Ab initio calculations of p-shell nuclei up to N2LO in chiral Effective Field Theory

TL;DR

This paper reports on ab-initio calculations of p-shell nuclei using chiral Effective Field Theory interactions up to N2LO, demonstrating high numerical accuracy and comparing results with experimental data.

Contribution

It provides the first comprehensive analysis of p-shell nuclei with chiral interactions up to N2LO, including three-body forces, and assesses the dependence on chiral order.

Findings

Ground state energies vary with chiral order.

Excitation spectra show good agreement with experiments.

Quantifiable uncertainties are achieved in calculations.

Abstract

Nuclear structure and reaction theory are undergoing a major renaissance with advances in many-body methods, realistic interactions with greatly improved links to Quantum Chromodynamics, the advent of high performance computing, and improved computational algorithms. State-of-the-art two- and three-nucleon interactions obtained from chiral Effective Field Theory provide a theoretical foundation for nuclear theory with controlled approximations. With highly efficient numerical codes, tuned to the current generation of supercomputers, we can perform ab-initio nuclear structure calculations for a range of nuclei to a remarkable level of numerical accuracy, with quantifiable numerical uncertainties. Here we present an overview of recent results for No-Core Configuration Interaction calculations of p-shell nuclei using these chiral interactions up to next-to-next-to-leading order, including three-body forces. We show the dependence of the ground state energies on the chiral order; we also present excitation spectra for selected nuclei and compare the results with experimental data.

Tables

Table 1. Table 1: Ground state energies of stable A = 4 𝐴 4 A=4 to 16 16 16 nuclei with χ 𝜒 \chi EFT interactions up to N 2 LO using R = 1.0 𝑅 1.0 R=1.0  fm  [ 13 , 14 ] . The uncertainty estimate is only the extrapolation uncertainty in the many-body calculation, and does not include the chiral truncation error, nor uncertainties in the LECs. Entries with an asterix ∗ indicate excited states for nuclei where the calculated and experimental ground states have different J P superscript 𝐽 𝑃 J^{P} . Experimental values are extracted from Ref.  [ 26 ] .

