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
"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