Review
Linear response theory with finite-range interactions

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Abstract

This review focuses on the calculation of infinite nuclear matter response functions using phenomenological finite-range interactions, equipped or not with tensor terms. These include Gogny and Nakada families, which are commonly used in the literature. Because of the finite-range, the main technical difficulty stems from the exchange terms of the particle–hole interaction. We first present results based on the so-called Landau and Landau-like approximations of the particle–hole interaction. Then, we review two methods which in principle provide numerically exact response functions. The first one is based on a multipolar expansion of both the particle–hole interaction and the particle–hole propagator and the second one consists in a continued fraction expansion of the response function. The numerical precision can be pushed to any degree of accuracy, but it is actually shown that two or three terms suffice to get converged results. Finally, we apply the formalism to the determination of possible finite-size instabilities induced by a finite-range interaction.

Introduction

The study of quantum systems having a large or even infinite number of fermions is of interest in many physical situations, involving fields as diverse as quantum chemistry, atomic physics, condensed matter, nuclear physics or astrophysics. For instance, liquid 3He or conduction electrons in metals are familiar examples of such systems in condensed matter [1], [2], [3], [4], [5], [6]. In nuclear physics, the concept of infinite nuclear matter as a homogeneous medium made of interacting nucleons is broadly used, because of its relative simplicity and related to the suppression of the boundaries. This idealised system is nevertheless connected to the physics of the inner part of atomic nuclei and also of some regions of compact stars [7], [8], [9], [10]. For these reasons, it is a very useful testing ground for various theories.

A usual way to get information about a physical system is by means of its response to an external probe [1], [2]. Some well-known examples among many of physical phenomena that require the knowledge of the response functions are the transition strengths, the inelastic cross-sections, the electron scattering by nuclei [11], the propagation of neutrinos in nuclear matter [12] and finite nuclei [13] or the study of vibrational modes in finite nuclei [14], [15], [16] and neutron stars (NS) [17], [18], [19]. When the interaction with the probe is small enough, the response of the system can be calculated in the linear approximation, also known as Random Phase Approximation (RPA) for historical reasons. Its basic ingredients are the particle–hole (ph) interaction and the ph Green’s function, whose poles represent the energies of excited states [5]. It is worth mentioning at this stage that, due to surface effects and long-range correlations by instance, response functions can be very different in infinite matter and finite nuclei. However, the objective of this review is not to examine these differences but to concentrate on the description of the different approximations used to calculate response functions in an infinite medium and to use them for several specific purposes as detailed below.

Two main approaches are introduced to get response functions in a non-relativistic frame: either by starting from a bare two-body nucleon–nucleon (NN) interaction, with the possible addition of three-body forces, or by using a phenomenological NN interaction. In the first approach, the many-body problem is treated as exactly as possible, using a variety of methods such as Monte-Carlo simulations [20], Coupled-Cluster [21], similarity Renormalisation Group [22], self-consistent Green’s function [23] or the Brueckner–Hartree–Fock method [24], [25], [26]. In the second approach, a mean-field approximation [27], [28] is used with an interaction whose parameters are adjusted so as to describe selected observables of finite nuclei and some bulk properties of nuclear matter as well. This second approach is the subject of the present review.

Several important informations about nuclear matter can be obtained by analysing the resulting response function. For instance, its singularities at zero energy can be related to a phase transition such as the spinodal one [29] or to instabilities such as the pion condensation [30], [31] or the ferromagnetic transition in neutron stars [32]. Moreover, if these instabilities appear at density values around the saturation density, ρ0, they can affect the description of finite nuclei properties. This connection between singularities in the infinite medium and instabilities in atomic nuclei has been investigated in Refs. [33], [34], [35], [36]. It has been shown that the appearance of such singularities at densities lower than 1.3ρ0 (with ρ0 the saturation density) may affect the convergence of the calculations in finite nuclei [37], [38]. To avoid such a problem, the response functions of nuclear matter are currently included into the fitting protocol used to adjust new interactions [33], [39], [40], [41].

Historically, phenomenological nuclear interactions have been limited to central, spin–orbit and density-dependent terms. However, in recent years, the importance of the tensor term has been stressed. Indeed, the bare nucleon–nucleon interaction contains an important tensor part, necessary to reproduce not only the phase shifts of the nucleon–nucleon scattering, but also the quadrupole moment of the deuteron. Several groups have worked on the introduction of a tensor term in effective interactions and its impact on the ground state properties of finite nuclei has been discussed for example in Refs. [42], [43], [44], [45].

