Large-scale exact diagonalizations reveal low-momentum scales of nuclei

Ab initio methods aim to solve the nuclear many-body problem with controlled approximations. Virtually exact numerical solutions for realistic interactions can only be obtained for certain special cases such as few-nucleon systems. Here we extend the reach of exact diagonalization methods to handle model spaces with dimension exceeding ${10}^{10}$ on a single compute node. This allows us to perform no-core shell model (NCSM) calculations for ${}^6\mathrm{Li}$ in model spaces up to $N_{\max }=22$ and to reveal the ${}^4\mathrm{He}$+d halo structure of this nucleus. Still, the use of a finite harmonic-oscillator basis implies truncations in both infrared (IR) and ultraviolet (UV) length scales. These truncations impose finite-size corrections on observables computed in this basis. We perform IR extrapolations of energies and radii computed in the NCSM and with the coupled-cluster method at several fixed UV cutoffs. It is shown that this strategy enables information gain also from data that is not fully UV converged. IR extrapolations improve the accuracy of relevant bound-state observables for a range of UV cutoffs, thus making them profitable tools. We relate the momentum scale that governs the exponential IR convergence to the threshold energy for the first open decay channel. Using large-scale NCSM calculations we numerically verify this small-momentum scale of finite nuclei.

Figure 1

Choice of Jacobi coordinates for the deuteron (a), the triton (b), and ${}^6\mathrm{Li}$ (c) such that ${\vec{\rho }}_1$ corresponds to the channel with the lowest separation energy.

Figure 2

Scaling plots for the ${}^6\mathrm{Li}$ nuclear many-body problem ($M=1$) as a function of the NCSM model space truncation $N_{\max }$. (a) Model space dimension; (b) Number of nonzero matrix elements (with two-body interactions only). The right axes displays the corresponding size (in TB) assuming that the elements are explicitly stored in double-precision floating-point format. Extrapolated data is shown as open symbols.

Figure 3

Many-body states in the proton and neutron sub-spaces factorize into blocks according to their $J_z$ projection. $A$-particle states with fixed $J_z=M$ are product states $|I\rangle =|\pi \rangle \otimes |\nu \rangle$ with proton- and neutron states from corresponding blocks.

Figure 4

Extrapolated energy $E_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^3\mathrm{H}$ with the ${\mathrm{NNLO}}_{\mathrm{opt}}\mathit{NN}$ interaction. The different panels correspond to different NCSM model space truncations from $max\left(N_{\max }\right)=16$ to $max\left(N_{\max }\right)=36$. The bands estimate uncertainties from subleading IR corrections. The squares denote the minimum energy computed with the NCSM as a function of $\mathrm{\Lambda }$.

Figure 5

Extrapolated ground-state (point-proton) radii $r_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^3\mathrm{H}$ with the ${\mathrm{NNLO}}_{\mathrm{opt}}\mathit{NN}$ interaction. The different panels correspond to different NCSM model space truncations from $max\left(N_{\max }\right)=16$ to $max\left(N_{\max }\right)=36$. The bands estimate uncertainties from subleading IR corrections. The squares denote the maximum radius computed with the NCSM as a function of $\mathrm{\Lambda }$ and model space truncation.

Figure 6

Fit parameters $a_0$ (a), $k_{\infty }\left(\mathrm{\Lambda }\right)$ for ${}^3\mathrm{H}$ energy extrapolation (b), and $k_{\infty }\left(\mathrm{\Lambda }\right)$ for radius extrapolation (c) for different NCSM model space truncations from $max\left(N_{\max }\right)=16$ to $max\left(N_{\max }\right)=36$. Open symbols denote results for which UV corrections are expected to be larger than IR ones, and the corresponding fits are unreliable. The lowest, theoretical separation momentum is given as a dashed line with an uncertainty band.

Figure 7

Recommended results for the ${}^3\mathrm{H}$ energy (upper panel) and radius (lower panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=16$ to $max\left(N_{\max }\right)=36$.

Figure 8

Computed ground-state energies (upper panel) and point-proton radii (lower panel) for ${}^4\mathrm{He}$ as a function of the oscillator spacing $\hslash \omega$ in model spaces of size $N_{\max }$ as indicated. Solid lines connect data points with equal $N_{\max }$. Dashed lines connect data points with equal UV cutoff $\mathrm{\Lambda }$, starting at $\mathrm{\Lambda }=750$ MeV to $\mathrm{\Lambda }=1450$ MeV (from left to right in steps of 50 MeV).

Figure 9

Extrapolated energy $E_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^4\mathrm{He}$. The different panels correspond to different NCSM model space truncations from $max\left(N_{\max }\right)=10$ to $max\left(N_{\max }\right)=20$. See caption of Fig. 4 for further details.

Figure 10

Extrapolated ground-state (point-proton) radii $r_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^4\mathrm{He}$. The different panels correspond to different NCSM model space truncations from $max\left(N_{\max }\right)=10$ to $max\left(N_{\max }\right)=20$. See caption of Fig. 5 for further details.

Figure 11

Fit parameter $k_{\infty }\left(\mathrm{\Lambda }\right)$ for ${}^4\mathrm{He}$ energy extrapolation (left panel) and radius extrapolation (right panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=10$ to $max\left(N_{\max }\right)=20$. The lowest, theoretical separation momentum is given as a dashed line with an uncertainty band.

