Importance-truncated No-core Shell Model Research Articles

Ab initio benchmarks of neutrinoless double- β decay in light nuclei with a chiral Hamiltonian

We report ab initio benchmark calculations of nuclear matrix elements (NMEs) for neutrinoless double-beta ($0\nu\beta\beta$) decays in light nuclei with mass number ranging from $A=6$ to $A=22$. We use the transition operator derived from light-Majorana neutrino exchange and evaluate the NME with three different methods: two variants of in-medium similarity renormalization group (IMSRG) and importance-truncated no-core shell model (IT-NCSM). The same two-plus-three-nucleon interaction from chiral effective field theory is employed, and both isospin-conserving ($\Delta T=0$) and isospin-changing ($\Delta T=2$) transitions are studied. We compare our resulting ground-state energies and NMEs to those of recent ab initio no-core shell model and coupled-cluster calculations, also with the same inputs. We show that the NMEs of $\Delta T=0$ transitions are in good agreement among all calculations, at the level of 10%. For $\Delta T=2$, relative deviations are more significant in some nuclei. The comparison with the exact IT-NCSM result allows us to analyze these cases in detail, and indicates the next steps towards improving the IMSRG-based approaches. The present study clearly demonstrates the power of consistent cross-checks that are made possible by ab initio methodology. This capability is crucial for providing meaningful many-body uncertainties in the NMEs for the $0\nu\beta\beta$ decays in heavier candidate nuclei, where quasi-exact benchmarks are not available.

Light neutron-rich hypernuclei from the importance-truncated no-core shell model

We explore the systematics of ground-state and excitation energies in singly-strange hypernuclei throughout the helium and lithium isotopic chains — from HeΛ5 to HeΛ11 and from LiΛ7 to LiΛ12 — in the ab initio no-core shell model with importance truncation. All calculations are based on two- and three-baryon interaction from chiral effective field theory and we employ a similarity renormalization group transformation consistently up to the three-baryon level to improve the model-space convergence. While the absolute energies of hypernuclear states show a systematic variation with the regulator cutoff of the hyperon–nucleon interaction, the resulting neutron separation energies are very stable and in good agreement with available data for both nucleonic parents and their daughter hypernuclei. We provide predictions for the neutron separation energies and the spectra of neutron-rich hypernuclei that have not yet been observed experimentally. Furthermore, we find that the neutron drip lines in the helium and lithium isotopic chains are not changed by the addition of a hyperon.

Induced Hyperon-Nucleon-Nucleon Interactions and the Hyperon Puzzle.

We present the first abinitio calculations for p-shell hypernuclei including hyperon-nucleon-nucleon (YNN) contributions induced by a similarity renormalization group transformation of the initial hyperon-nucleon interaction. The transformation including the YNN terms conserves the spectrum of the Hamiltonian while drastically improving model-space convergence of the importance-truncated no-core model, allowing a precise extraction of binding and excitation energies. Results using a hyperon-nucleon interaction at leading order in chiral effective field theory for lower- to mid-p-shell hypernuclei show a good reproduction of experimental excitation energies while hyperon separation energies are typically overestimated. The induced YNN contributions are strongly repulsive and we show that they are related to a decoupling of the Σ hyperons from the hypernuclear system, i.e., a suppression of the Λ-Σ conversion terms in the Hamiltonian. This is linked to the so-called hyperon puzzle in neutron-star physics and provides a basic mechanism for the explanation of strong ΛNN three-baryon forces.

Sensitivities and correlations of nuclear structure observables emerging from chiral interactions

Starting from a set of different two- and three-nucleon interactions from chiral effective field theory, we use the importance-truncated no-core shell model for ab initio calculations of excitation energies as well as electric quadrupole (E2) and magnetic dipole (M1) moments and transition strengths for selected p-shell nuclei. We explore the sensitivity of the excitation energies to the chiral interactions as a first step towards and systematic uncertainty propagation from chiral inputs to nuclear structure observables. The uncertainty band spanned by the different chiral interactions is typically in agreement with experimental excitation energies, but we also identify observables with notable discrepancies beyond the theoretical uncertainty that reveal insufficiencies in the chiral interactions. For electromagnetic observables we identify correlations among pairs of E2 or M1 observables based on the ab initio calculations for the different interactions. We find extremely robust correlations for E2 observables and illustrate how these correlations can be used to predict one observable based on an experimental datum for the second observable. In this way we circumvent convergence issues and arrive at far more accurate results than any direct ab initio calculation. A prime example for this approach is the quadrupole moment of the first 2^+ state in C-12, which is predicted with an drastically improved accuracy.

