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Symmetry-unrestricted self-consistent time-dependent density matrix theory with a Skyrme force.

Matthew C. Barton

Published Feb 28, 2018• 3 citations

Abstract

My research has been focused on time-dependent aspects of nuclear physics both at the mean-field and at the beyond-mean-field level. At the mean field level, the objective of my PhD has been to understand how the introduction of the tensor part of the Skyrme interaction affects heavy ion collisions and giant magnetic resonances, in a self consistent and symmetry unrestricted manner. The introduction of the tensor force redistributes the strength of the giant magnetic resonances within the same energy range. Within the study of heavy ion collisions of 16O+16O the introduction of the tensor decreased the amount of dissipation in the system. At the beyond mean field level, the objective of my PhD was to implement a time dependent density matrix (TDDM) theory, self consistently, without symmetry restrictions using the full Skyrme force. TDDM allows an order by order truncation of the Bogoliubov-Born-��Green-��Kirkwood-��Yvon (BBGKY) hierarchy, which relates the evolution of many body densities. If two-body correlations are assumed to dominate the dynamics of the system, the resulting equations incorporate one-particle-one-hole and two-particle-two-hole correlations. A variety of different nuclei below A=40 were chosen to study the formation of correlations for different nuclear ground states. Two body correlations were found to have a noticeable effect on the ground state properties of these nuclei. For example on average 4 - 5 % of the total energy is due to correlations. When time dependent calculations were performed with these correlated nuclei, computational limitations led to problems with conservation laws.

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Takashi Nakatsukasa

We present the basic concepts and our recent developments in density functional approaches with Skyrme functionals for describing nuclear dynamics at low energy.Time-dependent density functional theory (TDDFT) is utilized for the exact linear response with an external perturbation.For a description of collective dynamics beyond the perturbative regime, we present a theory of a decoupled collective submanifold to describe a slow motion based on TDDFT.Selected applications are shown to demonstrate the quality of their performance and feasibility.The advantages and disadvantages of the numerical aspects are also discussed.. . . . . .

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My research has been focused on time-dependent aspects of nuclear physics both at the mean-field and at the beyond-mean-field level. At the mean field level, the objective of my PhD has been to understand how the introduction of the tensor part of the Skyrme interaction affects heavy ion collisions and giant magnetic resonances, in a self consistent and symmetry unrestricted manner. The introduction of the tensor force redistributes the strength of the giant magnetic resonances within the same energy range. Within the study of heavy ion collisions of 16O+16O the introduction of the tensor decreased the amount of dissipation in the system. At the beyond mean field level, the objective of my PhD was to implement a time dependent density matrix (TDDM) theory, self consistently, without symmetry restrictions using the full Skyrme force. TDDM allows an order by order truncation of the Bogoliubov-Born-��Green-��Kirkwood-��Yvon (BBGKY) hierarchy, which relates the evolution of many body densities. If two-body correlations are assumed to dominate the dynamics of the system, the resulting equations incorporate one-particle-one-hole and two-particle-two-hole correlations. A variety of different nuclei below A=40 were chosen to study the formation of correlations for different nuclear ground states. Two body correlations were found to have a noticeable effect on the ground state properties of these nuclei. For example on average 4 - 5 % of the total energy is due to correlations. When time dependent calculations were performed with these correlated nuclei, computational limitations led to problems with conservation laws.

The Bogoliubov-Born-Green-Kirkwood-Yvon or time-dependent density matrix (TDDM) hierarchy of equations for higher density matrices is truncated at the three-body level in approximating the three-body correlation function by a quadratic form of two-body ones, closing the equations in this way. The procedure is discussed in detail and it is shown in nontrivial model cases that the approximate inclusion of three-body correlation functions is very important to obtain precise results. A small amplitude approximation of this time-dependent nonlinear equation for the two-body correlation function is performed (STDDM*-b) and it is shown that the one-body sector of this generalized nonlinear second random phase approximation (RPA) equation is equivalent to the self-consistent RPA (SCRPA) approach which had been derived previously by different techniques. It is discussed in which way SCRPA also contains the three-body correlations. TDDM and SCRPA are tested versus exactly solvable model cases.

A correlated ground state of

{}^{16}\mathrm{O}

and an

E2

giant resonance built on it are calculated using an extended version of the time-dependent Hartree-Fock theory called the time-dependent density-matrix (TDDM) theory. The Skyrme III force is used as an effective interaction for the calculation of both a mean field and two-body correlations. It is found that a TDDM calculation with the Skyrme force gives reasonable ground-state correlations and a large spreading width of the

E2

giant resonance.

The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of

^{40}

Ca and

^{48}

Ca and the other the fusion reactions of

^{16}

O +

^{16}

O.

A configuration-interaction time-dependent density functional theory (CI-TDDFT) for nuclear dynamics is developed. In this framework, the correlated nuclear many-body wave function is expanded in terms of time-dependent many-particle configurations built from a common set of orthonormal single-particle states. The equations of motion for both the expansion coefficients and the single-particle states are derived self-consistently using the Dirac-Frenkel time-dependent variational principle. This formulation extends conventional time-dependent density functional theory (TDDFT) by incorporating configuration mixing and beyond-mean-field correlations, while preserving energy and particle-number conservation. As an illustrative application, the method is implemented using the relativistic point-coupling functional PC-PK1 in the particle-hole channel and a monopole pairing interaction in the particle-particle channel, and is applied to the study of isoscalar giant monopole resonance in

^{58}

Ni and

^{60}

Ni. Numerical tests show that both the total energy and particle number are conserved, with relative deviations within

4\times 10^{-4}

during the time evolution. Compared with conventional TDDFT, CI-TDDFT yields broader strength distributions for giant monopole resonances while keeping the main peak positions close to those from TDDFT. This broadening is associated with configuration mixing in the valence space and suggests a coupling of the monopole oscillation to additional collective degrees of freedom. These results demonstrate the potential of CI-TDDFT as a quantum, microscopic beyond-mean-field framework for nuclear dynamics.

Linear density response functions are calculated for symmetric nuclear matter of normal density by time-evolving two-time Green's functions in real time. Of particular interest is the effect of correlations. The system is therefore initially time-evolved with a collision term calculated in a direct Born approximation until fully correlated. An external time-dependent potential is then applied. The ensuing density fluctuations are recorded to calculate the density response. This method was previously used by Kwong and Bonitz for studying plasma oscillations in a correlated electron gas. The energy-weighted sum-rule for the response function is guaranteed by using conserving self-energy insertions as the method then generates the full vertex-functions. These can alternatively be calculated by solving a Bethe -Salpeter equation as done in works by Bozek et al. The (first order) mean field is derived from a momentum-dependent (non-local) interaction while 2 nd order self-energies are calculated using a particle-hole two-body effective (or residual) interaction given by a gaussian local potential. We show results of calculations of the response function S(ɷ, q 0 ) for q 0 = 0.2, 0.4 and 0.8 fm -1 . Comparison is made with the nucleons being un-correlated i.e. with only the first order mean field included. We discuss the relation of our work with the Landau quasi-particle theory as applied to nuclear systems by Babu and Brown and followers.

We present the basic concepts and our recent developments in density functional approaches with Skyrme functionals for describing nuclear dynamics at low energy.Time-dependent density functional theory (TDDFT) is utilized for the exact linear response with an external perturbation.For a description of collective dynamics beyond the perturbative regime, we present a theory of a decoupled collective submanifold to describe a slow motion based on TDDFT.Selected applications are shown to demonstrate the quality of their performance and feasibility.The advantages and disadvantages of the numerical aspects are also discussed.. . . . . .