Moving away from singly-magic nuclei with Gorkov Green's function theory
EEPJ manuscript No. (will be inserted by the editor)
Moving away from singly-magic nuclei with Gorkov Green’sfunction theory
V. Som`a , C. Barbieri , T. Duguet , , and P. Navr´atil IRFU, CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom KU Leuven, Institut voor Kern- en Stralingsfysica, 3001 Leuven, Belgium TRIUMF, 4004 Westbrook Mall, Vancouver, BC, V6T 2A3, Canada
Abstract.
Ab initio calculations of bulk nuclear properties (ground-state energies, root mean square chargeradii and charge density distributions) are presented for seven complete isotopic chains around calcium, fromargon to chromium. Calculations are performed within the Gorkov self-consistent Green’s function approachat second order and make use of two state-of-the-art two- plus three-nucleon Hamiltonians, NN +3 N (lnl)and NNLO sat . An overall good agreement with available experimental data is found, in particular fordifferential energies (charge radii) when the former (latter) interaction is employed. Remarkably, neutronmagic numbers N = 28 , ,
34 emerge and evolve following experimental trends. In contrast, pairing gapsare systematically underestimated. General features of the isotopic dependence of charge radii are alsoreproduced, as well as charge density distributions. A deterioration of the theoretical description is observedfor certain nuclei and ascribed to the inefficient account of (static) quadrupole correlation in the presentmany-body truncation scheme. In order to resolve these limitations, we advocate the extension of theformalism towards incorporating breaking of rotational symmetry or, alternatively, performing a stochasticsampling of the self-energy.
A leap forward in ab initio calculations of atomic nucleioccurred about 15 years ago with the (re)introduction, innuclear structure theory, of so-called correlation expansion methods [1, 2]. As opposed to virtually exact approaches,which do not impose any formal approximation on thesolution of the many-body Schr¨odinger equation and scaleexponentially or factorially with the system size, corre-lation expansion techniques achieve a polynomial scalingat the price of an approximate, yet controlled and sys-tematically improvable, solution. Combined with the avail-ability of “softer” Hamiltonians, obtained via similarityrenormalisation group (SRG) transformations [3], such afavourable scaling progressively enabled the extension offirst-principle calculations beyond the region of light nucleitraditionally targeted by ab initio practitioners. Nowadays,systems up to mass number A ∼
70 can be routinely ac-cessed [4, 5, 6, 7, 8], with a few attempts reaching out toneutron-deficient tin ( A ∼ A ∼ A (cid:46) a r X i v : . [ nu c l - t h ] S e p V. Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory
A third route, followed here, relies on the use of areference product state breaking one or several symmetriesof the underlying Hamiltonian. In doing so, one can tradethe degeneracy with respect to particle-hole excitationscharacterising open-shell systems for a degeneracy withrespect to transformations of the associated symmetrygroup. As a result, the particle-hole degeneracy is lifted anda well-defined many-body expansion on top of a “deformed”reference product state can be designed. This trade-offallows one to access open-shell systems while maintaininga polynomial cost and the intrinsic simplicity of single-reference expansion methods. The handling of the pseudoGoldstone mode associated with the manifold of degeneratestates, necessary to restore the broken symmetry, can besafely postponed to a later stage [21, 22, 23].Largely employed in the context of nuclear energy den-sity functional [24], this approach was imported in abinitio nuclear structure about a decade ago. First, Gorkovself-consistent Green’s function (GSCGF) theory was de-veloped [25]. Few years later, coupled cluster theory wasextended to the use of a Bogolyubov reference state [26].More recently, Bogolyubov many-body perturbation theory(BMBPT) was introduced as a generalisation of standardMøller-Plesset theory [27, 28]. All these techniques relysolely on the breaking of the U(1) symmetry related toparticle-number conservation and are thus designed to effi-ciently account for static pairing correlations. In order todeal with the other source of strong correlations in nuclei,i.e. the quadrupole correlations typically associated withnuclear deformation, one would need to correspondinglybreak rotational SU(2) symmetry. Although work in thisdirection is in progress (see e.g. [29, 30]), the latter fea-ture is currently unavailable in nearly all state-of-the-artimplementations. As a consequence, the above methodsare preferentially applied to singly open-shell (i.e., semi-magic) nuclei, where the role of quadrupole correlationsis not predominant. Indeed, GSCGF and BMBPT havesuccessfully addressed complete semi-magic isotopic chains,e.g. oxygen, calcium or nickel [31, 32, 33, 27, 8, 28]. Thelimits of applicability of U(1)-breaking, SU(2)-conservingcorrelation expansion methods, however, have never beensystematically probed. Therefore, it is worthwhile to pushsuch calculations away from semi-magic nuclei in order toempirically identify if and where such a strategy eventuallyfails, i.e. the point beyond which an explicit breaking ofSU(2) symmetry will become mandatory.Recently, specific medium-mass doubly open-shell sys-tems, e.g. some titanium [34] or sulfur and argon [35, 36]isotopes, have been computed within the GSCGF approach.In this paper, we extend these works to a systematic studyof several isotopic chains around semi-magic calcium forwhich results were not available before. In particular, wecompute ground-state energies, charge radii and selectedcharge density distributions for chains ranging from argon( Z = 18) to chromium ( Z = 24) and compare to avail-able experimental data. Calculations were performed usingthe recently introduced N N +3 N (lnl) Hamiltonian [8]. Forcharge radii and densities, additionally, the NNLO sat [37] Hamiltonian was employed. Overall, the goals of the presentstudy can be summarised as follows:1. Assess the performance of state-of-the-art ab initio cal-culations on bulk properties of medium-mass nuclei.
In this respect, the present work follows up on theresults of Ref. [8], in which the novel
N N +3 N (lnl)interaction was benchmarked on semi-magic oxygen,calcium and nickel isotopes. Here it is shown that theglobal satisfactory agreement with experimental datafound in Ref. [8] extends to doubly open-shell isotopesaround calcium. Remarkably, neutron magic numbers N = 28 , ,
34 emerge and evolve following experimen-tal trends. While the neutron dripline is not addressedhere, the proton dripline is found at or near the ex-perimental one. As already remarked in Ref. [8] forcalcium, charge radii computed with
N N +3 N (lnl) aretoo small compared to the experimental values. In con-trast, NNLO sat provides a good overall description ofexisting data. Nevertheless, even our best calculationsfail to reproduce some finer details, e.g. the steep risebetween N = 28 and N = 32 and the parabolic-likebehaviour between N = 20 and N = 28. The latter canbe in part ascribed to many-body truncations. Inter-estingly, for both interactions, a second, smaller, kinkis observed at N = 34.2. Analyse pairing properties in nuclei within a first-principle description.
The ability of accessing ground-state energies of odd-even nuclei enables the investigation of pairing effectse.g. by considering three-point mass differences in even- Z isotopic chains. The resulting pairing strength turnsout to be underestimated compared to experimentalobservations, which possibly points to missing many-body correlations.3. Probe the limits of SU(2)-conserving correlation expan-sion methods in the description of doubly open-shellnuclei.
