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Physical Review C

Extended random phase approximation in a solvable model

Daisuke Shindo, Kazuo Takayanagi

Published Jul 29, 2003• 7 citations
Publisher

Abstract

We propose an extended Lipkin model that can describe decay processes of particle-hole states and is still solvable. We examine several RPA-type theories using the model. We show explicitly the roles played by the self-energies of particle-hole propagators, and clarify how the extended RPA theory describes collective states better than other RPA-type theories.

DOI
10.1103/physrevc.68.014312

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We propose an extended Lipkin model that can describe decay processes of particle-hole states and is still solvable. We examine several RPA-type theories using the model. We show explicitly the roles played by the self-energies of particle-hole propagators, and clarify how the extended RPA theory describes collective states better than other RPA-type theories.
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.
An extended Lipkin-Meshkov-Glick model for testing the proton-neutron random-phase approximation (pnRPA)(pn\mathrm{RPA})(pnRPA) method is developed, taking into account explicitly proton and neutron degrees of freedom. Besides the proton and neutron single-particle terms two types of residual proton-neutron interactions, one simulating a particle-particle and the other a particle-hole interaction, are included in the model Hamiltonian so that the model is exactly solvable in an isospin $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ basis. The behavior of the first excited (collective) state obtained by (i) exact diagonalization of the Hamiltonian matrix and (ii) with the pnRPA\mathrm{pn}\mathrm{RPA}pnRPA is studied as a function of the model parameters and the two results are compared with each other. Furthermore, charge-changing operators simulating nuclear beta decay and their action on eigenfunctions of the model Hamiltonian are defined and transition amplitudes of them are calculated using exact, the Tamm-Dancoff, and pnRPA\mathrm{pn}\mathrm{RPA}pnRPA eigenfunctions.
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 role of the small-norm amplitudes in extended random-phase approximation (RPA) theories such as the particle-particle and hole-hole components of one-body amplitudes and the twobody amplitudes other than two-particle/two-hole components are investigated for the onedimensional Hubbard model using an extended RPA derived from the time-dependent density matrix theory.It is found that these amplitudes cannot be neglected in strongly interacting regions where the effects of ground-state correlations are significant.............
It is shown that the random-phase approximation (RPA) method with its nonlinear higher generalization, which was previously considered as approximation except for a very limited case, reproduces the exact solutions of the Lipkin model. The nonlinear higher RPA is based on an equation nonlinear on eigenvectors and includes many particle-many hole components in the creation operator of the excited states. We demonstrate the exact character of solutions analytically for the particle number [Formula: see text] and, numerically, for [Formula: see text]. This finding indicates that the nonlinear higher RPA is equivalent to the exact Schrödinger equation, which opens up new possibilities for realistic calculations in many-body problems.
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.