Introduction of the one-body correlation operator in the unitary-model-operator approach

In the earlier unitary-model-operator approach (UMOA), one-body correlations have been taken into account approximately by the diagonalization of unitary-transformed Hamiltonians in the $0\text{p}0\text{h}$ and $1\text{p}1\text{h}$ space. With this prescription, the dependence of the harmonic-oscillator energy ($\hslash \omega$) on calculated observables is not negligible even at larger model spaces. In the present work, we explicitly introduce the one-body correlation operator so that it optimizes the single-particle basis states and then reduces the $\hslash \omega$ dependence. For an actual demonstration, we calculate the energy and radius for the ${}^4\mathrm{He}$ ground state with the softened nucleon-nucleon ($NN$) interactions from Argonne v18 (AV18) and chiral effective field theory ($\chi \mathrm{EFT}$) up to the next-to-next-to-next leading order ($\mathrm{N}{}^3\mathrm{LO}$). As a result, we obtain practically $\hslash \omega$-free results at sufficiently large model spaces. The present results are reasonably close to those by the other ab initio calculations with the same $NN$ interactions. This methodological development enables more systematic analysis of calculation results in the UMOA. We also discuss qualitatively the origin of the $\hslash \omega$ dependence on calculated observables in a somewhat simplified way.

Figure 1

Cancellations of bubble-diagram contributions for the one-body (a) and two-body (b) parts.

Figure 2

Illustrations of the model space and its decoupling employed in the previous work. The first-step decoupling (a) denotes the decoupling between the outside and inside (shaded area) of our model space. The second-step decoupling (b) means the decoupling of $2\text{p}2\text{h}$ excitations with $0\text{p}0\text{h}$ state (shaded area).

Figure 3

Illustrations of the model space and its decoupling employed in the present work. The shaded area indicates our reference state. The arrow means the decoupling of $1\text{p}1\text{h}$ and $2\text{p}2\text{h}$ excitations with the reference state.

Figure 4

Flow chart of the actual calculation. The steps surrounded with the dashed line (1, 3, and 6) are related to the procedure on the $1\text{p}1\text{h}$ decoupling newly added this time. See text for details.

Figure 5

Schematic representations of the transformed Hamiltonians without [left; panel (a)] and with [right; panel (b)] the one-body correlation operator.

Figure 6

Ground-state energies [upper panels (a) and (b)] and point-nucleon root-mean-square matter radii [lower panels (b) and (d)] of the ${}^4\mathrm{He}$ nucleus as a functions of the HO energy $\hslash \omega$. In panels (a) and (c), one-body correlations are approximately taken into account through the diagonalization in $0\text{p}0\text{h}$ and $1\text{p}1\text{h}$ space as done in earlier works [21, 22, 28, 29]. In panels (b) and (d), one-body correlations are explicitly included. Note that the results from “FY” (solid line) and “CCSD” are taken from Refs. [41] and [2], respectively.

Figure 7

Ground-state energies (upper panel) and point-nucleon root-mean-square matter radii (lower panel) of the ${}^4\mathrm{He}$ nucleus as a functions of the HO energy $\hslash \omega$. Circles and triangles are given with the Hamiltonian in Figs. 5 and 5, respectively. Squares are obtained after the diagonalization of the Hamiltonian in Fig. 5 in the $0\text{p}0\text{h}$ and $1\text{p}1\text{h}$ space. The dashed curves are obtained with the $0\text{p}0\text{h}$ HO reference state.

Figure 8

The total (a), one-body (b), two-body (c), and three-body energies (d) for ${}^4\mathrm{He}$ as functions of $\hslash \omega$ from the top to the bottom. The $NN$ interaction is the $V_{\mathrm{low}\mathrm{k}}$ interaction derived from AV18 interaction [38] at $\mathrm{\Lambda }=1.9 {\mathrm{fm}}^{-1}$. Circles (triangles) correspond to the results with (without) $S^{\left(1\right)}$. The dashed and dotted lines are for the HO reference and HF states, respectively.

Figure 9

The squared overlaps between the ground and reference states as functions of $\hslash \omega$. The employed $NN$ interaction is the $V_{\mathrm{low}\mathrm{k}}$ interaction derived from AV18 interaction at $\mathrm{\Lambda }=1.9 {\mathrm{fm}}^{-1}$. Circles (triangles) are obtained with the UMOA with $S^{\left(1\right)}$ and HF (UMOA without $S^{\left(1\right)}$) states. The dashed line is given with the HF and HO reference states.

Figure 10

One-body density matrix $\gamma$ of $s1/2$ orbitals for ${}^4\mathrm{He}$ calculated without [left; panels (a), (c), and (e)] and with [right; panels (b), (d), and (f)] the one-body correlation operator. The top, middle, and bottom panels are calculated at $e_{\max }=12$ and $\hslash \omega =$ 20, 28, and 36 MeV, respectively.

Figure 11

Comparisons for the ground-state energies of ${}^4\mathrm{He}$ in the UMOA and in the other ab initio calculations. The displayed UMOA results are obtained at $e_{max}=14$ and $\hslash \omega$ = 20 MeV. The error bands of the UMOA energies are estimated from the size of the three-body cluster term corrections. The CCSD, FY, importance-truncated no-core shell model (IT-NCSM), and experimental results are taken from Refs. [2, 41, 42, 43].