Lifetime measurements ofC17excited states and three-body and continuum effects

Author(s): Smalley, D; Iwasaki, H; Navratil, P; Roth, R; Langhammer, J; Bader, VM; Bazin, D; Berryman, JS; Campbell, CM; Dohet-Eraly, J; Fallon, P; Gade, A; Langer, C; Lemasson, A; Loelius, C; Macchiavelli, AO; Morse, C; Parker, J; Quaglioni, S; Recchia, F; Stroberg, SR; Weisshaar, D; Whitmore, K; Wimmer, K | Abstract: We studied transition rates for the lowest 1/2+ and 5/2+ excited states of C17 through lifetime measurements with the GRETINA array using the recoil-distance method. The present measurements provide a model-independent determination of transition strengths giving the values of B(M1;1/2+→3/2g.s.+)=1.04-0.12+0.03×10-2μN2 and B(M1;5/2+→3/2g.s.+)=7.12-0.96+1.27×10-2μN2. The quenched M1 transition strength for the 1/2+→3/2g.s.+ transition, with respect to the 5/2+→3/2g.s.+ transition, has been confirmed with greater precision. The current data are compared to importance-truncated no-core shell model calculations addressing effects due to continuum and three-body forces.

Ab Initio calculations of even oxygen isotopes with chiral two-plus-three-nucleon interactions.

We formulate the in-medium similarity renormalization group (IM-SRG) for open-shell nuclei using a multireference formalism based on a generalized Wick theorem introduced in quantum chemistry. The resulting multireference IM-SRG (MR-IM-SRG) is used to perform the first ab initio study of all even oxygen isotopes with chiral nucleon-nucleon and three-nucleon interactions, from the proton to the neutron drip lines. We obtain an excellent reproduction of experimental ground-state energies with quantified uncertainties, which is validated by results from the importance-truncated no-core shell model and the coupled cluster method. The agreement between conceptually different many-body approaches and experiment highlights the predictive power of current chiral two- and three-nucleon interactions, and establishes the MR-IM-SRG as a promising new tool for ab initio calculations of medium-mass nuclei far from shell closures.

Extrapolation uncertainties in the importance-truncated no-core shell model

Background: The importance-truncated no-core shell model (IT-NCSM) has recently been shown to extend theoretical nuclear structure calculations of $p$-shell nuclei to larger model (${N}_{\mathrm{max}}$) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a ``large'' ${N}_{\mathrm{max}}$ basis space, thus making the calculation tractable and reasonably quick to perform. Initial results indicate that the procedure agrees well with the NCSM, in which a complete basis is constructed for a given ${N}_{\mathrm{max}}$.Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete ${N}_{\mathrm{max}}$ space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete ${N}_{\mathrm{max}}$ space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to ${N}_{\mathrm{max}}=14$. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to ${N}_{\mathrm{max}}=\ensuremath{\infty}$ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for ${N}_{\mathrm{max}}=\ensuremath{\infty}$ extrapolations.Method: We report on ${}^{6}\mathrm{Li}$ calculations performed with the IT-NCSM and compare them to full NCSM calculations. We employ the Entem and Machleidt chiral two-body next-to-next-to-next leading order (N3LO) interaction (regulated at 500 MeV/$c$), which has been modified to a phase-shift equivalent potential by the similarity renormalization group (SRG) procedure. We investigate the dependence of the procedure on the technique employed to extrapolate to the complete ${N}_{\mathrm{max}}$ space, the harmonic oscillator energy ($\ensuremath{\hbar}\ensuremath{\Omega}$), and investigate the dependence on the momentum-decoupling scale ($\ensuremath{\lambda}$) used in the SRG. We also investigate the use of one or several reference states from which the truncated basis is constructed.Results: We find that the uncertainties generated from various extrapolating functions used to extrapolate to the complete ${N}_{\mathrm{max}}$ space increase as ${N}_{\mathrm{max}}$ increases. The extrapolation uncertainties range from a few keV for the smallest ${N}_{\mathrm{max}}$ spaces to about 50 keV for the largest ${N}_{\mathrm{max}}$ spaces. We note that the difference between extrapolated IT-NCSM and NCSM ground-state energies, however, can be as large as 100--250 keV depending on the chosen harmonic oscillator energy ($\ensuremath{\hbar}\ensuremath{\Omega}$). IT-NCSM performs equally well for various SRG momentum-decoupling scales, $\ensuremath{\lambda}=2.02$ fm${}^{\ensuremath{-}1}$ and $\ensuremath{\lambda}=1.50$ fm${}^{\ensuremath{-}1}$.Conclusions: In the case of ${}^{6}$Li, when using the softened chiral nucleon-nucleon N3LO interaction, we have determined the difference between extrapolated ${N}_{\mathrm{max}}=\ensuremath{\infty}$ IT-NCSM and full NCSM calculations to be about 100--300 keV. As $\ensuremath{\hbar}\ensuremath{\Omega}$ increases, we find that the agreement with NCSM deteriorates, indicating that the procedure used to choose the basis states in IT-NCSM depends on $\ensuremath{\hbar}\ensuremath{\Omega}$. We also find that using multiple reference states leads to a better ground-state description than using only a single reference state.

