Importance-truncated large-scale shell model

We propose an importance-truncation scheme for the large-scale nuclear shell model that extends its range of applicability to larger valence spaces and midshell nuclei. It is based on a perturbative measure for the importance of individual basis states that acts as an additional truncation for the many-body model space in which the eigenvalue problem of the Hamiltonian is solved numerically. Through a posteriori extrapolations of all observables to vanishing importance threshold, the full shell-model results can be recovered. In addition to simple threshold extrapolations, we explore extrapolations based on the energy variance. We apply the importance-truncated shell model for the study of ${}^{56}\mathrm{Ni}$ in the $pf$ valence space and of ${}^{60}\mathrm{Zn}$ and ${}^{64}\mathrm{Ge}$ in the $pf{g}_{9/2}$ space. We demonstrate the efficiency and accuracy of the approach, which pave the way for future applications of valence-space interactions derived in ab initio approaches in larger valence spaces.

Figure 1

Dimension of the importance-truncated model space (a) and ground-state energy relative to the core (b) for ${}^{56}\mathrm{Ni}$ in the $pf$ valence space as a function of the importance threshold for reference thresholds and $T_{max}=16$ using the gxpf1a interaction. The model space has been constructed for the simultaneous description of the six lowest eigenstates. For the threshold extrapolation we use polynomials of order two and three. The red lines denote the full $m$-scheme dimension and the ground-state energy of the full SM [2].

Figure 2

Energy-variance extrapolation of the ground-state energy relative to the core for ${}^{56}\mathrm{Ni}$ obtained in IT-SM using the gxpf1a interaction. In panel (a) results for different reference thresholds for $T_{max}=16$ are shown. In panel (b) calculations for different truncations with $C_{\text{min}}=1\times {10}^{-4}$ are depicted. The red lines denote the exact ground-state energy extracted from Ref. [2].

Figure 3

Energy-variance (a) and threshold (b) extrapolation of the energies of the six lowest natural-parity states of ${}^{56}\mathrm{Ni}$ using the gxpf1a interaction with and $T_{max}=16$. For the variance and threshold extrapolations, polynomials of order two and three have been employed, respectively. The red lines show the full SM results extracted from Ref. [2].

Figure 4

Natural-parity spectrum of ${}^{56}\mathrm{Ni}$ as a function of $T_{max}$ in terms of absolute energies relative to the core computed in the IT-SM with $C_{\text{min}}=1\times {10}^{-4}$ using the gxpf1a interaction. The right-hand columns show the results of an energy-variance extrapolation ($\mathrm{\Delta }{E}^2$) and the full SM energies extracted from Ref. [2].

Figure 5

Threshold dependence and extrapolation for the quadrupole moment of the $2_1^{+}$ state (a) and the $B(\text{E2}:2_1^{+}\to 0_1^{+})$ transition strength (c) for ${}^{56}\mathrm{Ni}$. The wave functions have been obtained in an IT-SM calculation using the gxpf1a interaction for $T_{max}=8$ and the reference thresholds . The red lines represent the full SM results obtained with the Antoine code [1, 35, 36]. Panels (b) and (d) illustrate the corresponding energy-variance extrapolations with respect to the target $T_{\text{max}}=8$ model space using linear and quadratic fit functions. For the transition strength, the mean energy variance of the states considered is used.

Figure 6

Lowest natural-parity states of ${}^{60}\mathrm{Zn}$ (a) and ${}^{64}\mathrm{Ge}$ (b) computed in the IT-SM for $C_{\text{min}}=2\times {10}^{-4}$ using the pfg9b3 interaction with subsequent threshold extrapolation for different values of $T_{max}$. The right-hand columns show the results obtained from the energy-variance extrapolation ($\mathrm{\Delta }{E}^2$) of the $T_{max}=6$ results. The dashed line shows an approximation for the energy of the $2^{+}$ state calculated from the excitation energy obtained in the IT-SM for $T_{max}=10$ and the $\mathrm{\Delta }{E}^2$-extrapolated ground-state energy. For ${}^{64}\mathrm{Ge}$, the MCSM results [32, 33] are shown for comparison.