The linear response theory (LRT) obtained from the linearized time-dependent mean field method is an important tool for calculating properties of excited states of many-fermion systems, such as nuclear giant resonances. In its charge-changing version, it can also give access to the beta-decay strengths. This method is especially important in heavy nuclei, where the shell-model or configuration-interaction approaches are intractable. An advantage is also that LRT does not require the knowledge of an interaction and can therefore be used both within density functional theory (DFT) and phenomenological energy density functional (EDF) approaches, giving rise to a set of equations of RPA type. Below, for simplicity we refer to this method and associated equations simply as RPA method/equations. Strength functions obtained in this way probe new aspects of the EDFs and thus have a potential of constraining parameters in phenomenological nuclear EDFs.
The purpose of the present study is to present an implementation of an efficient RPA algorithm that is based on the local nuclear EDF. For electronic systems, similar methods have been used since many years, see, e.g., the recent Ref. [1] for a review, and they also constitute parts of standardized computer packages such as GAMESS [2,3]. There are two essential elements of these methods, which are at the heart of their efficiency and scalability, namely, (i) the RPA equations are solved iteratively and (ii) the RPA matrix does not have to be explicitly calculated. The second of these elements is particularly important; it is based on the observation that the action of the RPA matrix on the vector of RPA amplitudes can proceed through the calculation of the mean fields corresponding to these amplitudes.
In nuclear physics context, probably the first study that used the concept of mean fields in the RPA method was that by P.-G. Reinhard [4]. Iterative solutions of the RPA equations were introduced by Johnson, Bertsch, and Hazelton [5], and applied to the case of separable interactions, but in fact these methods can also be applied in more complicated situations, as we show here. Strangely enough, these very efficient methods have not yet been used in practical applications. Only very recently, Nakatsukasa et al. [6,7] have implemented the analogous approach within the so-called finite amplitude method (FAM).
Our present implementation pertains to the spherical symmetry with neglected pairing correlations - thus it constitutes only a proof-of-principle study. The real challenge is in solving the quasiparticle RPA (QRPA) problem in deformed nuclei. Although at present, a few implementations that are based on solving the standard QRPA equations already exist [8,9] or begin to emerge [10,11], such a route is bound to be blocked by the shear dimensionality of the problem. On the other hand, as we show here, methods based on the iterative solutions using mean fields have much better scalability properties and are potentially very promising.
The paper is organized as follows. In Secs. 2 and 3 we lay down the essential features of the method by presenting the use of mean fields and iterative solution of the RPA method, respectively. Then, in Sec. 4 we present the method to remove the spurious RPA states, which is tailored to be used within the iterative approach. Sections 5 and 6 present the convergence and scalability properties of our method, respectively, and summary and conclusions are given in Sec. 7.