In-medium similarity renormalization group with chiral two- plus three-nucleon interactions

We use the recently proposed in-medium similarity renormalization group (IM-SRG) to carry out a systematic study of closed-shell nuclei up to ${}^{56}\mathrm{Ni}$, based on chiral two- plus three-nucleon interactions. We analyze the capabilities of the IM-SRG by comparing our results for the ground-state energy to coupled cluster calculations, as well as to quasiexact results from the importance-truncated no-core shell model. Using chiral two- plus three-nucleon Hamiltonians whose resolution scales are lowered by free-space SRG evolution, we obtain good agreement with experimental binding energies in ${}^{4}\mathrm{He}$ and the closed-shell oxygen isotopes, while the calcium and nickel isotopes are somewhat overbound.

New Horizons in Ab Initio Nuclear Structure Theory

Nuclear interactions derived within chiral effective field theory enable nuclear structure and reaction calculations solidly rooted in QCD. Using chiral two- and three-body interactions in ab initio studies of a variety of nuclear observables is, both, promising and challenging. Particularly the inclusion of chiral 3N interactions into exact and approximate many-body calculations is demanding and computationally expensive. We present recent key developments that facilitate ab initio calculations of ground and low-lying excited states of p- and sd-shell nuclei with full 3N interactions in the Importance-Truncated No-Core Shell Model using consistent Similarity Renormalization Group transformations of the NN+3N Hamiltonian. The treatment of 3N interactions without approximations allows for first studies of the dependence of nuclear-structure observables on the details of chiral interactions. Moreover, we present results for heavy nuclei obtained in Coupled Cluster calculations using a normal-ordered two-body approximation of the 3N interaction.

Ab Initio Nuclear Structure Theory with Chiral NN+3N Interactions

Low-energy nuclear theory has entered an era of ab initio nuclear structure and reaction calculations based on QCD. One of the most promising paths from QCD to nuclear observables employs Hamiltonians constructed within chiral effective field theory as starting point for precise ab initio studies. However, the full inclusion of chiral two- plus three-nucleon (NN+3N) interactions in exact and approximate many-body calculations poses a formidable challenge. We discuss recent breakthroughs that allow for ab initio calculations for ground states and spectra of nuclei throughout the p- and the lower sd-shell with full 3N interactions using consistent Similarity Renormalization Group (SRG) transformations and the Importance Truncated No-Core Shell Model (IT-NCSM). Moreover, we discuss extensions of these ab initio calculations to heavy nuclei within coupled cluster theory using a normal ordering approximation of the 3N interaction.

Center-of-mass problem in truncated configuration interaction and coupled-cluster calculations

The problem of center-of-mass (CM) contaminations in ab initio nuclear structure calculations using configuration interaction (CI) and coupled-cluster (CC) approaches is analyzed. A rigorous and quantitative scheme for diagnosing the CM contamination of intrinsic observables is proposed and applied to ground-state calculations for 4He and 16O. The CI and CC calculations for 16O based on model spaces defined via a truncation of the single-particle basis lead to sizable CM contaminations, while the importance-truncated no-core shell model based on the NmaxℏΩ space is virtually free of CM contaminations.

