Evolving nuclear many-body forces with the similarity renormalization group

In recent years, the Similarity Renormalization Group has provided a powerful and versatile means to soften interactions for ab initio nuclear calculations. The substantial contribution of both induced and initial three-body forces to the nuclear interaction has required the consistent evolution of free-space Hamiltonians in the three-particle space. We present the most recent progress on this work, extending the calculational capability to the $p$-shell nuclei and showing that the hierarchy of induced many-body forces is consistent with previous estimates. Calculations over a range of the flow parameter for ${}^6\mathrm{Li}$, including fully evolved $\mathit{NN}$ $+$ 3$N$ interactions, show moderate contributions due to induced four-body forces and display the same improved convergence properties as in lighter nuclei. A systematic analysis provides further evidence that the hierarchy of many-body forces is preserved.

Figure 1

Ground-state energy of ${}^3\mathrm{H}$ as a function of the basis size $N_{\max }$ for an ${\mathrm{N}}^3$LO $\mathit{NN}$ interaction [29] with an initial NNN interaction [12, 32]. Unevolved (“bare”) and Lee-Suzuki (L-S) results with $\hslash \Omega =28$ MeV are compared with SRG evolved to $\lambda =2.0$ $\hspace{0.5em}{\text{fm}}^{-1}$ and 1.5 $\hspace{0.5em}{\text{fm}}^{-1}$ with $\hslash \Omega =20$ MeV.

Figure 2

Ground-state energy of ${}^4\mathrm{He}$ as a function of the basis size $N_{\max }$ for an ${\mathrm{N}}^3$LO $\mathit{NN}$ interaction [29] with an initial NNN interaction [12, 32]. Unevolved ($\lambda$) results are compared with Lee-Suzuki (L-S) and SRG evolved to $\lambda =2.0$ and 1.2 $\hspace{0.5em}{\text{fm}}^{-1}$ at $\hslash \Omega =28$ and 36 MeV. The black arrow indicates the experimental value.

Figure 3

Ground-state energy of ${}^4\mathrm{He}$ for select $\lambda$ as a function of the basis size $N_{\max }$ for the (a) AV18 [27] and (b) CD-Bonn [28] interactions. Results are shown for $\hslash \Omega =44$ MeV with $N_{\mathrm{A}3\max }=40$. Dotted lines indicate current best results for these potentials [34].

Figure 4

Ground-state energy of ${}^6\mathrm{Li}$ as a function of the basis size $N_{\max }$ for an initial ${\mathrm{N}}^3$LO $\mathit{NN}$ interaction [29] and an initial ${\mathrm{N}}^2$LO NNN interaction [12, 32] evolved to various $\lambda$. Note the large scale and complete lack of convergence for unsoftened Hamiltonians ($\lambda =100$ $\hspace{0.5em}{\text{fm}}^{-1}$). The dotted line is the best Lee-Suzuki result.

Figure 5

Ground-state energy of ${}^6\mathrm{Li}$ as a function of the basis size $N_{\max }$ for an initial ${\mathrm{N}}^3$LO $\mathit{NN}$ interaction [29] and an initial ${\mathrm{N}}^2$LO NNN interaction [12, 32] evolved to various $\lambda$. Here the initial $\mathit{NN}$ potential was included up to $N_{\mathrm{A}2\max }=300$ and the NNN up to $N_{\mathrm{A}3\max }=40$. Results are compared with the best Lee-Suzuki shown by the thin black dotted line (see text). The dotted lines extending from the curves are examples of the extrapolations made throughout.

Figure 6

Ground-state energy of ${}^6\mathrm{Li}$ as a function of the oscillator frequency, $\hslash \Omega$. Again we use a ${\mathrm{N}}^3$LO $\mathit{NN}$ [29] up to $N_{\mathrm{A}2\max }=300$ and ${\mathrm{N}}^2$LO NNN [12, 32] up to $N_{\mathrm{A}3\max }=40$. Panels (a)–(d) are $\mathit{NN}$-only while (e) and (f) include both induced and initial NNN. The black arrows indicate the experimental value. Note for comparison, the left is shifted relative to the center due to the difference in initial interactions.

