Generalized seniority on a deformed single-particle basis

Recently, I proposed a fast computing scheme for generalized seniority on a spherical single-particle basis [J. Phys. G: Nucl. Part. Phys. 42, 115105 (2015)]. This work redesigns the scheme to make it applicable to deformed single-particle basis. The algorithm is applied to the rare-earth-metal nucleus ${}_{64}{}^{158}\mathrm{Gd}_{94}$ for intrinsic (body-fixed frame) neutron excitations under the low-momentum NN interaction $V_{\mathrm{low}\text{−}k}$. By allowing as many as four broken pairs, I compute the lowest 300 intrinsic states of several multipolarities. These states converge well to the exact ones, showing generalized seniority is very effective in truncating the deformed shell model. Under realistic interactions, the picture remains approximately valid: The ground state is a coherent pair condensate and the pairs gradually break up as excitation energy increases.

Figure 1

Memory requirements to store the $\chi$ and $t$ tables in three different model spaces. $2\mathrm{\Omega }$ is the dimension of the single-particle space, $2N$ is the number of particles, and $S$ is the generalized seniority quantum number.

Figure 2

Part of the Nilsson diagram. The 31 asterisks represent 31 consecutive pairs of Nilsson levels. The horizontal and vertical axes show their numbering and energy (zero of energy is arbitrary). The blue solid line is the Fermi surface. The Nilsson levels between the two red dashed (green dash-dotted) lines compose the valence space 1 (2).

Figure 3

Amplitudes $P\left(s\right)$ of each generalized seniority $S=2s$ vs the excitation energy of the $K=0$ eigenstates by calculation 1. The left (right) panels plot the lowest 300 eigenstates with positive (negative) parity. Therefore, each panel has 300 data points. The vertical dotted line is $E=E_{s=1,K^{\pi }=0^{+}}^{<}$ for the left panels and $E=E_{s=1,K^{\pi }=0^{-}}^{<}$ for the right panels.

Figure 4

Amplitudes of each generalized seniority vs the excitation energy of the $K=2$ eigenstates by calculation 1. The $P(s=0)$ amplitudes vanish owing to symmetry, and are not plotted.

Figure 5

Amplitudes of each generalized seniority vs the excitation energy of the $K=6$ eigenstates by calculation 1.

Figure 6

Energies of selected states by four different diagonalizations. There are 17 states labeled by $K_i^{\pi }$ as listed in the horizontal axis. The $S=8$ staircase curve corresponds to the left vertical axis and shows the eigenenergies of the 17 states by the $S=8$ diagonalization (choosing the $0_1^{+}$ energy as the zero of energy). The $S=2,4,6$ points correspond to the right vertical axis (in logarithmic scale) and show the errors of the eigenenergies by the $S=2,4,6$ diagonalizations, respectively, relative to the $S=8$ results.

Figure 7

Amplitudes of each generalized seniority vs the excitation energy of the $K=0$ eigenstates by calculation 2.

Figure 8

Computer time cost vs subspace dimension, for calculating the Hamiltonian matrix in different subspaces. The calculations are done in parallel on two octa-core CPUs (Intel Xeon E5-2620v4 @ 2.1 GHz). “MS1” stands for model space 1. See text for details.