Effective operators from exact many-body renormalization

We construct effective two-body Hamiltonians and $E2$ operators for the $p$ shell by performing $16\hslash \Omega$ ab initio no-core shell model (NCSM) calculations for $A=5$ and $A=6$ nuclei and explicitly projecting the many-body Hamiltonians and $E2$ operator onto the $0\hslash \Omega$ space. We then separate the effective $E2$ operator into one-body and two-body contributions employing the two-body valence cluster approximation. We analyze the convergence of proton and neutron valence one-body contributions with increasing model space size and explore the role of valence two-body contributions. We show that the constructed effective $E2$ operator can be parametrized in terms of one-body effective charges giving a good estimate of the NCSM result for heavier $p$-shell nuclei.

Figure 1

NCSM results for the excitation energies of the $J^{\pi }$ states and the ground-state energy of ${}^6\mathrm{Li}$, calculated in $N_{\max }\hslash \Omega$ spaces with the CD-Bonn potential and $\hslash \Omega =20$ MeV. The experimental spectra and the ground-state energy are shown for comparison.

Figure 2

The one-body effective charges, $e_1^{\rho }(2j_a,2j_b)$, for $p$-shell space are shown as a function of increasing model space $N_{\max }$.

Figure 3

The total effective charges $e_t^{\rho }(2j_a,2j_b)=e_1^{\rho }(2j_a,2j_b)+e_2^{\rho }(2j_a,2j_b)$ are shown as a function of $N_{\max }$. The dashed lines indicate the corresponding values of the one-body charges at $N_{\max }=14$ (see Fig. 2).

Figure 4

The one- and two-body contributions, as a function of increasing model space size $N_{\max }$, are shown in gray and white, respectively, for the isoscalar transition $2_1^{+}(T=0)\to 1_1^{+}(T=0)$ matrix element.

Figure 5

The one- and two-body contributions, as a function of increasing model space, are shown in gray and white, respectively, for the isovector transition $3_1^{+}(T=0)\to 1_3^{+}(T=1)$ matrix element.

Figure 6

The quadrupole moment of the ground state for ${}^6\mathrm{Li}$ [$1^{+}(T=0)$] is shown in terms of one- and two-body contributions as a function of increasing model space size. The one- and two-body contributions and total quadrupole moment are depicted as white, gray, and black histograms, respectively. The experimentally measured quadrupole moment [18] is listed on the figure for comparison.

Figure 7

The absolute values of quadrupole moment for the $J^{\pi }={\frac{3}{2}}_1^{-}$ state and the ${\frac{3}{2}}_1^{-}\to {\frac{1}{2}}_1^{-}E2$ transition matrix element for ${}^7\mathrm{Li}$ using the $A7$ effective interaction and corresponding effective charges. The left-most column (criss-crossed filling) refers to a SSM calculation using only the one-body effective charges. The middle column (zig-zag filling) refers to a SSM calculation using the total effective charges. The right-most column (solid black filling) refers to the full NCSM calculation for the same transition at $N_{\max }=6$.

Figure 8

The absolute values of quadrupole moment ($2J$) and $E2$ transition matrix elements ($2J_i-2J_f$) for ${}^9\mathrm{Li}$ yrast states using the $A9$ effective interaction and corresponding effective charges. The left-most column (criss-crossed filling) refers to a SSM calculation using only the one-body effective charges. The middle column (zig-zag filling) refers to a SSM calculation using the total effective charges. The right-most column (solid black filling) refers to the NCSM calculation for the same transition.