Nuclear ground states in a consistent implementation of the time-dependent density matrix approach

Background: Time-dependent techniques in nuclear theory often rely on mean-field or Hartree-Fock descriptions. Beyond-mean-field dynamical calculations within the time-dependent density matrix (TDDM) theory have often invoked symmetry restrictions and ignored the connection between the mean field and the induced interaction.

Purpose: We study the ground states obtained in a TDDM approach for nuclei from $A=12$ to $A=24$, including examples of even-even and odd-even nuclei with and without intrinsic deformation. We overcome previous limitations using three-dimensional simulations and employ density-independent Skyrme interactions self-consistently.

Methods: The correlated ground states are found starting from the Hartree-Fock solution, by adiabatically including the beyond-mean-field terms in real time.

Results: We find that, within this approach, correlations are responsible for $\approx 4–5\%$ of the total energy. Radii are generally unaffected by the introduction of beyond-mean-field correlations. Large nuclear correlation entropies are associated with large correlation energies. By all measures, ${}^{12}\mathrm{C}$ is the most correlated isotope in the mass region considered.

Conclusions: Our work is the starting point of a consistent implementation of the TDDM technique for applications into nuclear reactions. Our results indicate that correlation effects in structure are small, but beyond-mean-field dynamical simulations could provide insight into several issues of interest.

Figure 1

Time evolution of the mean-field (open symbols) and the total energy (filled symbols) of ${}^{12}\mathrm{C}$ from a HF to a correlated ground state. Data are provided for $N_{\text{max}}=14$ (squares) and $N_{\text{max}}=20$ (circles).

Figure 2

Occupation numbers of (a) neutron and (b) proton hole states as a function of time for the adiabatic switching of ${}^{12}\mathrm{C}$ with $N_{\text{max}}=20$. Panels (c) and (d) give the corresponding occupations of particle states in a logarithmic scale. States that are degenerate in spin are not shown for simplicity. The index $\alpha$ denotes energy levels in increasing order.

Figure 3

Evolution of the single-particle energies ${\epsilon }_{\alpha \alpha }$ with time for (a) ${}^{12}\mathrm{C}$ and (b) ${}^{16}\mathrm{O}$. Particle and hole states are shown in different colors and line styles. These results are obtained with the SV interaction with $N_{\text{max}}=14$.

Figure 4

Same as Fig. 1 for ${}^{16}\mathrm{O}$.

Figure 5

Same as Fig. 2 for ${}^{16}\mathrm{O}$ with $N_{\text{max}}=14$.

Figure 6

Same as Fig. 1 for ${}^{20}\mathrm{Ne}$.

Figure 7

Correlation entropy of ${}^{16}\mathrm{O}$ as a function of time for two different model spaces with $N_{\text{max}}=14$ (dashed line) and 20 (solid line).