\br     Nucleus			LO	NLO	N2LO	N2LO including 3NFs
	$J^P$	NN-only	NN-only	NN-only	$\alpha =0.04$ fm4	$\alpha =0.08$ fm4	expt.
\mr4He	$0^{+}$	$-45.453\left(6\right)$	$-28.533\left(4\right)$	$-28.11\left(1\right)$	$-28.202\left(5\right)$	$-28.298\left(2\right)$	$-28.296$
6He	$0^{+}$	$-43.2\left(2\right)\text{0}\text{0}$	$-28.7\left(2\right)\text{0}\text{0}$	$-27.9\left(2\right)\text{0}$	$-28.55\left(15\right)$	$-28.79\left(8\right)$	$-29.27\text{0}$
6Li	$1^{+}$	$-46.7\left(1\right)$	$-31.6\left(2\right)$	$-31.0\left(2\right)$	$-31.49\left(16\right)$	$-31.72\left(6\right)$	$-31.99$
7Li	${\frac{3}{2}}^{-}$	$-57.1{\left(2\right)}^{\ast }$	$-38.7\left(3\right)$	$-38.0\left(4\right)$	$-38.72\left(16\right)$	$-38.99\left(6\right)$	$-39.24$
8He	$0^{+}$	$-39.8\left(6\right)$	$-29.7\left(5\right)$	$-27.8\left(6\right)$	$-29.5\left(3\right)$	$-29.9\left(2\right)$	$-31.41$
8Li	$2^{+}$	$-55.7\left(5\right)$	$-40.3\left(7\right)$	$-39.0\left(8\right)$	$-40.4\left(4\right)$	$-40.7\left(2\right)$	$-41.28$
8Be	$0^{+}$	$-87.7\left(4\right)$	$-56.0\left(7\right)$	$-55.4\left(9\right)$	$-55.6\left(5\right)$	$-56.1\left(3\right)$	$-56.50$
9Li	${\frac{3}{2}}^{-}$	$-57.1\left(4\right)$	$-43.9\left(7\right)$	$-41.7\left(8\right)$	$-43.9\left(4\right)$	$-44.0\left(2\right)$	$-45.34$
9Be	${\frac{3}{2}}^{-}$	$-84.7\left(7\right)$	$-58.0\left(1.4\right)$	$-56.4\left(1.5\right)$	$-57.5\left(5\right)$	$-58.0\left(3\right)$	$-58.16$
10Be	$0^{+}$	$-92.2\left(8\right)$	$-65.2\left(1.5\right)$	$-62.8\left(1.7\right)$	$-64.1\left(9\right)$	$-64.9\left(5\right)$	$-64.98$
10B	$3^{+}$	$-88.1{\left(1.2\right)}^{\ast }$	$-64.6{\left(1.5\right)}^{\ast }$	$-62.3{\left(1.7\right)}^{\ast }$	$-64.3\left(8\right)$	$-64.9\left(5\right)$	$-64.75$
10B	$1^{+}$	$-93.9\left(8\right)$	$-64.9\left(1.8\right)$	$-63.1\left(1.9\right)$	$-63.1{\left(1.0\right)}^{\ast }$	$-64.1{\left(8\right)}^{\ast }$	$-{64.03}^{\ast }$
\mr		SRG evolved to $\alpha =0.04$ fm4			$\alpha =0.04$ fm4	$\alpha =0.08$ fm4
\mr11Be	${\frac{1}{2}}^{+}$	—	—	—	$-64.7{\left(1.3\right)}^{\ast }$	$-65.4{\left(8\right)}^{\ast }$	$-65.48$
11Be	${\frac{1}{2}}^{-}$	—	—	—	$-65.8\left(1.2\right)$	$-65.7\left(8\right)$	$-{65.16}^{\ast }$
11B	${\frac{3}{2}}^{-}$	$-108.(1.)$	$-76.8\left(6\right)$	$-73.9\left(7\right)$	$-77.2\left(9\right)$	$-77.7\left(5\right)$	$-76.21$
12Be	$0^{+}$	—	—	—	$-68.9\left(1.4\right)$	$-69.8\left(9\right)$	$-68.65$
12B	$1^{+}$	$-111.{(1.)}^{\ast }$	$-82.6\left(8\right)$	$-78.6\left(8\right)$	$-81.9{\left(9\right)}^{\ast }$	$-82.5{\left(5\right)}^{\ast }$	$-79.58$
12B	$2^{+}$	$-111.{(1.)}^{\ast }$	$-82.3{\left(9\right)}^{\ast }$	$-77.8{\left(7\right)}^{\ast }$	$-82.8\left(9\right)$	$-83.2\left(5\right)$	$-{78.63}^{\ast }$
12C	$0^{+}$	$-139.(1.)$	$-95.5\left(7\right)$	$-92.7\left(6\right)$	$-94.7\left(1.0\right)$	$-95.5\left(5\right)$	$-92.16$
13B	${\frac{3}{2}}^{-}$	—	—	—	$-89.5\left(1.0\right)$	$-90.3\left(7\right)$	$-84.45$
13C	${\frac{1}{2}}^{-}$	—	—	—	$-104.7\left(1.0\right)$	$-104.4\left(4\right)$	$-97.11$
14C	$0^{+}$	—	—	—	$-116.0\left(1.3\right)$	$-116.1\left(5\right)$	$-105.28$
14N	$1^{+}$	—	—	—	$-117.3\left(1.3\right)$	$-117.4\left(4\right)$	$-104.66$
15N	${\frac{1}{2}}^{-}$	—	—	—	$-130.4\left(1.6\right)$	$-131.0\left(6\right)$	$-115.49$
16O	$0^{+}$	$-223.2\left(4\right)$	$-152.(1.)$	$-146.(1.)$	$-144.(2.)$	$-145.2\left(8\right)$	$-127.62$
\br

Equations

$$\hat{\mathbf{H}}$$

$$\hat{\mathbf{H}}\Psi ({\vec{r}}_1,\dots ,{\vec{r}}_A)$$

$$E(N_{\max })$$

Full text

1 Ab Initio Nuclear Structure and High Performance Computing

A microscopic theory for the structure and reactions of atomic nuclei poses formidable challenges for high-performance computing. A nucleus with $Z$ protons and $N$ neutrons is a self-bound quantum many-body system with $A=N+Z$ strongly interacting nucleons. The interactions feature both attractive and repulsive contributions along with significant spin and angular momentum dependence. Furthermore there are both short-range and long-range terms in the interaction, and in addition to nucleon-nucleon (NN) interactions, one also needs suitable three-nucleon forces (3NFs), and possibly even higher many-body interactions. The corresponding Hamiltonian can be written as

[TABLE]

where $m$ is the nucleon mass, which we take to be equal for protons and neutrons. The nuclear wave functions are the solutions of the many-body Schrödinger equation

[TABLE]

at discrete energy levels $E$.

