example we apply to metallic clusters a simplified version of the approach which represents, however, a significant improvement over previous attempts to go towards a self-consistent RPA. We show that our method allows one to correct for the inadequacy of standard RPA in cases where the use of QBA becomes questionable.
The second random-phase approximation (SRPA) is the simplest and most natural extension of the RPA. It enlarges the space of the elementary modes introduced to describe the collective states by adding 2 particle -2 hole excitations to the 1 particle -1 hole ones of the RPA. In deriving the SRPA equations, use is made, as in the RPA, of the so-called quasi-boson approximation (QBA) where expectation values in the ground state of the system are approximated by their values in the uncorrelated reference state. This, however, has been shown to imply a degree of approximation worse than that in the RPA. It is, therefore, necessary to improve the QBA by considering a reference state which contains some correlations. Having in mind to perform such calculations for realistic systems, we consider a simple extension of the SRPA in which the reference state contains 2 particle - 2 hole correlations. The quality of such an extension is tested by applying it to a solvable three-level model and found to be good.
We present a new extension of the random-phase approximation method: the quasiboson approximation is avoided and correlations are included in the ground state without resorting to renormalized operators or renormalized matrix elements; the configuration space is enlarged by considering also elementary excitations corresponding to the annihilation of a particle (hole) and the creation of another particle (hole) on the correlated ground state, together with the particle-hole ones. Two new and relevant advantages of this method with respect to the existing extensions of random-phase approximation are highlighted: (i) the energy weighted sum rules are completely satisfied; (ii) the problem of the existence of nonphysical states in the spectrum, related to the inclusion of particle-particle and hole-hole configurations, is solved: a way to unambiguously disentangle physical from nonphysical states in the excitation spectrum is presented. The method is applied here to a three-level Lipkin model where its quality can be judged by comparing with the exact results. Both advantages (i) and (ii) shall lead to feasible future applications of this extended RPA to several realistic cases.
The ground-state energy and static and dynamic correlation functions are investigated in the inhomogeneous-Hartree-Fock (IHF) plus random-phase-approximation (RPA) approach applied to a one-dimensional spinless-fermion model showing self-trapped doping states at the mean-field level. Results are compared with those obtained using homogeneous HF and exact diagonalization. RPA fluctuations added to the generally inhomogeneous-HF ground state allow the computation of dynamical correlation functions that compare well with exact diagonalization results. The RPA correction to the ground-state energy agrees well with the exact results at strong- and weak-coupling limits. We also compare it with a related quasiboson approach. The instability towards self-trapped behavior is signaled by a RPA mode with frequency approaching zero.