Tensor-optimized high-momentum antisymmetrized molecular dynamics with a bare interaction, and its application to the ${}^4\mathrm{He}$ nucleus

We formulate the “tensor-optimized high-momentum antisymmetrized molecular dynamics” (TO-HMAMD) framework for ab initio calculations of nuclei by hybridizing the tensor-optimized (TO) and high-momentum (HM) AMD approaches. This hybrid approach has advantages in both analytical simplicity and numerical efficiency compared with other AMD-based methods which treat the bare interaction, especially for heavier nuclear systems. In this work, the $s$-shell nucleus ${}^4\mathrm{He}$ is calculated with TO-HMAMD by including up to double product of nucleon-nucleon ($NN$) correlations, described by using high-momentum pairs and spatial correlation functions of nucleons. The total energy and radius of the ${}^4\mathrm{He}$ nucleus are well reproduced using the $\mathrm{AV}{8}^{\prime }$ interaction. The spin-isospin channel dependence is also discussed for $NN$ correlations, which are found to be mostly contributed in the even-state channels, especially the triplet-even channel. Analyses of the analytical formation and numerical results suggest that TO-HMAMD could be a promising framework for $p$-shell nuclear systems.

Figure 1

Some of the correlation diagrams for the ${}^4\mathrm{He}$ wave functions in HMAMD, TOAMD, and TO-HMAMD methods. Vertical lines indicate the particles, numbered from the left side as 1, 2, 3, 4. The red connections labeled HM denote the $NN$ correlations described by the high-momentum pairs of nucleons. The blue connections labeled $F$ denote the $NN$ correlations described by the TOAMD correlation functions in the central or tensor channels. The green connection labeled “$F$ or HM” stands for either $F$ or HM. The label in square brackets below each diagram indicates the configuration of particle correlations.

Figure 2

Diagrams of cluster expansions in calculating matrix elements for the two-body operator $V$. The diagram (a) is for the operator $V$. The diagrams (b)–(d) correspond to the cluster expansion of the operator $F^{†}V$. The diagrams (e)–(p) correspond to the cluster expansion of the operator $F^{†}VF$. The blue solid connections indicate the TOAMD correlation functions $F_D$ or $F_S$. The dotted connections indicate the two-body operator $V$. For each diagram, the corresponding integral kernels should be derived analytically.

Figure 3

The energy of the ${}^4\mathrm{He}$ nucleus calculated with TO-HMAMD using the bare interaction $\mathrm{AV}{8}^{\prime }$ by successively adding the high-momentum pairs (+HM) and correlation functions $F_D$ and $F_S$.

Figure 4

Hamiltonian components of the ${}^4\mathrm{He}$ nucleus calculated with TO-HMAMD using the bare interaction $\mathrm{AV}{8}^{\prime }$ by successively adding the high-momentum pairs (+HM) and correlation functions $F_D$ and $F_S$. K/2 denotes a half of a kinetic component. C, T,and LS denote central, tensor, and spin-orbit components, respectively.

Figure 5

The energy convergence of ${}^4\mathrm{He}$ with respect to the spin-isospin channels of the correlation functions $F_D$ and $F_S$ in TO-HMAMD and single TOAMD calculations.

Figure 6

The energy convergence of ${}^3\mathrm{H}$ with respect to the spin-isospin channels of the correlation functions $F_D$ and $F_S$ in TO-HMAMD and single TOAMD calculations.

Figure 7

Energy convergence of ${}^4\mathrm{He}$ with respect to the spin-isospin combinations of the high-momentum pairs in TO-HMAMD and single HMAMD calculations.

Figure 8

Energy convergence of ${}^3\mathrm{H}$ with respect to the spin-isospin combinations of the high-momentum pairs in TO-HMAMD and single HMAMD calculations.