It is observed that the description of experimental datadeteriorates for certain sets of nuclei away from singly-magic calcium. It is conjectured that this might signalthe onset of significant quadrupole correlations, i.e.static deformation. A careful scrutiny indeed reveals acorrelation between the inaccuracy of the results (quan-tified in terms of deviation from experimental data)and an estimate of the deformation.Developing the above points, the manuscript is organised asfollows. First, the theoretical and computational scheme isbriefly recalled in Sec. 2. Section 3 is devoted to the study ofground-state energies, in the form of either total (Sec. 3.1)or differential (Secs. 3.2 and 3.3) binding energies. Further,a discussion of three-point mass differences is presented inSec. 3.4. The impact of (expected) nuclear deformation oncalculated ground-state energies is investigated in Sec. 3.5.Finally, a systematic survey of nuclear radii and a selectionof representative charge density distributions are presentedin Sec. 4. Conclusions and perspectives follow in Sec. 5. . Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory 3
All calculations presented here were performed within theGorkov self-consistent Green’s function approach at secondorder in the algebraic diagrammatic construction expan-sion [ADC(2)] [25, 38]. An extensive study of oxygen, cal-cium and nickel isotopes has been recently carried out inthe same computational scheme and published in Ref. [8].Hence, only the most salient features are recalled here andthe reader is referred to [8] for more computational andtechnical details.Two different two- plus three-nucleon (2N+3N) Hamil-tonians were employed in the present study. The first one,labelled
N N +3 N (lnl), is based on the next-to-next-to-next-to-leading order (N LO) nucleon-nucleon potentialfrom Entem and Machleidt [39, 40] complemented with theN LO 3 N interaction for which a combination of local andnonlocal regulators is used [8]. Low-energy constants werefitted to A = 2 , , λ = 2 fm − . Thesecond one, labelled NNLO sat , was introduced in Ref. [37]with the explicit goal of providing an improved descriptionof saturation properties. Here, in contrast to N N +3 N (lnl),low-energy constants were simultaneously fitted to few-body systems as well as selected ground-state energies andradii of carbon and oxygen isotopes. This Hamiltonian isSRG-unevolved.Three-nucleon forces are treated following the formal-ism developed in Ref. [41]. In practice, the three-bodyHamilton operator is self-consistently convoluted with thecorrelated one-body density matrix and contributes toone- and two-body effective interactions [42]. The contri-butions resulting from contracting two- and many-bodydensity matrices were seen to be negligible for our pur-poses [43, 44]. Note that we discard interaction-irreduciblediagrams containing three-body vertices. The formalismneeded to include these at the ADC(3) level was presentedin Ref. [45] and their contribution is estimated to be compa-rable, in terms of both importance and required computingresources, to ADC(5) computations with only two-nucleoninteractions. The procedure used in this paper generatesan A -dependent symmetry-conserving Hamiltonian thatcan be viewed as a generalisation of the particle-number-conserving normal-ordered two-body approximation dis-cussed in Ref. [46].As for (many-body) operators, they are expanded ona one-body harmonic oscillator (HO) basis (or productsthereof) including states up to e max ≡ max (2 n + l ) =13. Three-body operators are further restricted to three-body basis states characterised by e = 16 < e max .For some representative isotopes, a variation of the HOfrequency (cid:126) ω was performed in order to locate the opti-mal value for the binding energy and radius, which areobservables considered in this work. Based on this analysis,we performed calculations using (cid:126) ω = 18 MeV for bothenergies and radii for N N +3 N (lnl), and (cid:126) ω = 20 MeVand (cid:126) ω = 14 MeV for energies and radii respectively forNNLO sat . All results presented here were obtained withthese model space parameters, unless otherwise stated. In Ref. [8] theoretical uncertainties associated to model-space and many-body truncations were investigated forcalcium isotopes. For total ground-state energies, such un-certainties were estimated to be respectively 0.5% and 2%(1% and 3%) for N N +3 N (lnl) (NNLO sat ). For charge radii,a combined (model space plus many-body) uncertainty of0.1 fm was assumed for both interactions. A refined uncer-tainty analysis for charge radii is presented in Sec. 4. Evenif the estimates on many-body truncation will have to becorroborated by explicit GSCGF-ADC(3) calculations inthe future, one can suppose that the above values givea reasonable indication also for the (doubly) open-shellnuclei considered here. Let us start by analysing total ground-state energies alongthe seven isotopic chains studied in this work, i.e. argon( Z = 18), potassium ( Z = 19), calcium ( Z = 20), scan-dium ( Z = 21), titanium ( Z = 22), vanadium ( Z = 23)and chromium ( Z = 24). The current implementation ofGSCGF theory is based on the assumption that J Π = 0 + for targeted ground states and is therefore well suited for
14 16 18 20 22 24 26 28 30 32 34 36 38 40 42-600-550-500-450-400-350-300-250-200 N E [ M e V ] ArScTiCaKVCr
Full symbols: experimental data
Empty symbols: extrapolated data
Symbols + line: theory
Fig. 1.
Total binding energies along Z = 18 −
24 isotopicchains computed at the ADC(2) level with the NN +3 N (lnl)interaction (symbols joined by solid lines). For comparison,experimental data (measured [47, 34, 48, 49, 36], full symbolsand extrapolated [47], empty symbols) are displayed. Bothcalculated and experimental values are shifted by (20 − Z ) × even-even nuclei. The ground-state energy of odd-evensystems can be computed via [50] E A odd-even = ˜ E A + ω , (1)where ˜ E A is the ground-state energy of the odd-even nu-cleus computed as if it had J Π = 0 + , i.e. as a fully pairedeven-number-parity state forced to have the right odd num-ber of particles on average, and ω is the lowest one-nucleonseparation energies in the latter calculation. Further detailscan be found in Refs. [25, 38]. A more direct but similarapproach is to use the addition and separation energiesencoded in the spectral function but to recompute theeven-even isotope with the center of mass corrections for A ±
1, as done in Ref. [42]. As a result, one can access theground-state energy of all isotopes with even Z and thatof odd-even isotopes with odd Z . Other observables, e.g.radii or densities, are instead available only for even-evensystems. Further developments, e.g. involving the use ofHellmann-Feynman theorem, are needed to extend theircalculation to odd-even systems.Computed ground-state energies are presented in Fig. 1and compared to experimental (measured and extrapo-lated) data. The global behaviour is well captured by thecalculated energies across all values of Z and N , althoughunderbinding with respect to experiment is observed for allchains. The deviation per nucleon is roughly of the samemagnitude for all nuclei (as also visible from Fig. 8, dis-cussed below). This points to a global effect like e.g. missingspecific many-body correlations. Indeed, ADC(3) results forcalcium isotopes [8], displayed in Fig. 1 as horizontal bars,show that excellent agreement with experimental valuesis reached once a more refined truncation schemes is used.These findings confirm the good performance achieved bythe N N +3 N (lnl) Hamiltonian for ground-state energiesin this mass region [8]. Systematically accessing successive nuclides along isotopicor isotonic chains allows to investigate some of the mostfundamental properties of atomic nuclei such as the limitsof their existence as bound states or the emergence (andevolution) of magic numbers. Such properties are beststudied by looking at total ground-state energy differences.Two-neutron separation energies S ( N, Z ) ≡ | E ( N, Z ) | − | E ( N − , Z ) | (2)are first considered. Their values computed from the to-tal energies of Fig. 1 are shown in Fig. 2, together withavailable and extrapolated experimental data. The overallagreement with experiment is remarkable, with computedvalues following the main trends of measured data. Thetwo neutron magic numbers N = 20 and N = 28, associ-ated with sudden drops of S , are visible in all theoreticalcurves. The N = 28 gap is very well reproduced across allisotopic chains, with the good description carrying over tolarger neutron numbers for most chains. On the contrary,
16 18 20 22 24 26 28 30 32 34 36 38 40-100102030405060 N S [ M e V ] ArScTiCaKVCr
Symbols + line: theory
Full symbols: experimental data
Empty symbols: extrapolated data
Fig. 2.