Roth and Navrátil Reply:

In their comment on our recent Letter [1] Dean et al. [2] criticize the calculations for the ground-state energy of {sup 40}Ca within the importance truncated no-core shell model (NCSM). In particular they address the role of configurations beyond the 3p3h level, which have not been included in the {sup 40}Ca calculations for large N{sub max} {h_bar}{Omega} model spaces. Before responding to this point, the following general statements are in order. For the atomic nucleus as a self-bound system, translational invariance is an important symmetry. The only possibility to preserve translational invariance when working with a Slater determinant basis is to use the harmonic oscillator (HO) basis in conjunction with a basis truncation according to the total HO excitation energy, i.e. N{sub max} {h_bar}{Omega}, as done in the ab initio NCSM. This is important not only for obtaining proper binding or excitation energies, but also for a correct extraction of physical wavefunctions. The spurious center-of-mass components can be exactly removed only if the HO basis and the N{sub max} {h_bar}{Omega} truncation are employed. The minimal violation of the translational invariance was one of the main motivations for developing the importance-truncation scheme introduced in the Letter. In this scheme, we startmore » with the complete N{sub max} {h_bar}{Omega} HO basis space and select important configurations via perturbation theory. All symmetries are under control and our importance-truncated NCSM calculations are completely variational and provide an upper bound of the ground-state energy of the system. The restriction to the 3p3h level, made for computational reasons in the N{sub max} > 8 calculations for {sup 40}Ca, is not inherent to the importance truncation scheme. The explicit inclusion of 4p4h configurations--though computationally more demanding--is straight-forward, even for the largest N{sub max} {h_bar}{Omega} model spaces discussed. To demonstrate this fact we have performed full 4p4h calculations for {sup 40}Ca in a 14{h_bar}{Omega} no-core model space at {h_bar}{Omega} = 24 MeV using the V{sub low k} interaction employed in the Letter. The resulting ground state energy of E{sub 4p4h} = -471.0 MeV can be compared with E{sub 3p3h} = -461.2 MeV for the 3p3h calculation reported in Fig. 5(b) with an uncertainty of typically 1 MeV due to the extrapolation {kappa}{sub min} {yields} 0. Thus the 4p4h configurations change the resulting ground-state energy of {sup 40}Ca by approximately 2%. In addition to the explicit inclusion of 4p4h or higher-order configurations in the importance truncated space, there are various other means to account for their effect on the energy. We are presently developing a perturbative method for the inclusion of up to 6p6h configurations using the ground state obtained from the NCSM diagonalization as unperturbed state. Furthermore, there are even simpler methods, such as multireference Davidson corrections, which are employed successfully in quantum chemistry [3], for estimating the contribution of excluded configurations to the energy. They even restore size extensivity in an approximate way. The coupled-cluster method (CCM) used by the authors of the Comment lacks the above discussed features important for the nuclear many-body problem, in particular it violates translational invariance from the very beginning and it does not fulfill the variational principle. The problem of a spurious center-of-mass contamination of the many-body states in CCM is not resolved and often not even mentioned (see e.g. Ref. [4]). Moreover, non-iterative triples corrections like CCSD(T), as referred to in the comment, tend to overestimate the correlation energy or even collapse. It is claimed that the CCM is very accurate. A closer inspection of recently published nuclear many-body results does not support this statement. In Ref. [5], the {sup 4}He binding energy with the chiral N{sup 3}LO NN potential was determined with an uncertainty of several MeV. The same is obtained in the ab initio NCSM with accuracy of 10 keV [6, 7]. The CCM binding energy results for {sup 16}O with the identical chiral N{sup 3}LO NN potential obtained in Refs. [8] and [9] differ by more than 5 MeV. According to Ref. [9], the CD-Bonn NN potential overbinds {sup 16}O. This is contrary to results obtained in Ref. [10] and also contrary to expectations as the CD-Bonn underbinds lighter nuclei. The CCSD(T) ground state energy for {sup 40}Ca with V{sub low k} reported in Ref. [4] is about 30 MeV lower than our 4p4h-result (CCSD is about 20MeV lower). Taking into account the small 2% difference of our 3p3h and 4p4h results and the violation of the variational principle and the translational invariance in the CCM method, we believe that the CCM result overestimates the exact ground-state energy in this case.« less