Structure of $p$-shell nuclei using three-nucleon interactions evolved with the similarity renormalization group

The similarity renormalization group (SRG) is used to soften interactions for ab initio nuclear structure calculations by decoupling low- and high-energy Hamiltonian matrix elements. The substantial contribution of both initial and SRG-induced three-nucleon forces requires their consistent evolution in a three-particle basis space before applying them to larger nuclei. While, in principle, the evolved Hamiltonians are unitarily equivalent, in practice the need for basis truncation introduces deviations, which must be monitored. Here we present benchmark no-core full configuration calculations with SRG-evolved interactions in $p$-shell nuclei over a wide range of softening. These calculations are used to assess convergence properties, extrapolation techniques, and the dependence of energies, including four-body contributions, on the SRG resolution scale.

Figure 1

Ground-state energy of ${}^4$He as a function of $\lambda$ for the three calculations in Table . The results at all $\lambda$ are converged at the 2-keV level or better. The dotted line is the unevolved result.

Figure 2

Ground-state energy of the triton for the unevolved chiral EFT Hamiltonian in different three-body basis sizes ($N_{\mathrm{A}3\max }$) with a large, fixed two-body basis ($N_{\mathrm{A}2\max }=300$).

Figure 3

Ground-state energy of ${}^8$Be for a fixed many-body basis size of $N_{\max }=8$ for Hamiltonians evolved to $\lambda =1.5{\text{fm}}^{-1}$ in different three-body basis sizes ($N_{\mathrm{A}3\max }$) with a fixed, large two-body basis ($N_{\mathrm{A}2\max }=300$).

Figure 4

Attempted extrapolation in $N_{\mathrm{A}3\max }$ at fixed $N_{\max }=8$ of ${}^8$Be ground-state energies from Fig. 3 at several values of $\hslash \Omega$.

Figure 5

Ground-state energy of ${}^7$Li for $NN$-only evolved Hamiltonians at $\lambda =2.5$, $2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. See text for further details.

Figure 6

Ground-state energy of ${}^7$Li for $NN+NNN$-induced evolved Hamiltonians at $\lambda =2.5$, $2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. The gray shadowing signifies an unreliable region; see text for further details.

Figure 7

Ground-state energy of ${}^7$Li for $NN+NNN$ evolved Hamiltonians at $\lambda =2.5$, $2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. The gray shadowing signifies an unreliable region; see text for further details.

Figure 8

Ground-state energy of ${}^{10}$B for $NN$-only evolved Hamiltonians at $\lambda =2.5$, $2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. See text for further details.

Figure 9

Ground-state energy of ${}^{10}$B for $NN+NNN$-induced Hamiltonians evolved to $\lambda =2.5$, $2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. The gray shadowing signifies an unreliable region; see text for further details.

Figure 10

Ground-state energy of ${}^{10}$B for $NN+NNN$ evolved Hamiltonians at $\lambda =2.5$, $2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. The gray shadowing signifies an unreliable region; see text for further details.

Figure 11

Ground-state energy of ${}^{12}$C for $NN$-only evolved Hamiltonians at $\lambda =2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. See text for further details.

Figure 12

Ground-state energy of ${}^{12}$C for $NN+NNN$-induced Hamiltonians evolved to $\lambda =2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. The gray shadowing signifies an unreliable region; see text for further details.

Figure 13

Ground-state energy of ${}^{12}$C for $NN+NNN$ evolved Hamiltonians at $\lambda =2.0$, $1.5$, and $1.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1]. The gray shadowing signifies an unreliable region; see text for further details.

Figure 14

Ground-state energy extrapolations of ${}^7$Li as a function of $N_{\max }$ with an N${}^3$LO $NN$ interaction [21] evolved to $\lambda =2.5{\text{fm}}^{-1}$. The symbols are the calculated points. The curves show single extrapolations using Eq. (3) with $N_{\max }=2–6$ (dashed), $4–8$ (dotted), and $6–10$ (solid) at (blue circles) $\hslash \Omega =26\text{MeV}$, which minimize the amount of extrapolation, and at (red diamonds) $\hslash \Omega =36\text{MeV}$, which minimize the numerical error estimate. The horizontal dotted and solid lines, with the band indicating the associated error bars, are the result from a constrained fit following the procedure of Ref. [1] for five $\hslash \Omega$ values from 22 to $30\text{MeV}$.