Two-neutron separation energies along Z = 18 −
24 iso-topic chains computed with the NN +3 N (lnl) interaction (sym-bols joined by solid lines), compared to experimental (measured,full symbols and extrapolated, empty symbols) data. Both cal-culated and experimental values are shifted by ( Z − × the gap at N = 20 turns out to be overestimated, withthe comparison to experiment worsening when departingfrom proton magic number Z = 20. The description deteri-orates also in other regions, e.g. for argon isotopes between N = 20 and N = 28 or more generally for chromium iso-topes. In such systems both protons and neutrons have anopen-shell character. The absence of a closed shell is likelyto induce strong quadrupole correlations that are difficultto capture in the present calculations, based on expandingover a spherical reference state.The neutron dripline, i.e. the position of the last boundsystem in a given isotopic chain, can be also read fromtwo-neutron separation energies as unbound nuclei are char-acterised by negative values of S . None of the computedneutron rich isotopes shown in Fig. 2 results unbound, i.e.the dripline is predicted to be located beyond N = 40 forall considered chains . The smallest S value are reachedfor − Ar and are as low as 100 keV. However, one mustremark that continuum coupling is likely to play an impor-tant role when binding energies are so close to the neutronemission threshold. Presently, the continuum is crudely in-cluded via the discretised harmonic oscillator basis, whichdoes not ensure correct asymptotic properties. In futurestudies, in order to reliably determine the position of the Present calculations could not be extended beyond N = 40due to convergence issues, see discussion in Ref. [8] for moredetails.. Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory 5 neutron dripline, particular care will have to be devotedto a more proper treatment of this aspect.The coupling to the particle continuum plays a lesserrole around the proton dripline because of Coulomb repul-sion. Given that present calculations span several neigh-bouring chains, the proton dripline can be investigatedwithin this theoretical setting. Here, the key quantities areone-proton and two-proton separation energies, definedrespectively as S ( N, Z ) ≡ | E ( N, Z ) | − | E ( N, Z − | (3)and S ( N, Z ) ≡ | E ( N, Z ) | − | E ( N, Z − | . (4)For a given element, the most proton-rich isotope for whichboth S > S > S and S are displayed as a function of neutron numberfor the isotopic chains considered in this study . Experi- For potassium only S can be computed, while for argonnone of the two separation energies is available in the presentcalculations.
14 18 22
N(f) Cr Exp . Theory S S S S (a) K [ M e V ] Ca (b) Sc (c)
14 18 22051015 Ti (d) N [ M e V ]
14 18 22 V (e) N Fig. 3.
One- and two-proton separation energies displayedas a function of neutron number for different Z . Calculationsperformed with the NN +3 N (lnl) interaction (symbols joinedby solid lines) are compared to existing data (measured, fullsymbols and extrapolated, empty symbols, all joined by dashedlines). The solid coloured (dashed black) arrows at the topof each panel mark the computed (experimental) driplines. Insome cases (K, Ca, Sc, V) the theoretical dripline can not bedetermined unambiguously from the calculations, hence thetwo possible values are shown.
20 21 22 23 24-15-10-5051015 Z S [ M e V ] Full symbols: experimental data
Empty symbols: extrapolated data
Symbols + solid line: theory
N=20N=18N=16N=14
Fig. 4.
Two-proton separation energies displayed as a functionof proton number for different isotonic chains. Calculationsperformed with the NN +3 N (lnl) interaction (symbols joinedby solid lines) are compared to existing data (measured, fullsymbols and extrapolated, empty symbols, all joined by dottedlines). mentally, for these elements, the proton dripline has beendetermined up to vanadium, with the last bound isotopesbeing K, Ca, Sc, Ti and V. For chromium, the lastknown isotope is Cr. Theoretical curves generally followthe experimental trends yielding an overall correct quali-tative description of both S and S . Looking more indetail, one observes that calculations tend to overestimatethe measured separation energies in potassium and calcium,provide an excellent reproduction of scandium isotopes andunderestimate titanium, vanadium and chromium. As aresult, the position of the proton dripline is found at toosmall N (with a difference of two or three neutrons) forthe first two elements. In scandium, as well as vanadium,the dripline is correctly determined at N = 19 and N = 20respectively. In titanium and chromium, it is also found re-spectively at N = 19 and N = 20, in this case one neutronaway from what observed experimentally.The cause of this small discrepancy can be tracedback to the poor reproduction of the Z = 20 gap bythe N N +3 N (lnl) Hamiltonian, as evident in Fig. 4. Here,two-proton separation energies are plotted as a function ofproton number for different isotonic chains. One noticesthat, similarly to what observed in Fig. 2 for N = 20, the Z = 20 gap is overestimated by at least 5 MeV in all con-sidered isotones. The disagreement becomes more severefor low neutron numbers, which impacts the determina-tion of the proton dripline in lighter isotopes. In spite ofthese shortcomings, this detailed analysis proves the overallremarkable quality of present ab initio calculations, not Experimentally, the dripline is typically established bymeans of a void observation of one or several isotopes ratherthan by determining a negative value of S or S . V. Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory
16 18 20 22 24 26 28 30 32 34 36 38-20-100102030405060 N Δ S [ M e V ] ArScTiVCaKCr
Full symbols: experimental data
Empty symbols: extrapolated data
Symbols + line: theory
Fig. 5.
Two-neutron shell gaps along Z = 18 −
24 isotopicchains computed with the NN +3 N (lnl) interaction (symbolsjoined by solid lines), compared to experimental (measured,full symbols and extrapolated, empty symbols) data. Boththeoretical and experimental values are shifted by ( Z − × dissimilar from what emerges from the systematic studyreported in Ref. [51]. A finer insight regarding the magic character of specificneutron numbers can be gained by looking at so-calledtwo-neutron shell gaps, defined as ∆ ( N, Z ) ≡ S ( N, Z ) − S ( N + 2 , Z ) (5)and displayed in Fig. 5. As for the S , one first noticesan overall excellent agreement with experiment, with theclear exception of the N = 20 peak and its vicinity. Whilein semi-magic calcium isotopes calculations only fail toreproduce the height of the peak, experimental data forother isotopes show a displacement of the peak, linked to apossible disappearance of the N = 20 magic number, whichis not reproduced by the present calculations. In contrast,the N = 28 peak is very well reproduced up to Z = 22,with the description only slightly deteriorating for Z = 23and Z = 24. The emergence of the N = 32 subclosureis nicely visible in lighter elements, as well as the one at N = 34 in argon, potassium and calcium. When goingtowards higher proton number their evolution is poorly
18 19 20 21 22 23 24 02468 Z N=34(d)
18 19 20 21 22 23 2402468 Z Δ S [ M e V ] N=32(c)
TheoryExtrap. dataExp. data
N=30(b) Δ S [ M e V ] N=28(a)
Fig. 6.
Two-neutron shell gaps along four isotonic chains com-puted with the NN +3 N (lnl) interaction (circles), compared toexperimental (measured, full squares, and extrapolated, emptysquares) data. Results for N = 28 , ,
32 and 34 are shown inpanels (a), (b), (c) and (d) respectively. described starting with N = 34 in scandium and N = 32 invanadium. The behaviour becomes even more inconsistentfor chromium. Again, this might signal the importance ofcertain ingredients (e.g. quadrupole correlations) that aremissing in the present theoretical framework.In spite of these deficiencies, remarkably, all magic num-bers as well as their qualitative evolution emerge “fromfirst principles”, i.e. starting solely from inter-nucleon in-teractions whose coupling constants have been adjustedonly in few-body systems. Let us stress that, indeed, noad hoc information about the magic character of theseisotopes is inserted at any stage of the calculation. Theemergence of this feature can be better appreciated inFig. 6 where neutron gaps are compared to experimental(measured and extrapolated) data along N = 28 , ,
32 and34 isotonic chains. While there is room for improvementin Z = 22 , ,
24 isotones for reasons discussed above, theoverall description is very reasonable. In addition, calcula-tions of the N = 28 gaps were recently extended down tochlorine and sulfur [36] where an excellent agreement withnovel precision mass measurement was also found. One of the longstanding challenges in low-energy nuclearphysics relates to the microscopic description of nuclearsuperfluidity [52]. The microscopic origin of nucleonic pair-ing, i.e. how it originates in the context of a first-principlecalculation and the role played by different types of many-body correlations, remains to be elucidated [53]. A fun-damental, yet unresolved, question relates to how much . Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory 7
14 16 18 20 22 24 26 28 30 32 34 36 38 40-5051015 N Δ ( ) [ M e V ] ArTiCaCr
Full symbols + dotted line: experimental data
Empty symbols + dotted line: extrapolated data
Symbols + solid line: theory
Fig. 7.
