Entanglement and seniority

Background: In quantum mechanics, entanglement is the most striking phenomenon which has no counterpart in classical systems. Although different approaches have already been developed to study correlations in the case of indistinguishable particles, the exploration of the so-called mode entanglement is still in its initial stage in nuclear physics.

Purpose: We aim to study mode entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority $\nu =0,1,2$, and 3, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. In addition, we compare the predictions of the seniority model with the results of the shell model to gain insight into the structure and correlations of the ground and lowest yrast states.

Methods: In the seniority model the analytic results are calculated using the quasispin formalism. The numerical shell model calculations are carried out in the standard shell model framework using an inert core. The applied realistic effective interactions are derived earlier using the $G$-matrix formalism. The density matrix renormalization group method is also applied in order to directly calculate the mode entropies.

Results: In the seniority model simple analytical expressions are given for the mode entropies. The peculiar behavior of the half-filled shells and the seniority-zero states is revealed. Numerical results are presented for the lightest stable calcium isotopes and for ${}^{94}\mathrm{Ru}$ nucleus.

Conclusions: For ${}^{94}\mathrm{Ru}$, the seniority model accounts for the $0g_{9/2}$ mode entropies, but seniority mixing is important for certain yrast states. Interaction induced quantum fluctuations decrease the occupation of the $0f_{5/2}$, $1p_{3/2}$, and $1p_{1/2}$ shells, and result in non-negligible mode entropies on these shells, too, clearly outside the scope of the simple ${\left(0g_{9/2}\right)}^4$ seniority model. The $0f_{7/2}$ shell based seniority model is more accurate for the $\mathrm{Ca}$ isotopes, but seniority mixing is substantial for some ${}^{44}\mathrm{Ca}$ yrast states, too. Mode and one-body entanglement entropies are useful tools to investigate the structure of quantum correlations in nuclei.

Figure 1

Squared overlap of the CI and SEN model wave functions, as a function of the total angular momentum for ${}^{42}\mathrm{Ca}$, ${}^{44}\mathrm{Ca}$, and ${}^{46}\mathrm{Ca}$ nuclei. Full CI calculations were performed with the GXPF1 interaction. The ground states and the yrast states are very well captured by $\nu =0$ seniority and $\nu =2$ seniority states, respectively, excepting the $J=4$ state of ${}^{44}\mathrm{Ca}$, where a mixing with seniority $\nu =4$ excitations is apparent.

Figure 2

Mode entropies of the sp orbital $0f_{7/2}$ for ${}^{44}\mathrm{Ca}$ ground and yrast states. The results of CI and DMRG calculations performed with the GXPF1 interaction are compared with predictions of the SEN model (ground state $\nu =0$, yrast states $\nu =2$).

Figure 3

Mode entropies of the sp orbital $0f_{7/2}$ in the $J=7/2$ ground state of ${}^{43}\mathrm{Ca}$, and in its $J=5/2$ yrast state, as determined by CI/DMRG calculations (red squares) and predicted by the SEN model (black continuous lines). As a reference, we also display the ground state entropies of ${}^{42}\mathrm{Ca}$ (empty triangles).

Figure 4

Total correlations (dashed lines, empty symbols) and one-body entanglement entropies (continuous lines, filled symbols) as a function of total angular momentum for ground and yrast states of the nuclei ${}^{42}\mathrm{Ca}$, ${}^{44}\mathrm{Ca}$, and ${}^{46}\mathrm{Ca}$. Full CI/DMRG calculations were carried out with the GXPF1 interaction.

Figure 5

Ground state mode entropies of ${}^{94}\mathrm{Ru}$ sp orbitals obtained by CI/DMRG (symbols, dashed lines), compared with the SEN model prediction (black continuous line).

Figure 6

Mode entropies of the sp orbital $0g_{\frac{9}{2}}$ in the yrast excited states of ${}^{94}\mathrm{Ru}$, constructed using full CI and DMRG (red squares), and compared with seniority $\nu =2$ states of the SEN model (black continuous lines).

Figure 7

Mode entropies of the ${}^{94}\mathrm{Ru}$ sp orbital $0g_{9/2}$, calculated for the yrast state $J^{\pi }=4^{+}$ using full CI/DMRG computations, and compared with the $0g_{9/2}$ shell SEN model. Mixing with the so-called $\alpha$ seniority-four state accounts for the observed structures, and is needed to yield satisfactory agreement.