Methods to get infinite nuclear matter response functions using the zero-range Skyrme interaction [46] have been reviewed in Ref. [47]. The case of finite-range interactions is the subject of the present review, concentrating on the Gogny [48], [49], [50] and the Nakada [51], [52] interactions. Although these are not the only finite-range interactions available in the literature [53], [54], they are currently the most commonly used to study finite nuclei at the mean field level [27], [50], [52].

The most difficult technical aspect in calculating response functions is related to the exchange term of the ph-interaction. This difficulty is overcome if a zero-range interaction is used, and it is thus convenient to briefly summarise some results obtained with the well-known family of zero-range Skyrme interactions. We recall that the standard form of the Skyrme interaction, fixed by Vautherin and Brink [55], contains central, spin–orbit and density-dependent terms only, with a quadratic momentum dependence which simulates finite-range effects. It has been shown [56] that both its zero-range and its simple momentum dependence allow one to get exact and relatively simple algebraic expressions for the response functions and other related quantities. These results were originally presented for symmetric nuclear matter (SNM), and afterwards extended to asymmetric nuclear matter and pure neutron matter (PNM) including the special case of charge-exchange operators [57], [58], [59], [60], [61]. They have been applied to a variety of problems, including neutrino transport in neutron stars [62], [63]. The original form of the Skyrme interaction [46] also contains other terms, namely zero-range tensor terms, which are relevant for the spin and spin–isospin channels. However, apart from some exploratory studies [64], these terms have been omitted in most calculations of finite nuclei in recent years, they have been included either perturbatively to existing central ones [65], [66], [67], [68], or with a complete refit of the parameters [42], [43], [69], [70], [71]. Again, due to their zero-range and their particular momentum dependence, it is possible to get the exact response functions, although the tensor terms make algebraic expressions rather cumbersome. It has been shown [47] that the tensor interaction has a very strong impact on the response functions for both spin channels due to spin–orbit coupling.

Skyrme interactions are reasonably well controlled around the saturation density ρ0 of SNM, for moderate isospin asymmetries and zero temperature. However, it has a certain number of drawbacks. For instance, most Skyrme parametrisations predict that the isospin asymmetry energy becomes negative when the density is increased [72], [73], [74], [75], [76], [77]. Consequently, the symmetric system would be unstable at some density beyond the saturation one, preferring a largely asymmetric system made by an excess of either protons or neutrons. Another type of instability refers to the magnetic properties of neutron matter. Most Skyrme interactions predict that even in the absence of a magnetic field a spontaneous magnetisation arises in pure neutron matter at some critical density [32], [78], [79]. Actually, it has been shown [80] that any reasonable Skyrme parametrisation predicts instabilities of nuclear matter beyond some critical density. Another inconvenient of zero-range interactions refers to pairing properties. Indeed, it leads to ultraviolet divergency [81] that needs to be treated with either an explicit regularisation [82] or via a cutoff of the available phase space [83]. Moreover, the UNEDF-SciDAC scientific collaboration [84] has investigated the role of the optimisation procedure on the quality and predictive power of the standard Skyrme functional [85], concluding that there is no more room to improve spectroscopic qualities by simply acting on the optimisation procedure [86]. To overcome this inconvenient, higher order momenta terms [87] have been considered, namely N2LO and N3LO Skyrme generalisations, which simulate finite-range effects [88], [89], [90]. Even if the response function formalism is more complex in this case, it is still possible to obtain analytical results and to use them to avoid finite-size instabilities during the adjustment of the parameters of the interaction [41], [91]. However, the use of a limited number of momentum powers to simulate finite-range effects may restrict the applicability of Skyrme-like interactions to describe the response functions at high values of the momentum transfer. In principle, all above mentioned difficulties should be removed by using a finite-range interaction.

The simplest way to deal with the exchange term of the ph interaction is to consider the limit of zero transferred momentum by placing the particle and hole momenta over the Fermi sphere. This is named Landau approximation of the ph-interaction. In this case, the form of the ph interaction becomes universal and it is possible to find analytical expressions for the response function [92]. Since q=0 could be a too strong approximation, some authors have tried to keep an explicit momentum dependence only over the direct term and perform a Landau approximation for the exchange term [93]. Either meson-exchange [31], [94] or effective [93] interactions have been used in this approach. A general method to obtain response functions consists in using a partial wave expansion of both the ph interaction and the RPA ph propagator [93]. This method leads to a system of coupled integral equations that need to be solved numerically. Its validity depends on the convergence of the expansion. For all the considered interactions, a few multipoles suffice to get the converged response. As an alternative to the multipolar expansion, some authors have investigated a method based on the continuous fraction (CF) approximation to the RPA response function [95], [96], [97], [98], [99], [100]. In both cases, the exchange term is treated exactly, and the calculations can be carried out up to any degree of accuracy. Actually, two or three terms suffice to get converged results. The interest of the CF method is that the formalism is the same for both infinite matter and finite nuclei. The CF convergence can be assessed by comparing with the multipolar expansion for nuclear matter, and the results could be hopefully translated to finite nuclei.