Three-point mass differences along Z = 18 , ,
22 and24 isotopic chains computed with the NN +3 N (lnl) interac-tion (symbols joined by solid lines), compared to experimen-tal (measured, full symbols and extrapolated, empty symbols)data. Both calculated and experimental values are shifted by( Z − × of the pairing gap in finite nuclei is accounted for at low-est order [54, 55] and how much is due to higher-orderprocesses, i.e. to the induced interaction associated withthe exchange of collective medium fluctuations betweenpaired particles [56, 57, 58, 59]. By treating normal andanomalous propagators consistently and at the same levelof approximation, GSCGF many-body scheme is in an ex-cellent position to contribute to this quest. In finite nuclei,the odd-even mass staggering is a good measure of nucle-onic, e.g. neutron, pairing. In particular, the three-pointmass difference formula ∆ (3) ( N, Z ) ≡ ( − N E ( N − , Z ) − E ( N, Z )+ E ( N +1 , Z )](6)successively evaluated for even and odd N closely encom-passes the pairing gap [50, 60] as long as N does notcorrespond to a shell closure . Calculated three-point massdifferences for argon, calcium, titanium and chromiumare compared to available experimental data in Fig. 7. In Note that ∆ (3) corresponds to half of the energy differencebetween the lowest unoccupied quasiparticle and the highestoccupied quasihole states, that is the particle-hole neutron gapat the Fermi surface. At subshell closures, this is dominatedby the gap among different nuclear orbits. However, for openneutron shells only the pairing contribution remains. spite of a reasonable general trend, the pairing strengthgenerated in the present ab initio calculations is too lowcompared to experiment. This feature is particularly visiblefor N ∈ [21 ,
27] isotopes in all considered chains, as well asbeyond N = 34 for calcium and titanium. Keeping in mindthe possible deficiency of the currently used Hamiltonian,this result likely points to missing higher-order correla-tions [56, 57, 58, 59]. The ADC(2) truncation scheme usedhere already includes both the lowest-order pairing termand the induced interaction resulting from the exchange ofunperturbed particle-hole excitations. However, it does notaccount for collective vibrations. The extension of GSCGFto the ADC(3) level is envisaged in the near future, know-ing that such a truncation does indeed seize importantfeatures of collective fluctuations and of their effect onsuperfluidity.In titanium and chromium, theoretical and experimen-tal three-point mass differences show further qualitativedifferences. In addition to the average value of ∆ (3) beingtoo low, the increase of its oscillation between N = 20 and N = 28 compared to calcium isotopes along with the shell-closure disappearances at N = 28 , ,
34 are not captured.The oscillation of ∆ (3) around its average is not relatedto the anomalous part of the self-energy (i.e. the pairinggap) but rather to its normal part (i.e. the effective mean-field) [50, 60]. The qualitative evolution of this staggeringfrom calcium to titanium and chromium pointed out aboveis thus a fingerprint of increased quadrupole correlations onthe normal self-energy. The absence of this evolution in ourtheoretical calculation confirms the need to include thesecorrelations consistently in both normal and anomalouschannels. While extending GSCGF to the ADC(3) levelshould help better describing the staggering of ∆ (3) , anexplicit treatment of deformation will probably be the most
14 16 18 20 22 24 26 28 30 32 34 36 38 400.10.150.20.250.30.350.40.45 ArCaTiCr N Δ E / A [ M e V ] Fig. 8.
Relative errors (theory - experiment) on total bindingenergies per nucleon along Z = 18 , ,
22 and 24 isotopic chains.Both calculations and experimental data are taken from Fig. 1. V. Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory
16 20 24 28 32 36 40 00.10.20.3 Cr β N (d)
16 20 24 28 32 36 40 Ti N (c) Ti
16 20 24 28 32 36 40
CaN (b) β Δ E/A
16 20 24 28 32 36 4000.10.20.30.40.5 Ar Δ E / A [ M e V ] N (a) Fig. 9.
Relative errors on total binding energies per nucleon for Z = 18 , ,
22 and 24 isotopes (full symbols and solid lines, takenfrom Fig. 8) and corresponding deformation parameter β computed via HFB calculations with the Skyrme SLy4 interaction [61](empty symbols and dashed lines). efficient way to reach a quantitative agreement wheneverquadrupole fluctuations become truly collective, i.e. as onemoves significantly away from semi-magic systems. For several of the quantities discussed above, the pooreragreement with theoretical data when departing from semi-magic calcium has been ascribed to an inefficient descrip-tion of quadrupole correlations. To substantiate this ob-servation, differences between computed and experimentalground-state energies per nucleon are displayed in Fig. 8for four isotopic chains. Unsurprisingly, the best agreementwith experimental values is found for calcium isotopes.For this chain, the error is dominated by the many-bodytruncation at ADC(2) and by possible flaws of the adoptednuclear Hamiltonian. Other chains perform generally worse,with the quality of the description deteriorating in partic-ular for neutron-rich argon and chromium isotopes. In allcases a clear minimum is visible at N = 20 and a maxi-mum around N = 24, which suggests a correlation withthe closed- or open-shell character of the neutrons and theassociated absence or presence of static deformation.This hypothesis is examined in Fig. 9, where the fourcurves of Fig. 8 are plotted separately and compared tothe deformation parameter β obtained in (single-reference)energy density functional calculations [61]. The correla-tion between the two quantities is striking for all chains.This observation supports the intuition that the collectivequadrupole correlations arising in doubly-open shell sys-tems can hardly be captured by present SU(2)-conservingcalculations.Even if in principle all correlations can be accountedfor in the current theoretical scheme, one would need toinclude very high orders in the expansion in order to graspsuch quadrupole static correlations. Indeed, these are typi-cally associated with the coherent superposition of manyparticle-many hole excitations that are not included inthe low-order many-body truncation schemes currently atreach. Extending beyond the ADC(3) approximation in-volves a factorial increase in the numbers of diagrams and would need a shift of paradigm in which all contributionsare dealt with at once through stochastic sampling [62].An alternative solution is the extension of existing expan-sion methods towards SU(2)-breaking schemes that willenable an efficient description of static deformation fromthe outset. Among the basic nuclear properties addressed by ab initiocalculations in the past few years, the size of medium-massnuclei has typically represented (and, to a good extent,still represents) one of the main challenges. The first setsof calculations that successfully reproduced ground-stateenergies of oxygen isotopes failed to provide, at the sametime, a good description of charge radii [33]. The NNLO sat
Hamiltonian, specifically introduced to cure this issue [37],very much improved the description of radii although dis-crepancies for neutron-rich systems have been shown topersist [33, 63]. An unsatisfactory account of nuclear sizesremains for several Hamiltonians that are currently em-ployed in state-of-the-art calculations [64, 8]. Very recently,new generations of chiral interactions have been proposedand shown to provide promising results for charge radii ofclosed-shell [65] as well as some open-shell [30] medium-mass nuclei. The behaviour along isotopic chains aroundcalcium remains however to be investigated. In Ref. [8]charge radii of oxygen, calcium and nickel isotopes havebeen systematically investigated with the
N N +3 N (lnl)and NNLO sat Hamiltonians. The study confirmed the goodperformance of NNLO sat up to the nickel chains. Here, inaddition to a more refined analysis of calcium isotopes,charge radii along argon, titanium and chromium chainsare presented.Mean square (m.s.) charge radii are computed startingfrom m.s. point-proton radii (cid:104) r (cid:105) as follows (cid:104) r (cid:105) = (cid:104) r (cid:105) + (cid:104) R (cid:105) + NZ (cid:104) R (cid:105) + 3 (cid:126) m c . (7)The last term corresponds to the relativistic Darwin-Foldycorrection [66] amounting to 0 .