Figure 15

Ground-state energy of ${}^7$Li for $NN+NNN$ evolved Hamiltonians at $\lambda =2.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1] (solid line and points with error bars) and on the combined IR/UV correction formula Eq. (5) that yields $E_{\infty }$ (dashed line) and individual corrections for each $\hslash \Omega$ and $N_{\max }$ combination (points near the dashed line) based on the single set of best-fit parameters. The gray shadowing signifies an unreliable region (see text).

Figure 16

Ground-state energy of ${}^{12}$C for the $NN+NNN$ evolved Hamiltonians at $\lambda =2.0{\text{fm}}^{-1}$, plus extrapolations based on Extrapolation B from Ref. [1] (solid line and points with error bars) and on the combined IR/UV correction formula Eq. (5) that yields $E_{\infty }$ (dashed line) and individual corrections for each $\hslash \Omega$ and $N_{\max }$ combination (points near the dashed line) based on the single set of best-fit parameters. The gray shadowing signifies an unreliable region (see text).

Figure 17

Ground-state energy of ${}^7$Li for the $NN+NNN$ evolved Hamiltonians at $\lambda =2.0{\text{fm}}^{-1}$, with IR (vertical dashed lines) and UV (vertical dotted lines) corrections from Eq. (5) that add to predicted $E_{\infty }$ values (points near the horizontal dashed line, which is the global $E_{\infty }$). The gray shadowing signifies an unreliable region (see text).

Figure 18

Ground-state energy of ${}^{10}$B for the $NN$-only evolved Hamiltonians at $\lambda =2.0{\text{fm}}^{-1}$ for $N_{\max }=4–8$ and $\hslash \Omega =28–32\text{MeV}$ plus $N_{\max }=10$ and $\hslash \Omega =26,28\text{MeV}$ with an IR-only fit using Eq. (6). The fit value of $E_{\infty }$ is the dashed line.

Figure 19

Extrapolated ground-state energy of ${}^7$Li as a function of $\lambda$ for each SRG calculation. The initial interaction was N${}^3$LO $NN$ [21] included up to $N_{\mathrm{A}2\max }=300$ and N${}^2$LO $NNN$ [29, 39] up to $N_{\mathrm{A}3\max }=40$. The dashed curves connect data points and error bars obtained using the extrapolations described in the text. The small black arrow on the left shows the experimental value.

Figure 20

Same as Fig. 19 but for the ${}^8$Be ground state, as well as twice the ${}^4$He ground-state energy (solid curves).

Figure 21

Same as Fig. 19 but for the ${}^{10}$B ground state. The small black arrow on the left shows the experimental value.

Figure 22

Same as Fig. 19 but for the ${}^{12}$C ground state, as well as three times the ${}^4$He ground-state energy (solid curves).

Figure 23

Lowest two excited states ${}^7$Li as a function of $\hslash \Omega$ for each SRG $\lambda$ value at $N_{\max }=8$. The initial interaction was N${}^3$LO $NN$ [21] included up to $N_{\mathrm{A}2\max }=300$ and N${}^2$LO $NNN$ [29, 39] up to $N_{\mathrm{A}3\max }=40$. The small black arrows on the left show the experimental values.

Figure 24

Lowest two excited states ${}^8$Be as a function of $\hslash \Omega$ for each SRG $\lambda$ value at $N_{\max }=8$. The small black arrows on the left show the experimental values.

Figure 25

Lowest excited states ${}^{10}$B as a function of $\hslash \Omega$ for each SRG $\lambda$ value at $N_{\max }=8$. The small black arrows on the left show the experimental values.