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Physical Review B

Particle-hole excitations within a self-consistent random-phase approximation

D. Gambacurta, F. Catara

Published May 23, 2008• 14 citations
Publisher

Abstract

We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the $X$ and $Y$ amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.

DOI
10.1103/physrevb.77.205434

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We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the XXX and YYY amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.
We develop a procedure recently proposed for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The one-body density matrix in the correlated ground state is now no more assumed to be diagonal in the Hartree-Fock basis. An expression is rather provided where this quantity is expressed in terms of the X and Y amplitudes of the particle-hole phonon operators. A set of RPA-like equations depending only on the one-body density matrix is then constructed by means of the method of linearized equations of motion and solved via an iterative procedure. The procedure is employed to calculate spectra and strength distributions of several neutral and charged Na and Li metal clusters within the jellium approximation. Improvements are observed with respect to standard RPA calculations.
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 self-consistent RPA method is presented for N-particle Fermi systems with two- and three-body forces. The RPA equations for the generalised 'particle-hole' and 'particle-particle' problem, respectively, are derived by minimising the mean excitation energies defined analogously to mean single-particle energies. This corresponds to the equation-of-motion method. For the generalised 'one-particle/two-particle-one-hole problem' the Green-function scheme is used in order to establish a connection between two possible ways of treating the problem. It is shown that the RPA Hamiltonians have almost the same structure as in the case of pure two-body forces, and that Rowe's general RPA equations are contained as a special case of the treatment.
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 present an approximation scheme for the calculation of the principal excitation energies and transition moments of finite many-body systems. The scheme is derived from a first-order approximation to the self-energy of a recently proposed extended particle-hole Green's function. A Hermitian eigenvalue problem is encountered of the same size as the well-known random phase approximation (RPA). We find that it yields a size consistent description of the excitation properties and removes an inconsistent treatment of the ground-state correlation by the RPA. By presenting a Hermitian eigenvalue problem the new scheme avoids the instabilities of the RPA and should be well suited for large-scale numerical calculations. These and additional properties of the new approximation scheme are illuminated by a very simple exactly solvable model.