033 fm . (cid:104) R (cid:105) and (cid:104) R (cid:105) . Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory 9
34 36 38 40 42 44 46 48 50 52 54 56 58 60-0.1-0.0500.050.10.150.2 Exp. A Ca Δ ⟨ r c h2 ⟩ / [f m ] (b) NNLO sat
NN+3N(lnl) ⟨ r c h2 ⟩ / [f m ] (a) Fig. 10. (a) Absolute root mean square charge radii of calciumisotopes and (b) differential ones relative to Ca computed withthe NN +3 N (lnl) and NNLO sat interactions. Available experi-mental data from Refs. [67, 63, 71] are displayed. Dark (light)symbols were obtained using a value (cid:104) R (cid:105) = 0 . [72]( (cid:104) R (cid:105) = 0 .
770 fm [68]) in Eq. (7). Error bars account for theuncertainty associated to model space truncation (see text fordetails). represent the m.s. charge radius of the proton and theneutron respectively. While the latter is relatively well es-tablished, (cid:104) R (cid:105) = − . [67], the determinationof the former has been debated and revised in the past fewyears. In the past, the value of (cid:104) R (cid:105) (cid:39) .
77 fm inferredfrom electron scattering experiments was commonly usedand included in the CODATA compilation [68]. Recentexperiments, including electronic and muonic hydrogenLamb shift measurements, favour a lower m.s. radius of (cid:104) R (cid:105) (cid:39) .
70 fm [69]. As a result, the CODATA value wasupdated to (cid:104) R (cid:105) (cid:39) . [70]. This value is adoptedin the present work and used in Eq. (7), unless specifiedotherwise. Given the large variation of (cid:104) R (cid:105) found in theliterature, however, it is worth investigating its impact oncomputed charge radii, specially in comparison with othersources of theoretical error in the calculation.Figure 10 shows root m.s. (r.m.s.) charge radii alongcalcium isotopes computed with the N N +3 N (lnl) andNNLO sat Hamiltonians, either as absolute, panel (a), or rel-ative to Ca, panel (b). For each interaction, the two setsof points (dark and light symbols) were obtained with twodifferent values of the proton radius in Eq. (7), respectively (cid:104) R (cid:105) = 0 . [72] and (cid:104) R (cid:105) = 0 .
770 fm [68]. The twovalues are representative of the two sets of experimentalresults discussed above. For each set of points, error bars conservatively account for the uncertainty coming fromtruncation of the one-body basis in the calculation. Specif-ically, they are obtained from the variation associated todifferent values of the HO frequency (cid:126) ω around the optimalvalue (itself determined as the closest point to the inter-section of the different e max curves, see Fig. 5 of Ref. [8]).While such variation is sizeable for NNLO sat , it is generallysmaller than the size of the points for N N +3 N (lnl). InRef. [8] the effects of higher-order (specifically, ADC(3))correlations on charge radii was also assessed for selecteddoubly-closed isotopes. For both interactions, the associ-ated uncertainties were of the order of 1%, i.e. they arecomparable to the ones coming from model-space trunca-tion in the case of NNLO sat . Hence, one should virtuallyadd such error bars also to the N N +3 N (lnl) results. Forthe latter, another possible source of error comes from ne-glecting many-body radius operators induced by the SRGevolution. However, recent calculations performed with asimilar interaction have shown that the consistent inclu-sion of such operators does not impact significantly thefinal result [73]. In conclusion, for absolute r.m.s. chargeradii, the chosen value of (cid:104) R (cid:105) can lead to a 0.5% variation,whereas uncertainties associated to model-space and many-body truncations are each of the order of 1%, with thecaveat that many-body truncations have been estimatedonly on closed-shell (not deformed) isotopes. The situationis even more favourable for differential radii, as visible inFig. 10(b). Here most of the errors cancel out and one isleft with some sizeable model-space truncation uncertaintyonly for the most neutron-rich isotopes. These improvedcalculations do not present significant differences with re-spect to the ones discussed in Ref. [8]. Results obtainedwith N N +3 N (lnl) underestimate the experimental valuesby about 5% throughout the calcium chain. Although themain experimental trend is roughly captured by the theo-retical curves (see also Fig. 12 and associated discussion),two of its peculiar features, namely the parabolic behaviourbetween Ca and Ca and the steep rise beyond Ca,are missing.Let us now move to results for argon, titanium andchromium isotopes, displayed in Fig. 11. Globally, the be-haviour is similar to the one observed in the calcium chain,with NNLO sat calculations very close to experimental dataand
N N +3 N (lnl) underestimating experiment by about 5to 10%. In argon, see Fig. 11 (a), charge radii computedwith NNLO sat reproduce very well existing data, withthe notable exception of the most neutron-rich isotopeavailable, Ar. The trend presents a kink at this nucleus,after which a steady increase with neutron number is ob-served until N = 34 where a second kink appears. Resultsobtained with N N +3 N (lnl) follow a similar behaviourpast Ar, as one can appreciate by looking at relativecharge radii displayed in Fig. 11 (d). Below N = 28, how-ever, the N N +3 N (lnl) slope is somehow different fromNNLO sat and experimental data. Experimental points aremore scarce for titanium and chromium, with essentiallyonly stable or long-lived isotopes available. In titanium,Fig. 11 (b) and (e), isotopes with N = 22 −
26 are wellreproduced by NNLO sat , while Ti is overestimated, simi-
32 34 36 38 40 42 44 46 48 50 52 54 56 58-0.100.10.20.3 A Ar Δ ⟨ r c h2 ⟩ / [f m ] (d)3.23.43.63.8 Exp. ⟨ r c h2 ⟩ / [f m ] (a) NN+3N(lnl)NNLO sat
36 38 40 42 44 46 48 50 52 54 56 58 60 62 A Ti (e)(b)
38 40 42 44 46 48 50 52 54 56 58 60 62 64 A Cr (f)(c) Fig. 11.
Root mean square charge radii of (a) argon, (b) titanium and (c) chromium isotopes computed with the NN +3 N (lnl)and NNLO sat interactions. Experimental data are taken from Ref. [67]. Panels (d), (e) and (f) show corresponding differentialradii relative to Ar, Ti and Cr respectively. larly to Ar.
N N +3 N (lnl) follows the same relative trendaround stability, with slightly different slopes in the proton-and neutron-rich regions. Analogous behaviour is observedfor chromium, shown in Fig. 11 (c) and (f). Also in this casethe radius of the N = 28 isotope, Cr, is overestimated byNNLO sat calculations, which instead give an excellent re-production of neighbouring Cr and Cr. Curves obtainedwith
N N +3 N (lnl) present the same general features as inthe titanium chain.To better gauge the overall quality of the theoreticaldescription, m.s. charge radii along all four isotopic chainsare shown in Fig. 12. By examining available experimentaldata, one can identify three distinct regions :a) Below N = 20, a steady increase with mild odd-evenstaggering is observed for calcium and argon.b) Between N = 20 and N = 28, the slope of the experi-mental trend changes noticeably, going from positive(argon) to null (calcium) and negative (titanium andchromium). Moreover, this is superposed with an in-verse parabolic behaviour characterised by a markedodd-even staggering. The parabolic trend is weak inargon and titanium, but pronounced in calcium.c) Above N = 28, one finds a steep increase with smallor even absent signs of odd-even staggering and shellclosures.Computed charge radii do reproduce some but not all ofthese experimental trends. Below N = 20, the steady be-haviour is captured by the calculations, although a slightshift is present for calcium. In the central region, the change Notice that this differentiation also applies to odd- Z chainsaround calcium and extends up to iron, see Ref. [74].