The plan of this review is the following. In Section 2 we present the general formalism to calculate the response function in an infinite nuclear system. The relevant quantities are defined, as the ph propagator, the Bethe–Salpeter equation, and the response functions. In Section 3 we present the phenomenological finite-range interactions considered in this review and the connection between finite- and zero-range interactions. The simplest approximation to deal with the exchange term, using either a Landau or a Landau-like ph interactions is discussed in Section 4. Then we review the multipolar expansion method in Section 5, and the continued fraction method in Section 6, to get the response function as two different expansions. In Section 7 we review the use of the response function to detect finite-size instabilities of phenomenological interactions. Finally, we give some concluding remarks in Section 8. Some useful technical details and formulae are given in appendices.

Section snippets

Linear response formalism

In this section, we present a general description of the formalism used throughout this article to determine nuclear response functions and some related quantities. This formalism is based on the Green’s function and it has been discussed in great detail in several books and articles [1], [5], [101]. We summarise it here, mainly to fix the notations and signal some interesting points. Since we shall only consider homogeneous systems, the proper definitions of the strength function and of any

Phenomenological finite-range interactions

There are two popular types of finite-range interactions being currently used to describe nuclear structure, namely Gogny [48] and Nakada [51] interactions. Both types can be cast in the following general form V(r1,r2)=VC(r1,r2)+VDD(r1,r2)+VSO(r1,r2)+VT(r1,r2),which is a sum of central, density-dependent, spin–orbit and tensor terms. A formal difference between these interactions lies in the finite-range form factor, consisting of a superposition of either Gaussians or Yukawians, respectively.

Landau approximations to the ph interaction

The Landau’s theory [3], [101], [149], [150], [151], [152], [153] encompasses the basic properties of Fermi liquids. In this approach, the excitations of a strongly interacting normal Fermi system are described in terms of weakly interacting quasiparticles – or ph excitations – which are long-lived only near the Fermi surface. The quasiparticle interaction is solely characterised by a set of Landau parameters. In liquid helium, some of these parameters can be obtained directly from experiment,

A multipolar expansion

We present now a general method to solve directly the Bethe–Salpeter equation in the three-dimensional momentum space, proposed in Ref. [93]. It consists in expanding the Green’s functions and the ph interaction on a complete basis of spherical harmonics, thus transforming Eq. (12) into a set of coupled integral equations on the momentum modulus variable. The interest is that this is an exact method, with no approximations, provided the expansion is pushed up to the required degree of

Continued fraction approximation

All the previous approximations start from the GHF ph propagator of infinite matter, so that many technical details are not directly transposable to finite nuclei. We discuss now another method to properly deal with the exchange term, whose basic principles can be used in both infinite and finite systems. The idea is to express the RPA response function as a continued fraction (CF), where the full ph interaction appears in the form of multiple momentum averages weighted with GHF. Formally, the

Finite size instabilities

To conclude this review we consider two important applications related to possible finite-size instabilities of phenomenological interactions, whose presence can be detected through the response functions. The first one is connected to finite nuclei, the second one to the propagation of neutrinos in nuclear matter.

Concluding remarks

This review has focused on the methods for calculating the response function of symmetric nuclear matter based on the two most common families of phenomenological finite-range interactions, namely Gogny and Nakada. The core of the review has been the discussion of existing approximations and methods to deal with the exchange part of the ph interaction with particular attention to the inclusion of an explicit tensor term.

An analysis of some basic ground states properties of the infinite medium

Acknowledgements

The authors would like to thank Magda Ericson, Alexandros Gezerlis, Marcella Grasso, Marco Martini, Arnau Rios and Michael Urban for useful comments and discussions. This work has been supported by the FIS2017-84038-C2-1-P (MINECO, Spain) and Science and Technology Facilities Council (STFC) Grants No. ST/M006433/1 and No. ST/P003885/1. Numerical calculations were undertaken on the Viking Cluster, which is a high performance computing facility provided by the University of York.

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