Figure 7

Spectrum of ${}^6\mathrm{Li}$ as a function of $N_{\mathrm{A}3\max }$. Convergence in $N_{\max }$ is shown in (a) and (b) and convergence in $N_{\mathrm{A}3\max }$ is shown in (c) and (d). Absolute energies are shown in (a) and (c) while excitation energies are used in (b) and (d). All four panels are for $\lambda =1.8$ $\hspace{0.5em}{\text{fm}}^{-1}$ with $\mathit{NN}$ $+$ NNN at $\hslash \Omega =20$ MeV and $N_{\mathrm{A}2\max }=300$.

Figure 8

Ground-state energy of ${}^3\mathrm{H}$ as a function of the SRG evolution parameter, $\lambda$. See Table  for the nomenclature of the curves. The calculations use $N_{\mathrm{A}2\max }=196$, $N_{\mathrm{A}3\max }=36$, and $\hslash \Omega =20$ MeV.

Figure 9

Ground-state energy of ${}^4\mathrm{He}$ as a function of the SRG evolution parameter, $\lambda$. Note the comparison of two values for $\hslash \Omega$ (solid and dashed) and the best LS results (dotted). Here, $N_{\mathrm{A}2\max }=196$ and $N_{\mathrm{A}3\max }=32$ compared to $N_{\mathrm{A}2\max }=N_{\mathrm{A}3\max }=28$ for the prior work [3]. The results here are converged to within 10 keV of those from Ref. [3]. See Table  for the nomenclature of the curves. Note the comparison of two values for $\hslash \Omega$ (solid and dashed) and the best LS results (dotted). The thick arrow indicates the experimental value.

Figure 10

Ground-state energy of (a) ${}^3\mathrm{H}$ and (b) ${}^4\mathrm{He}$ as a function of $\lambda$, starting from Hamiltonians based on CD-Bonn [28], Argonne $V_{18}$ [27], and ${\mathrm{N}}^3$LO [29]. See Table  for the nomenclature of the curves. Here we used $\hslash \Omega =28/28/20$ and 44/44/28 for ${}^3\mathrm{H}$ and ${}^4\mathrm{He}$, respectively. The dotted lines show the converged LS results for each potential. For ${}^3\mathrm{H}$ these values are AV18 $=$ $-7.62$, CD-Bonn $=$ $-7.99$, and ${\mathrm{N}}^3$LO $=$ $-7.85$.

Figure 11

Extrapolated ground-state energy of ${}^6\mathrm{Li}$ as a function of the SRG evolution parameter, $\lambda$. Error bars are based on variation of the minimum in $\hslash \Omega$ as discussed in the text. The truncations used were $N_{\mathrm{A}2\max }=300$ and $N_{\mathrm{A}3\max }=40$.

Figure 12

Expectation value of the components of the nuclear interaction in the (a) ${}^3\mathrm{H}$ and (b) ${}^4\mathrm{He}$ ground states as a function of the SRG parameter $\lambda$. The initial interaction was ${\mathrm{N}}^3$LO $\mathit{NN}$ [29] and ${\mathrm{N}}^2$LO NNN at $\hslash \Omega$ $=$ 20 and 28 MeV (left and right) and $N_{\mathrm{A}3\max }$ $=$ 32 with the ${}^4\mathrm{He}$ final truncation at $N_{\max }$ $=$ 18. The insets show expanded details for the three- and four-body forces.

Figure 13

Contributions from individual terms to $d\langle V_{\lambda }^{(3)}\rangle /d\lambda$; the flow of the triton ground-state expectation value of the three-body potential. The calculation is done (a) without and (b) with an initial three-body interaction.

Figure 14

Contributions of the terms in Eq. (14) to $d\langle V_{\lambda }^{(4)}\rangle /d\lambda$, the evolving ${}^4\mathrm{He}$ ground-state expectation value of the four-body force. The calculation is done (a) without and (b) with an initial three-body interaction.

Figure 15

Various observables (a)–(e) in ${}^6\mathrm{Li}$ as a function of $N_{\max }$ for select $\lambda$’s. These results are with included initial NNN forces and $N_{\mathrm{A}2\max }$ $=$ 300 and $N_{\mathrm{A}3\max }$ $=$ 40 and $\hslash \Omega$ $=$ 20 MeV. The arrow shows the experimental value and the dotted line shows the best LS result.