14 16 18 20 22 24 26 28 30 32 34 36 38 401112131415
Ar Ca Ti Cr N ⟨ r ⟩ [f m ] Th.Exp.
Fig. 12.
Mean square charge radii of argon, calcium, titaniumand chromium isotopes computed with the NNLO sat interac-tion (coloured symbols and solid lines) compared to availableexperimental data [67, 63, 71] (dark grey symbols and dashedlines).. Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory 11
18 20 22 24 26 28 30 32 34 36-0.500.51 Ar Ca Ti Cr N δ ⟨ r ⟩ , N [f m ] Exp. Th.
26 30 34 38-0.500.51
Fig. 13.
Changes in m.s. charge radii for argon, calcium, tita-nium and chromium relative to N = 28. Results obtained withthe NNLO sat Hamiltonian (coloured symbols and solid lines)are compared to existing experimental data [67, 63, 71] (greysymbols and dashed lines). In the inset, changes in m.s. chargeradii relative to N = 34 are shown for argon and chromiumisotopes. in slope from argon to chromium is qualitatively repro-duced. In contrast, the parabolic behaviour is basicallyabsent in all calculated curves. We note that the chargeradii for calcium between N = 20 −
28 have been explainedin terms of coupling to collective modes in Ref. [75] andexcitations across the sd and pf orbits using the shellmodel approach [76]. In both cases, quadrupole excitationsto (possibly deformed) states are involved. The particle-vibration coupling at the origin of this mechanism is en-coded in the ADC(3) many-body truncation and in its twinapproach, the Faddeev Random phase approximation [77],which is slightly more sophisticated for collective modes.Thus, ADC(3) stands out as the minimum requirement tobe able to reproduce the inverted bell behaviour of radii inthe central region. However, the above early studies werebased on phenomenological interactions. For ab initio appli-cations, it is not clear a priori to what extent the ADC(3)will be sufficient to resolve the low-energy quadrupoledeformations with current soft chiral Hamiltonians.For all isotopes, the theoretical charge radius at N = 28is systematically larger than the measured one. This alsoaffects the slope beyond this point, which results less steepthan what observed in experimental data. This inabilityto reproduce the pronounced kink at N = 28 is commonto other ab initio calculations as initially discussed inRef. [63]. In order to analyse this feature in more details,Fig. 13 shows measured and computed m.s. charge radiirelative to N = 28. The two experimental curves extendingbeyond N = 28 do indeed present the same rise towards N = 30. Manganese ( Z = 25) and iron ( Z = 26), forwhich experimental data are available, also follow thistrend. The same behaviour, with a kink followed by asteep rise essentially independent of Z , is found at the Exp. r [fm] 𝜌 c h [f m - ] Cr ( × Cr Cr ( × Fig. 14.
Charge density distributions of three chromium iso-topes. NNLO sat calculations are compared to density profilesdetermined via electron scattering [80]. Curves relative to Crand Cr have been rescaled for better readability. N = 50 and N = 82 magic numbers [74]. Remarkably, thetheoretical curves capture this basic feature, yielding radiithat increase almost independently of Z beyond N = 28.As already remarked, however, the slope is less steep thanthe experimental one, which represents a challenge for mostof nuclear structure calculations. Interestingly, a second,less pronounced kink is visible at N = 34 (see inset ofFig. 13), suggesting the presence of a weak shell closure.A similar feature is observed in the charge radii computedwith the N N +3 N (lnl) Hamiltonian, see Figs. 10 and 11.For both interactions, the angle of the kink decreasessmoothly with Z , which is consistent with the evolutionof the N = 34 neutron gaps computed with N N +3 N (lnl)and reported in Fig. 6(d).To conclude the present section, some examples ofcharge density distributions in chromium isotopes areshown in Fig. 14. Theoretically, the charge distribution iscomputed starting from the point-proton distribution andfolding it with the charge distribution of the proton, asdetailed in Ref. [78]. Corrections associated with the centerof mass motion in the HO basis are seen to be negligiblefor masses above A = 16 [79]. In Fig. 14 distributionsof , , Cr computed with NNLO sat are compared tocharge profiles determined from electron scattering crosssections [80]. Theoretical distributions follow closely theexperimental curves in the region around and above r ch .In contrast, for all three isotopes the behaviour differ inthe nuclear interior, with the calculations displaying adip around 1.5 fm that is not present, or not probed, inthe experimental distributions. One does not observe aqualitatively different behaviour for Cr, whose value of r ch slightly departs from experiment. A similar level ofagreement between SCGF calculations and experiment wasrecently found for argon and calcium isotopes [35] and for Sn, although theoretical error bars were lager for thelatter case [11] .
Correlation expansion methods represent a promising long-term option to simulate the majority, if not all, of atomicnuclei from first principles. To this purpose, the choiceof the reference state, including the use of deformed ba-sis states and the possibility of breaking symmetries, iscrucial, notably to account for essential static correlationsfrom the outset. So far, ab initio approaches have mainlyexploited the breaking of U(1) symmetry associated toparticle number conservation to account for static pairingcorrelations. In the past few years, this strategy has en-abled computations of semi-magic, i.e. singly open-shellnuclei, where quadrupole correlations associated to nu-clear deformation are typically weak, i.e. predominantlydynamical. In the present work such U(1)-breaking, SU(2)-conserving calculations are pushed away from semi-magicnuclei in a systematic fashion for the first time. Resultsare overall encouraging, with many general experimentalfeatures captured by the ab initio simulations. At the sametime, a degradation of the description for certain groups ofnuclei signals the inefficient account of (static) quadrupolecorrelations and calls for a SU(2)-breaking extension ofthe present theoretical framework.Specifically, bulk nuclear properties (ground-state en-ergies, charge radii and density distributions) were com-puted along seven isotopic chains around calcium, fromargon to chromium. Calculations were performed withinthe Gorkov self-consistent Green’s function approach atsecond order and employed two state-of-the-art two- plusthree-nucleon Hamiltonians,
N N +3 N (lnl) and NNLO sat . N N +3 N (lnl) results provide a good global description ofground-state energies. Total energies are slightly underesti-mated, consistently with missing higher-order correlationsas discussed in detail in Ref. [8]. Differential energies, i.e.one- and two-nucleon separation energies as well as two-neutron shell gaps, are generally in excellent agreementwith experiment. In particular, neutron magic numbers N = 28 , ,
34 emerge and evolve following experimentaltrends. The largest discrepancy with experimental data isfound for the
N, Z = 20 gaps, both overestimated by thecalculations. This impacts the description of the protondripline, which however remains reasonably reproduced.In contrast, three-point mass differences along the vari-ous isotopic chains evidence that present calculations donot provide sufficient pairing strength. The future inclu-sion of higher-order, e.g. ADC(3), corrections accountingfor collective fluctuations might result instrumental for amore accurate description of pairing properties. While suchcomputations are routine in Dyson-SCGF (i.e., for closedshells), they become computationally challenging in theGorkov formalism due to the increase in the number of Bo-golyubov mean-field orbits resulting from SU(1)-breaking.A full Gorkov-ADC(3) will require improved algorithms,such as importance truncation, but it is within reach andcan be implemented in the foreseeable future.As remarked in Ref. [8],