In No-Core Configuration Interaction (NCCI) nuclear structure calculations [1] the wave function $\Psi$ of a nucleus consisting of $A$ nucleons is expanded in an $A$-body basis of Slater determinants $\Phi_{k}$ of single-particle wave functions $\phi_{nljm}(\vec{r})$. Here, $n$ is the radial quantum number, $l$ the orbital motion, $j$ the total spin from orbital motion coupled to the intrinsic nucleon spin, and $m$ the spin-projection. The Hamiltonian ${\bf\hat{H}}$ is also expressed in this basis and thus the many-body Schrödinger equation becomes a matrix eigenvalue problem; for $A>4$ and NN plus 3N interactions, this matrix is sparse. The eigenvalues of this matrix are approximations to the energy levels, to be compared to the experimental binding energies and spectra, and the corresponding eigenvectors to the nuclear wave functions. Although the wave functions themselves are not observable, they can be employed to evaluate additional physical observables.

Conventionally, one uses a harmonic oscillator (HO) basis with energy parameter $\hbar\omega$ for the single-particle wave functions. A convenient and efficient truncation of the complete (infinite-dimensional) basis is a truncation on the total number of HO quanta: the basis is limited to many-body basis states with $\sum_{A}N_{i}\leq N_{0}+N_{\rm max}$, with $N_{0}$ the minimal number of quanta for that nucleus and $N_{\rm max}$ the truncation parameter. (Even (odd) values of $N_{\rm max}$ provide results for natural (unnatural) parity.) Numerical convergence toward the exact results for a given Hamiltonian is obtained with increasing $N_{\rm max}$, and is marked by approximate $N_{\rm max}$ and $\hbar\omega$ independence. In practice we use extrapolations to estimate the binding energy in the complete (but infinite-dimensional) space [2, 3, 4, 5, 6], based on a series of calculations in finite bases.

The rate of convergence depends both on the nucleus and on the interaction. For realistic interactions, the dimension of the matrix needed to reach a sufficient level of convergence is in the billions, and the number of nonzero matrix elements is in the tens of trillions, which saturates available storage on current computing facilities. All NCCI calculations presented here were performed on the Cray XC30 Edison and Cray XC40 Cori at NERSC and the IBM BG/Q Mira at Argonne National Laboratory, using the code MFDn [7, 8].

2 Nuclear Interactions from Chiral Effective Field Theory

Chiral Effective Field Theory ($\chi$EFT) allows us to derive nuclear interactions (and the corresponding electroweak current operators) in a systematic way [9, 10, 11]. The chiral expansion is by no means unique: e.g. different choices for the functional form of the regulator and/or different choices for the degrees of freedom lead to different $\chi$EFT interactions. With the LENPIC collaboration [12, 13, 14] we use the same $\chi$EFT interactions for ab initio calculations ranging from nucleon-nucleon and nucleon-deuteron scattering to the structure of medium-mass nuclei. Specifically, here we use the semilocal coordinate-space regularized chiral potentials of Refs. [15, 16] to calculate the binding energies and spectra of $p$-shell nuclei. The leading order (LO) and next-to-leading order (NLO) contributions are given by NN-only potentials while 3NFs appear first at next-to-next-to-leading order (N2LO) in the chiral expansion [10, 11]. Four-nucleon forces are even more suppressed and start contributing at N3LO. The chiral power counting thus provides a natural explanation of the observed hierarchy of nuclear forces.

The Low-Energy Constants (LECs) in the NN-only potentials of Refs. [15, 16] have been fitted to nucleon-nucleon scattering, without any input from nuclei with $A>2$. The 3NFs at N2LO involve two LECs which govern the strength of the one-pion-exchange-contact term and purely contact 3NF contributions. Conventionally, these LECs are expressed in terms of two dimensionless parameters $c_{D}$ and $c_{E}$. Obviously, these LECs cannot be fixed from nucleon-nucleon scattering; they have to be fitted to select 3-body (or higher $A$-body) observables. We follow the commonly adopted practice [17, 18, 19, 20] and use the 3H binding energy as one of the observables; this gives us a correlation between $c_{D}$ and $c_{E}$.

