Program description

Nuclear Density Functional Theory

Nuclear Density Functional Theory Density functional theory (DFT) is built on theorems showing the existence of universal energy functionals for many-body systems, which include, in principle, all many-body correlations.

In nuclear physics, self-consistent methods based on the DFT, e.g., the Hartree-Fock-Bogoliubov theory with Skyrme parameterizations, have achieved a level of sophistication which permits analyses of experimental data for a wide range of properties and for arbitrarily heavy nuclei. However, the achieved accuracy and predictive power still leaves much to be desired. The quest for a truly universal DFT of nuclei, including dynamical effects and symmetry restoration, is one of the main themes of theoretical nuclear structure worldwide:
"For medium-mass and heavy nuclei, a critical challenge is the quest for the universal energy density functional, which will be able to describe properties of finite nuclei as well as extended asymmetric nucleonic matter."   (NSAC Nuclear Theory Report).

Condensed matter physicists and computational chemists have developed such functionals for the Coulomb interaction that describe properties of a wide range of systems with chemical accuracy. We believe that a concerted effort rooted in a fundamental understanding of internucleon interactions offers promise to achieve corresponding qualitative improvements in the accuracy and applicability for nuclear physics. Current nuclear functionals lack a sufficient understanding of density and isospin dependences and an adequate treatment of many-body correlations, which is required for robust and controlled extrapolations to low densities, large asymmetries, and higher temperatures. New challenges not commonly faced by Coulomb DFT are the essential roles of symmetry breaking and pairing, and the need for symmetry restoration in finite, self-bound systems. The functional should have a solid foundation based on microscopic inter-nucleon interactions with an ultimate goal of quantitative matching to microscopic theory (as in Coulomb DFT). Addressing these challenges will require us to exploit advances in the study of microscopic inter-nucleon interactions, in the development of many-body computational techniques, and in raw computer power, as well as to further develop DFT itself as applied to finite, self-bound systems.

Some specific questions that will be addressed during this program are:

1) What is the form of the nuclear energy density functional?

In principle the energy functional could be highly nonlocal in the density. The functionals in use up to now are rather simple, often completely local in the dependence on interaction effects. Revisiting the justification and limitations of these simple functionals may suggest new forms to treat interaction and correlation effects better. The correlation energies associated with symmetry breaking are particular difficult to incorporate into a functional. In this quest we may find that theoretical tools such as the Green's function formalism and the effective active formalism may be just as useful in nuclear physics as they have been in other branches of many-body theory.

2) What are the constraints on the nuclear energy density functional?

Aside for the intrinsic limitations of the functionals, they also are limited by the insufficient constraints from data employed to determine the parameters.

The density and gradient dependences of the isovector terms are poorly known, both for the ordinary densities and the pairing fields. To make a progress, a consorted effort will be required to study new functionals when applied to finite nuclei and infinite or semi-infinite nuclear matter. Another goal is to understand connections between the symmetry energy and isoscalar and isovector mean fields, and in particular the influence of effective mass and pair correlations on symmetry energy versus the isospin. Such understanding will allow us to better determine isospin corrections to nuclear mean fields and energy density functionals.

A promising approach to a systematic density expansion is given by Effective Field theory (EFT), now widely applied for few-nucleon systems. It offers as well a way to estimate the errors, which may be coupled to the optimization of parameters from experiment through efficient global fits, with systematic error and covariance analysis. Other ways to parameterize the nuclear interaction in the low energy domain such as RG method may be useful as well.

Time-odd fields certainly play a role in excited state spectroscopy and odd nuclei but they have been ignored for the most part in constructing functionals. Starting from nucleon interactions, we need to see how strong the effects are and construct the corresponding functionals. These terms are expected to play a significant role at very high spin when the nucleus is strongly polarized, but they should also influence properties of beta decay and ground states of odd-mass and odd-odd nuclei.

3) What is the form of the pairing functional?

 Pairing phenomonology tells us little about the detailed properties of the pairing interaction. Up to now, the microscopic theory of the pairing interaction has only seldom been applied in realistic calculations for finite nuclei. A "first-principle" derivation of pairing interaction from the bare NN force still encounters many problems such as, e.g., treatment of core polarization. Hence, phenomenological density-dependent pairing interactions are usually introduced. It is not obvious, how should the density dependence be parametrized although nuclear matter calculations and some experimental data (e.g., isotope shifts and odd-even mass staggering) strongly suggest that pairing is strongly affected by nuclear surface. This is why neutron-rich nuclei play such an important role in this discussion. Indeed, because of strong surface effects, the properties of these nuclei are sensitive to the density dependence of pairing. The investigation of the density and isospin dependence of pairing interactions is a significant part of this program.

A better understanding of the symmetry energy appears to be a key element in resolving the question of the proton-neutron (p-n) pairing. The isoscalar p-n pairing is our current best explanation for the additional binding of N=Z nuclei, the so-called Wigner energy. However, basic questions regarding the collectivity of such a phase still remain unanswered, and will be part of the scientific agenda of the program.

4) How to account for quantum correlations and symmetry breaking effects?

Spontaneous symmetry breaking effects are at the heart of the mean-field description of highly correlated many-body systems. A large part of those correlations can indeed be included by considering symmetry-breaking product states. Within the mean-field approach, one can understand many physical observables by directly employing broken-symmetry states, however, for finite systems, quantitative description often does require symmetry restoration. For this purpose, one can apply a variety of theoretical techniques, in particular projection methods and the generator coordinate method (see also Section on "Collective dynamics in exotic nuclei").

In practical applications, the mean-field approximation requires implementation of dynamical corrections, which account for correlations going beyond the simple product state. The most important are translational, rotational, vibrational, and particle-number corractions, but the fluctuations due to internally broken parity and isospin can also be significant in some nuclei.

Ideally, one would like to work out approximations that would allow avoiding full-scale collective calculations, but would be based on calculations performed on the top of self-consistent mean fields. In this way, we hope to develop the microscopic mass formula in which bo the the mean-field mass and the dynamical corrections would be obtained from the same energy density functional. In this context, it is important to note that the realistic energy density functional does not have to be related to any given effective force. This creates a problem if a symmetry is spontaneously broken. While the projection can be carried out in a straightforward manner for energy functionals that are related to a two-body potential, the restoration of spontaneously broken symmetries of a general density functional poses a conceptional dilemma which has not been properly addressed.

5) Computational techniques and error analysis

Reliable extrapolation is possible only with the establishment of theoretical error bars. Consequently, construction of new energy density functionals should supplemented by a complete error and covariance analysis. We believe that it is not sufficient to "predict" properties of exotic nuclei by extrapolating properties of those measured in experiment. We must also quantitatively determine errors related to such an extrapolation. Moreover, for an experimental work it is essential that an improvement gained by measuring one or two more isotopes be quantitatively known. From theoretical perspective, we must also know the confidence level with which the parameters of the functional are determined.