N N +3 N (lnl) calculations yieldcharge radii that underestimate the experimental measure-ments by about 5 to 10% throughout all considered chains.Still, relative trends are generally good, which points to some small systematic deficiencies in the Hamiltonian. Incontrast, NNLO sat provides an overall good reproductionof both absolute and relative charge radii. The main experi-mental trends below N = 20, between N = 20 and N = 28and above N = 28 are qualitatively described. The largestdiscrepancy with data is detected for N = 28 isotopes,whose radius is overestimated in all considered elements.As a consequence, the steep rise past N = 28 observedin calcium and chromium is not reproduced to a full ex-tent by the present calculations. The inability to correctlydescribe the charge radius difference between Ca and Ca is common to nearly all existing nuclear structurecalculations (with the notable exception of Ref. [81]) andcurrently represents a challenge in particular for ab initioapproaches.For some of the doubly open-shell nuclei consideredin this study, strong (i.e. static) quadrupole correlationsare expected to play an important role and lead to theonset of deformation. Indeed, a careful comparison be-tween computed and experimental ground-state energiesreveals a remarkable correlation between the error withrespect to experiment and the expected degree of defor-mation (quantified through the deformation parameter β obtained via EDF calculations [61]). Such correlationsare likely to impact the calculated observables but canbe hardly accounted for in the current scheme that uses(rotational) symmetry-conserving reference states and in-cremental extensions of the formalism. In fact, any increasein the ADC( n ) order beyond n = 3 would be impracticaldue to the factorial increase in diagrams and degrees offreedom. Besides, such a truncation scheme is unlikely toresolve deformation degrees of freedom until several ordersbeyond the current capabilities. To break through theselimitations different approaches could be envisaged, suchas the stochastic sampling of the self-energy or a SU(2)-breaking scheme. In the first case, one would still workin a standard (spherical or partially deformed) basis butdiagrams are summed to very high orders using bold dia-grammatic Monte Carlo techniques [62]. This approach isparticularly suited to address correlations at medium ener-gies that have been identified as key ingredients to deviseab initio nucleon-nucleus optical potentials [82]. In the sec-ond path, the extension towards a SU(2)-breaking schemewould impose nuclear deformation already at the level ofthe reference state and allow many-body truncations atlow ADC( n ) orders, still requiring a final projection ongood angular momentum. Both approaches will involvesophisticated extensions of the SCGF formalism and willbe long-term developments. Acknowledgements
The authors wish to thank R. Garcia Ruiz and F. Raimondifor useful discussions. Calculations were performed by us-ing HPC resources from GENCI-TGCC (Contracts No.A005057392, A007057392) and at the DiRAC Complexitysystem at the University of Leicester (BIS National E-infrastructure capital grant No. ST/K000373/1 and STFCgrant No. ST/K0003259/1). This work was supported by . Som`a et al.: Moving away from singly-magic nuclei with Gorkov Green’s function theory 13 the United Kingdom Science and Technology FacilitiesCouncil (STFC) under Grant No. ST/L005816/1 and inpart by the NSERC Grant No. SAPIN-2016-00033. TRI-UMF receives federal funding via a contribution agreementwith the National Research Council of Canada.
References
1. W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. , 377 (2004).2. K. Kowalski, D. J. Dean, M. Hjorth-Jensen, T. Papen-brock, and P. Piecuch, Phys. Rev. Lett. , 132501(2004).3. S. K. Bogner, R. J. Furnstahl, and A. Schwenk, Prog.Part. Nucl. Phys. , 94 (2010).4. S. Binder, J. Langhammer, A. Calci, and R. Roth,Physics Letters B , 119 (2014).5. H. Hergert, S. K. Bogner, T. D. Morris, S. Binder,A. Calci, J. Langhammer, and R. Roth, Phys. Rev. C , 041302 (2014).6. G. Hagen, G. R. Jansen, and T. Papenbrock, Phys.Rev. Lett. , 172501 (2016).7. R. Taniuchi et al. , Nature , 53 (2019).8. V. Som`a, P. Navr´atil, F. Raimondi, C. Barbieri, andT. Duguet, Phys. Rev. C , 014318 (2020).9. T. D. Morris, J. Simonis, S. R. Stroberg, C. Stumpf,G. Hagen, J. D. Holt, G. R. Jansen, T. Papenbrock,R. Roth, and A. Schwenk, Phys. Rev. Lett. ,152503 (2018).10. P. Gysbers et al. , Nature Phys. , 428 (2019).11. P. Arthuis, C. Barbieri, M. Vorabbi, and P. Finelli,(2020), arXiv:2002.02214 [nucl-th] .12. Z. Rolik, A. Szabados, and P. R. Surj´an, The Journalof Chemical Physics , 1922 (2003).13. P. R. Surj´an, Z. Rolik, A. Szabados, and D. K¨ohalmi,Annalen der Physik , 223 (2004).14. B. R. Barrett, P. Navratil, and J. P. Vary, Prog. Part.Nucl. Phys. , 131 (2013).15. E. Gebrerufael, K. Vobig, H. Hergert, and R. Roth,Phys. Rev. Lett. , 152503 (2017).16. H. Hergert, Phys. Scripta , 023002 (2017).17. A. Tichai, E. Gebrerufael, K. Vobig, and R. Roth,Phys. Lett. B , 448 (2018).18. S. K. Bogner, H. Hergert, J. D. Holt, A. Schwenk,S. Binder, A. Calci, J. Langhammer, and R. Roth,Phys. Rev. Lett. , 142501 (2014).19. G. R. Jansen, J. Engel, G. Hagen, P. Navratil, andA. Signoracci, Phys. Rev. Lett. , 142502 (2014).20. S. R. Stroberg, S. K. Bogner, H. Hergert, and J. D.Holt, Ann. Rev. Nucl. Part. Sci. , 307 (2019).21. T. Duguet, Journal of Physics G: Nuclear and ParticlePhysics , 025107 (2014).22. T. Duguet and A. Signoracci, J. Phys. G , 015103(2017).23. Y. Qiu, T. M. Henderson, T. Duguet, and G. E.Scuseria, Phys. Rev. C , 044301 (2019).24. M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev.Mod. Phys. , 121 (2003). 25. V. Som`a, T. Duguet, and C. Barbieri, Phys. Rev. C , 064317 (2011).26. A. Signoracci, T. Duguet, G. Hagen, and G. R. Jansen,Phys. Rev. C , 064320 (2015).27. A. Tichai, P. Arthuis, T. Duguet, H. Hergert, V. Som`a,and R. Roth, Phys. Lett. B , 195 (2018).28. A. Tichai, R. Roth, and T. Duguet, Front. in Phys. , 164 (2020).29. J. M. Yao, B. Bally, J. Engel, R. Wirth, T. R.Rodr´ıguez, and H. Hergert, Phys. Rev. Lett. ,232501 (2020).30. S. Novario, G. Hagen, G. Jansen, and T. Papenbrock,(2020), arXiv:2007.06684 [nucl-th] .31. V. Som`a, C. Barbieri, and T. Duguet, Phys. Rev. C , 011303 (2013).32. V. Som`a, A. Cipollone, C. Barbieri, P. Navr´atil, andT. Duguet, Phys. Rev. C , 061301 (2014).33. V. Lapoux, V. Som`a, C. Barbieri, H. Hergert, J. D.Holt, and S. R. Stroberg, Phys. Rev. Lett. , 052501(2016).34. E. Leistenschneider, M. P. Reiter, S. Ayet San Andr´es,B. Kootte, J. D. Holt, P. Navr´atil, C. Babcock,C. Barbieri, B. R. Barquest, J. Bergmann, J. Bollig,T. Brunner, E. Dunling, A. Finlay, H. Geissel, L. Gra-ham, F. Greiner, H. Hergert, C. Hornung, C. Jesch,R. Klawitter, Y. Lan, D. Lascar, K. G. Leach, W. Lip-pert, J. E. McKay, S. F. Paul, A. Schwenk, D. Short,J. Simonis, V. Som`a, R. Steinbr¨ugge, S. R. Stroberg,R. Thompson, M. E. Wieser, C. Will, M. Yavor, C. An-dreoiu, T. Dickel, I. Dillmann, G. Gwinner, W. R. Plaß,C. Scheidenberger, A. A. Kwiatkowski, and J. Dilling,Phys. Rev. Lett. , 062503 (2018).35. C. Barbieri, N. Rocco, and V. Som`a, Phys. Rev. C , 062501 (2019).36. M. Mougeot, D. Atanasov, C. Barbieri, K. Blaum,M. Breitenfeld, A. de Roubin, T. Duguet, S. George,F. Herfurth, A. Herlert, J. D. Holt, J. Karthein,D. Lunney, V. Manea, P. Navr´atil, D. Neidherr,M. Rosenbusch, L. Schweikhard, A. Schwenk, V. Som`a,A. Welker, F. Wienholtz, R. N. Wolf, and K. Zuber,Phys. Rev. C , 014301 (2020).37. A. Ekstr¨om, G. R. Jansen, K. A. Wendt, G. Hagen,T. Papenbrock, B. D. Carlsson, C. Forss´en, M. Hjorth-Jensen, P. Navr´atil, and W. Nazarewicz, Phys. Rev.C , 051301 (2015).38. V. Som`a, C. Barbieri, and T. Duguet, Phys. Rev. C , 024323 (2014).39. D. R. Entem and R. Machleidt, Phys. Rev. C ,041001 (2003).40. R. Machleidt and D. Entem, Physics Reports , 1(2011).41. A. Carbone, A. Cipollone, C. Barbieri, A. Rios, andA. Polls, Phys. Rev. C , 054326 (2013).42. A. Cipollone, C. Barbieri, and P. Navr´atil, Phys. Rev.C , 014306 (2015).43. A. Cipollone, C. Barbieri, and P. Navr´atil, Phys. Rev.Lett. , 062501 (2013).44. C. Barbieri, Journal of Physics: Conference Series ,012005 (2014).
