Variational approach with the superposition of the symmetry-restored quasiparticle vacua for nuclear shell-model calculations

We propose a variational calculation scheme utilizing the superposition of the angular-momentum, parity, number projected quasiparticle vacua, that is especially suitable for applying to medium-heavy nuclei in shell-model calculations. We derive a formula for the energy variance with quasiparticle vacua and apply the energy-variance extrapolation to the present scheme for further precise estimation of the exact shell-model energy. The validity of the method is presented for the shell-model calculation of ${}^{132}\mathrm{Ba}$ in the $50\le Z,N\le 82$ model space. We also discuss the feasibility of this scheme in the case of the ${}^{150}\mathrm{Nd}$ in the $50\le Z\le 82$ and $82\le Z\le 126$ model space and demonstrate that its neutrinoless double-$\beta$-decay matrix element is obtained showing good convergence.

Figure 1

(a) Energy expectation values $E_{N_b}$ of the $0_1^{+}$ (red), $2_1^{+}$ (blue), $4_1^{+}$ (green) states of ${}^{132}\mathrm{Ba}$ by the QVSM and the MCSM as a function of the number of the basis states $N_b$. The horizontal dotted lines show the exact shell-model energies. (b) The $0_1^{+}, 2_1^{+}, 4_1^{+}$ energies of ${}^{132}\mathrm{Ba}$ obtained by the conventional Lanczos method with $t=2$ truncation, by that with $t=4$ truncation, by the exact calculation without truncation, by the MCSM with 50 basis states, and by the QVSM with 20 basis states are shown from left to right.

Figure 2

(a) Excitation energies of the $0_2^{+}$ and $0_3^{+}$ states (red dashed lines), and $2_1^{+}, 2_2^{+}$, and $2_3^{+}$ states (blue solid lines) of ${}^{132}\mathrm{Ba}$ against the number of the basis states $N_b$. (b) Quadrupole moments of the $2_1^{+}, 2_2^{+}$, and $2_3^{+}$ states. (c) Square root of the $B\left(E2\right)$ transition probabilities from the ground state to the $2_1^{+}, 2_2^{+}$, and $2_3^{+}$ states. The triangles at the rightmost side denote the exact values.

Figure 3

Variance-energy plot of ${}^{132}\mathrm{Ba}$. The red circles, blue triangles, and green diamonds denote the energy expectation values against the energy variance of the $0_1^{+}, 2_1^{+}$, and $4_1^{+}$ states, respectively, obtained by the QVSM. The dashed curves are drawn to be fitted for the last 12 points of the QVSM results with a second-order polynomial. and their $y$ intercepts are the extrapolated values. The left panel shows the exact shell-model energies.

Figure 4

Variance-energy plot of ${}^{132}\mathrm{Ba}$. The red circles (blue triangles) denote the lowest three $0^{+}$ ($2^{+}$) states obtained by the QVSM. The black open circles (triangles) denote the energies and energy variances of the $0^{+}$ ($2^{+}$) states of the MCSM. The dashed curves are drawn to be fitted for the last 20 points of the QVSM results with a second-order polynomial. and their $y$ intercepts are the extrapolated values. The left panel shows the exact shell-model energies.

Figure 5

Variance-energy plot of ${}^{68}\mathrm{Ni}$ with the A3DA interaction [27]. The red filled circles, blue filled triangles, and green filled diamonds are the QVSM results of the $0_1^{+}, 2_1^{+}$, and $4_1^{+}$ states, respectively. The QVSM is obtained with $N_b=30$. The black symbols are the corresponding MCSM results with 120 basis states. The solid lines denote the fitted curves by a first-order polynomial for the last 13 QVSM points.

Figure 6

$0\nu \beta \beta$-decay NME of ${}^{76}\mathrm{Ge}$ obtained by the MCSM. These values are shown as a function against the number of the basis states $N_b$. The red, blue, green, orange lines with the solid circles denote the total, GT-type, Fermi-type, and tensor-type NMEs, respectively. The circles show the MCSM values with $N_b=5,10,15,20,30,40,50,60,70,90,120,$ and 150. The exact shell-model values are shown as the triangles at the rightmost.

Figure 7

$0\nu \beta \beta$-decay NME of the ${}^{76}\mathrm{Ge}$ against $N_b$ obtained by the QVSM. The exact values are shown as the triangles at the rightmost.

Figure 8

(a) Energy expectation values and (b) spectroscopic quadrupole moments of the yrast states of ${}^{150}\mathrm{Nd}$ obtained by the QVSM with $N_b=20$ and the MCSM with $N_b=100$ against the energy variance. The red circles, blue triangles, and green diamonds denote the QVSM results of the $0^{+}, 2^{+}$, and $4^{+}$ states, respectively. The dashed, solid, and dotted lines are the MCSM results of the $0^{+}, 2^{+}$, and $4^{+}$ states, respectively. The effective charges are taken as $(e_p,e_n)=(1.58,0.85)e$.

Figure 9

(a) Energy expectation values and (b) spectroscopic quadrupole moments of the yrast states of ${}^{150}\mathrm{Sm}$ obtained by the QVSM and the MCSM against the energy variance. See caption of Fig. 8 for details.

Figure 10

Excitation energies of the $2^{+}$ and $4^{+}$ states of (a) ${}^{150}\mathrm{Nd}$ and (b) ${}^{150}\mathrm{Sm}$ as a function of the number of the QVSM basis states.

Figure 11

$0\nu \beta \beta$-decay NME of the ${}^{150}\mathrm{Nd}$ obtained by the MCSM against the number of the basis states $N_b$. The red, blue, green, orange lines with the solid circles denote the total, GT-type, Fermi-type, and tensor-type NMEs, respectively.

Figure 12

$0\nu \beta \beta$-decay NME of the ${}^{150}\mathrm{Nd}$ obtained by the QVSM against the number of the basis states $N_b$. The red, blue, green, orange lines with the solid circles denote the total, GT-type, Fermi-type, and tensor-type NMEs, respectively.