A wide range of observables has been considered in the literature to constrain the remaining LEC. In Ref. [14] different ways to fix this LEC in the 3-nucleon sector were explored, and it was shown that it can be reliably determined from the minimum in the differential cross section in elastic nucleon-deuteron scattering at intermediate energies. This allows us to make parameter-free calculations for $A\geq 4$ nuclei. In these proceedings we present an overview of the ground state energies for all stable $p$-shell nuclei (excluding mirror nuclei), as well as excitation spectra for selected nuclei up to $A=12$, all obtained with the same semilocal regulator $R=1.0$ fm and the same LECs. Specifically, the LECs values for the 3NFs at N2LO are $c_{D}=7.2$ and $c_{E}=-0.671$, as determined in Ref. [14]. Application of these interactions to nucleon-deuteron scattering can be found in Refs. [12, 13] for NN-only potentials, along with selected properties of light- and medium-mass nuclei, and in Ref. [14] including the 3NFs at N2LO.

3 Ground State Energies for $p$-shell Nuclei

Here we present our results for the ground state energies of the stable $p$-shell nuclei, excluding mirror nuclei, all obtained with the same semilocal chiral interactions up to N2LO. In Fig. 1 we show the ground state ($J^{P}=1^{+}$) energy of 6Li as function of the HO basis parameter $\hbar\omega$ for a range of $N_{\rm max}$ values. With NN-only potentials, we can perform calculations up to $N_{\rm max}=18$ for $A=6$ nuclei. This is sufficient to achieve a reasonable level of convergence, as can be seen from the left three panels of Fig. 1. With 3NFs however, we are limited to significantly smaller bases, and in order to improve the numerical convergence with basis size we therefore first perform a Similarity Renormalization Group (SRG) transformation [21, 22, 23] on the Hamiltonian. The right-most panel of Fig. 1 shows results for the ground state energy of 6Li at N2LO including 3NFs at a very modest SRG flow parameter $\alpha=0.02$ fm4 (note that $\alpha=0$ correspond to the original Hamiltonian, without SRG), for calculations up to $N_{\rm max}=12$. Indeed, the convergence with increasing $N_{\rm max}$ is significantly improved with this SRG-evolved interaction compared to the bare NN-only interactions at NLO and N2LO. At $N_{\rm max}=12$ the level of convergence is already comparable to that of the bare NLO and N2LO potentials at $N_{\rm max}=16$. Also note that the variational minimum in $\hbar\omega$ shifts to lower values due to the SRG evolution.

In Fig. 2 we show the ground state energies of 7Li (left, $J^{P}=\frac{3}{2}^{-}$) and 10B (right, $J^{P}=3^{+}$) as function of $N_{\rm max}$ at fixed $\hbar\omega$ values close to the variational minimum with the N2LO interaction with and without explicit 3NFs. Based on these results in finite bases, we can use extrapolations to the complete (infinite-dimensional) basis. Here we use a three parameter fit at fixed $\hbar\omega$ at or just above the variational minimum

[TABLE]

which seems to work well for a range of interactions and nuclei [2, 24, 25]. The lines in Fig. 2 correspond to the extrapolating function fitted to the three highest available $N_{\rm max}$ values.

Again, with the SRG-evolved interactions the ground state energies converge more rapidly with $N_{\rm max}$ than with the bare (black dots and curves) NN-only N2LO interaction. However, as a consequence of the SRG transformation, our results do depend on the SRG flow parameter $\alpha$, because we do not incorporate any induced interactions beyond 3NFs. Without explicit 3NFs, this dependence seems to be negligible, and typically less than the extrapolation uncertainty – the bare NN-only N2LO interaction and the two SRG-evolved interaction with induced 3NFs extrapolate to approximately the same value. On the other hand, with explicit 3NFs there is a weak but noticeable dependence on the SRG parameter $\alpha$, as can be seen by the spread of the red extrapolation curves in Fig. 2. This $\alpha$ dependence is due to induced 4-body (and higher-body) interactions which we have neglected.