45. F. Raimondi and C. Barbieri, Phys. Rev. C , 054308(2018).46. J. Ripoche, A. Tichai, and T. Duguet, Eur. Phys. J.A , 40 (2020).47. W. Huang, G. Audi, M. Wang, F. G. Kondev, S. Naimi,and X. Xu, Chinese Physics C , 030002 (2017).48. S. Michimasa, M. Kobayashi, Y. Kiyokawa, S. Ota,D. S. Ahn, H. Baba, G. P. A. Berg, M. Dozono,N. Fukuda, T. Furuno, E. Ideguchi, N. Inabe, T. Kawa-bata, S. Kawase, K. Kisamori, K. Kobayashi, T. Kubo,Y. Kubota, C. S. Lee, M. Matsushita, H. Miya,A. Mizukami, H. Nagakura, D. Nishimura, H. Oikawa,H. Sakai, Y. Shimizu, A. Stolz, H. Suzuki, M. Takaki,H. Takeda, S. Takeuchi, H. Tokieda, T. Uesaka,K. Yako, Y. Yamaguchi, Y. Yanagisawa, R. Yokoyama,K. Yoshida, and S. Shimoura, Phys. Rev. Lett. ,022506 (2018).49. X. Xu et al. , Phys. Rev. C , 064303 (2019),arXiv:1905.12577 [nucl-ex] .50. T. Duguet, P. Bonche, P.-H. Heenen, and J. Meyer,Phys. Rev. C , 014311 (2001).51. J. Holt, S. Stroberg, A. Schwenk, and J. Simonis,(2019), arXiv:1905.10475 [nucl-th] .52. D. J. Dean and M. Hjorth-Jensen, Rev. Mod. Phys. , 607 (2003).53. T. Duguet, “Pairing in finite nuclei from low-momentum two- and three-nucleon interactions,” in Fifty Years of Nuclear BCS (2013) pp. 229–242.54. T. Duguet, T. Lesinski, K. Hebeler, and A. Schwenk,Mod. Phys. Lett. A , 1989 (2010).55. T. Lesinski, K. Hebeler, T. Duguet, and A. Schwenk,J. Phys. G , 015108 (2012).56. F. Barranco, R. Broglia, G. Colo, E. Vigezzi, andP. Bortignon, Eur. Phys. J. A , 57 (2004).57. G. Gori, F. Ramponi, F. Barranco, P. F. Bortignon,R. A. Broglia, G. Col`o, and E. Vigezzi, Phys. Rev. C , 011302 (2005).58. A. Pastore, F. Barranco, R. A. Broglia, and E. Vigezzi,Phys. Rev. C , 024315 (2008).59. A. Idini, F. Barranco, E. Vigezzi, and R. Broglia, J.Phys. Conf. Ser. , 092032 (2011).60. T. Duguet, P. Bonche, P.-H. Heenen, and J. Meyer,Phys. Rev. C , 014310 (2001).61. M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys.Rev. C , 034322 (2006).62. K. Van Houcke, F. Werner, E. Kozik, N. Prokof’ev,B. Svistunov, M. J. H. Ku, A. T. Sommer, L. W.Cheuk, A. Schirotzek, and M. W. Zwierlein, NaturePhysics , 366 (2012).63. R. F. Garcia Ruiz, M. L. Bissell, K. Blaum,A. Ekstr¨om, N. Fr¨ommgen, G. Hagen, M. Ham-men, K. Hebeler, J. D. Holt, G. R. Jansen,M. Kowalska, K. Kreim, W. Nazarewicz, R. Neu-gart, G. Neyens, W. N¨ortersh¨auser, T. Papenbrock,J. Papuga, A. Schwenk, J. Simonis, K. A. Wendt, andD. T. Yordanov, Nature Physics , 594 (2016).64. J. Simonis, S. R. Stroberg, K. Hebeler, J. D. Holt, andA. Schwenk, Phys. Rev. C , 014303 (2017). 65. T. H¨uther, K. Vobig, K. Hebeler, R. Machleidt, andR. Roth, (2019), arXiv:1911.04955 [nucl-th] .66. J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys.Rev. A , 4579 (1997).67. I. Angeli and K. Marinova, Atomic Data and NuclearData Tables , 69 (2013).68. P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod.Phys. , 633 (2008).69. H.-W. Hammer and U.-G. Meißner, Science Bulletin , 257 (2020).70. “CODATA recommended values of the fundamen-tal physical constants: 2018,” https://physics.nist.gov/cgi-bin/cuu/Value?rp .71. A. J. Miller, K. Minamisono, A. Klose, D. Garand,C. Kujawa, J. D. Lantis, Y. Liu, B. Maaß, P. F. Man-tica, W. Nazarewicz, W. N¨ortersh¨auser, S. V. Pineda,P. G. Reinhard, D. M. Rossi, F. Sommer, C. Sum-ithrarachchi, A. Teigelh¨ofer, and J. Watkins, NaturePhysics , 432 (2019).72. W. Xiong et al. , Nature , 147 (2019).73. T. Miyagi, T. Abe, M. Kohno, P. Navr´atil, R. Okamoto,T. Otsuka, N. Shimizu, and S. R. Stroberg, Phys. Rev.C , 034310 (2019).74. R. Garcia Ruiz and A. Vernon, Eur. Phys. J. A ,136 (2020).75. F. Barranco and R. Broglia, Physics Letters B ,90 (1985).76. E. Caurier, K. Langanke, G. Martnez-Pinedo,F. Nowacki, and P. Vogel, Physics Letters B ,240 (2001).77. C. Barbieri, D. Van Neck, and W. H. Dickhoff, Phys.Rev. A , 052503 (2007).78. T. Duguet, V. Som`a, S. Lecluse, C. Barbieri, andP. Navr´atil, Phys. Rev. C , 034319 (2017).79. N. Rocco and C. Barbieri, Phys. Rev. C , 025501(2018).80. H. De Vries, C. W. De Jager, and C. De Vries, Atom.Data Nucl. Data Tabl. , 495 (1987).81. P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C ,064328 (2017).82. A. Idini, C. Barbieri, and P. Navr´atil, Phys. Rev. Lett.123