Figure 12

Recommended results for the ${}^4\mathrm{He}$ energy (upper panel) and radius (lower panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=10$ to $max\left(N_{\max }\right)=20$.

Figure 13

Computed ground-state energies (upper panel) and point-proton radii (lower panel) for ${}^6\mathrm{Li}$ as a function of the oscillator spacing $\hslash \omega$ in model spaces of size $N_{\max }$ as indicated. Solid lines connect data points with equal $N_{\max }$. Dashed lines connect data points with equal UV cutoff $\mathrm{\Lambda }$, starting at $\mathrm{\Lambda }=750$ MeV to $\mathrm{\Lambda }=1400$ MeV (from left to right in steps of 50 MeV).

Figure 14

Extrapolated energy $E_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^6\mathrm{Li}$. The different panels correspond to different NCSM model space truncations from $max\left(N_{\max }\right)=12$ to $max\left(N_{\max }\right)=22$. See caption of Fig. 4 for further details.

Figure 15

Extrapolated ground-state (point-proton) radii $r_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^6\mathrm{Li}$. The different panels correspond to different NCSM model space truncations from $max\left(N_{\max }\right)=12$ to $max\left(N_{\max }\right)=22$. See caption of Fig. 5 for further details.

Figure 16

Fit parameter $k_{\infty }\left(\mathrm{\Lambda }\right)$ for ${}^6\mathrm{Li}$ energy extrapolation (left panel) and radius extrapolation (right panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=12$ to $max\left(N_{\max }\right)=22$. The lowest, theoretical separation momentum is given as a dashed line with an uncertainty band.

Figure 17

Recommended results for the ${}^6\mathrm{Li}$ energy (upper panel) and radius (lower panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=12$ to $max\left(N_{\max }\right)=22$.

Figure 18

Extrapolated ground-state energy $E_{\infty }\left(\mathrm{\Lambda }\right)\text{for}{}^6\mathrm{He}$. The different panels correspond to different NCSM model space truncations. See caption of Fig. 4 for further details.

Figure 19

Fit parameter $k_{\infty }\left(\mathrm{\Lambda }\right)$ for ${}^6\mathrm{He}$ energy extrapolation (left panel) and radius extrapolation (right panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=12$ to $max\left(N_{\max }\right)=18$.

Figure 20

Extrapolated energy $E_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^8\mathrm{He}$ from the NCSM (upper panels) and the coupled-cluster method (lower panels) with different model space truncations. The band estimates uncertainties from subleading IR corrections. The squares denote the minimum energy computed with the respective method as a function of $\mathrm{\Lambda }$.

Figure 21

Extrapolated energy $E_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^{16}\mathrm{O}$. The different panels correspond to different model space truncations. The bands estimate uncertainties from subleading IR corrections. The squares denote the minimum energy computed with the Lambda-CCSD(T) as a function of $\mathrm{\Lambda }$.

Figure 22

$\mathtt{pANTOINE}$ scaling plots for the ${}^6\mathrm{Li}$ nuclear many-body problem as a function of the NCSM model space truncation $N_{\max }$. The model space dimension is shown in Fig. 2. (a) Total storage required for implicit matrix construction and for 15 iterations of Lanczos vectors; (b) average read speed of implicit matrix data from disk. For $N_{\max }\ge 18$ the Lanczos vector is split in several blocks.

Figure 23

$\mathtt{pANTOINE}$ scaling plots for the ${}^6\mathrm{Li}$ nuclear many-body problem ($M=1$) as a function of the NCSM model space truncation $N_{\max }$. (a) Multiplication efficiency (defined as the number of nonzero matrix elements divided by the actual number of multiplications being performed); multiplication per clock cycle for (b) Intel(R) Xeon(R) E5 1650v2 and (c) Intel(R) Xeon(R) X5570 using a single thread (dotted line), multiple threads (solid line with circles), and multiple-threads with splitting of the Lanczos vector (solid line with square symbols).

Figure 24

Extrapolated energy $E_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^3\mathrm{He}$. See caption of Fig. 4 for further details.

Figure 25

Extrapolated ground-state (point-proton) radii $r_{\infty }\left(\mathrm{\Lambda }\right)$ (circles) for ${}^3\mathrm{He}$. The different panels correspond to different NCSM model space truncations from $max\left(N_{\max }\right)=16$ to $max\left(N_{\max }\right)=36$. See caption of Fig. 5 for further details.

Figure 26

Fit parameter $k_{\infty }\left(\mathrm{\Lambda }\right)$ for ${}^3\mathrm{He}$ energy extrapolation (left panel) and radius extrapolation (right panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=16$ to $max\left(N_{\max }\right)=36$. Open symbols denote results for which UV corrections are expected to be larger than IR ones, and the corresponding fits are unreliable. The lowest, theoretical separation momentum is given as a dashed line with an uncertainty band.

Figure 27

Recommended results for the ${}^3\mathrm{He}$ energy (upper panel) and radius (lower panel) for different NCSM model space truncations from $max\left(N_{\max }\right)=16$ to $max\left(N_{\max }\right)=36$.