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Physical Review C

Extension of the second random-phase approximation

D. Gambacurta, M. Grasso, F. Catara, Michelangelo Sambataro

Published Feb 28, 2006• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.73.024319

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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.
Collective excitations are one of the most common and interesting features of many-body systems. Of particular interest are those collective modes which can be interpreted in terms of vibrations. Nuclei show a large variety of such modes, both low-lying and high-lying. In particular the giant dipole resonance is due to the coherent motion of protons against neutrons. Its analogue in metal clusters is the dipole plasmon excitation, due to the oscillation of electrons against the positive ions. The Random Phase Approximation (RPA) is extensively used as a microscopic theory to study the basic properties of these collective excitations. In the derivation of RPA use is made of the Quasi Boson Approximation (QBA). It consists in replacing the expectation values in the correlated ground state with those in the uncorrelated (Hartree-Fock) one. Strictly related to QBA is the feature of RPA of predicting a harmonic spectrum. On the other hand, the existence of anharmonicities in the multiphonon spectra of nuclei and their relevance in various physical processes are well established. Overcoming QBA has represented the starting point of many attempts aiming at improving RPA. A recently developed approach in this line will be presented and discussed. A very nice property of RPA is to satisfy Energy Weighted Sum Rules (EWSR). However, some violations are present in several extensions of RPA. We will show how to cure this shortcoming.
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.
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.
A limitation common to all extensions of random-phase approximation including only particle-hole configurations is that they violate to some extent the energy weighted sum rules. Considering one such extension, the improved RPA (IRPA), already used to study the electronic properties of metallic clusters, we show how it can be generalized in order to eliminate this drawback. This is achieved by enlarging the configuration space, including also elementary excitations corresponding to the annihilation of a particle (hole) and the creation of another particle (hole) on the correlated ground state. The approach is tested within a solvable 3-level model.
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.
The exactly solvable Luttinger model for a system of one-dimensional, massless, two-component fermions interacting via general two-body forces is used as a test case for a systematically generated sequence of decoupling approximations commonly referred to as random-phase approximations (RPA) of which the simple RPA is the lowest. It is found, irrespective of both the nature and strength of the interaction, that the exact (collective) spectrum, as well as the exact correlation energy, can already be obtained in the simple RPA, and furthermore, that the next higher RPA gives a vanishing correction to these quantities. It is also shown, quite in contradistinction to the exact result, that the free-gas momentum distribution n‾k{\overline{n}}_{k}nk​ persists in the second RPA as well as in the first, thereby eliminating the higher RPA as a possible approach to the calculation of n‾k{\overline{n}}_{k}nk​. Thus, in the context of this particular model, although the first few orders of RPA are highly effective for calculating the excitation spectrum and correlation energy, they are sensibly powerless in their ability to predict any variation in structure beyond that of the free-gas distribution, a result which most probably holds as well in the realistic three-dimensional case.