aa r X i v : . [ nu c l - t h ] F e b A study of open shell nuclei using chiraltwo-body interactions.
G. PudduDipartimento di Fisica dell’Universita’ di Milano,Via Celoria 16, I-20133 Milano, ItalyFebruary 8, 2021
Abstract
We apply the Hybrid-Multi-Determinant method using the recent chi-ral two-body interactions of Entem-Machleidt-Nosyk (EMN) without renor-malization to few nuclei up to A=48. Mostly we use the bare fifth order NNinteraction N4LO-450. For M g and Cr the excitation energies of the +1 states are far larger than the corresponding experimental values. Pacs numbers : 21.10.-k,21.60.DeKeywords: nuclear many-body theory Introduction.
In the past several years we have witnessed the development of powerful ab-initio many-body techniques to solve the nuclear Schroedinger equation. Amongthese methods we mention the no-core shell model (NCSM) (ref.[1]), the coupled-cluster (CC) method (ref.[2]) and the in-medium similarity renormalization group(IM-SRG) (ref.[3]). While the NCSM can only be used for light nuclei becauseof the exponential increase of the size of the Hilbert space with the particle num-ber, for closed shells or around shell closure the CC method has been used upto medium mass nuclei. Quite recently, advances in the Multi-Reference IM-SRG (MR-IM-SRG) have been applied to doubly open shell medium mass nuclei(ref.[4]). Both CC method and (MR)-IM-SRG scale polynomially with the sizeof the single-particle space. This is both an advantage and a limitation. Thatis, from one hand a polynomial scaling allows to reach large single-particle ba-sis and medium mass nuclei, on the other hand the nuclear wave function hascomponents in the full Hilbert space which grows exponentially in size with thesize of the single-particle space. Presumably (or better hopefully) out of the fullHilbert space only a tiny fraction gives the most important contributions to ob-servables. The method we use, the Hybrid-Multi-determinant method (HMD)(ref.[5]), is rather different from the CC or IM-SRG, in the sense that no simplereference state is needed. We approximate the nuclear wave function as a lin-ear combination of the most generic Slater determinants and the coefficients ofthese Slater determinants, as well as the Slater determinants themselves, are de-termined variationally using rank-3 gradient methods (ref.[6])(very similar to thewell known BFGS method (ref.[7])). Also the HMD method uses a number ofcoefficients much smaller than the size of the Hilbert space. However analyti-2ally strongly founded extrapolation methods (refs. [8]-[10]) allow to estimatewith some uncertainty the energy at zero energy variance (as it should be for aneigenstate in the full Hilbert space). More precisely, suppose that we have anapproximate eigenstate | ψ > of the Hamiltonian, then the expectation value ofthe Hamiltonian is related to the energy variance obtained with this state by therelation < ˆ H > − E gs = a < ( ˆ H − < ˆ H > ) > , where ˆ H is the many-bodyHamiltonian, a is a constant and E gs is the ground state energy in the full Hilbertspace, provided the state | ψ > is sufficiently close to the exact eigenstate. A setof approximate wave functions would allow us to extract the ground state energy E gs . This energy-variance-of-energy (EVE) method allows for a bridge between arelatively small parametrization of the nuclear wave function and the full Hilbertspace. We performed this extrapolation only for M g using major shells.This extrapolation is not necessary for the evaluation of the excitation energies, asdescribed below.The HMD method is equally applicable to both closed shell and open shellnuclei. Although in this work we do not include a genuine NNN interaction, it isnonetheless interesting to see what predictions a reasonably soft NN interactiongives for excitation energies in the case of open shells nuclei, especially wherecollective behavior appears, without any renormalization.As the NN interaction we consider the recently introduced chiral interaction byEntem, Machleidt and Nosyk (ref.[11]) without additional renormalization. Theoutline of this paper is as follows. In section 2 we briefly recap the HMD method.In section 3 we present the numerical results and in section 4 some conclusionsand outlook. 3 A brief recap of the the HMD method.
The key idea of the HMD method is to expand the nuclear wave function as alinear combination of many generic Slater determinants (with exact or partialrestoration of good quantum numbers using projectors) and to determine theseSlater determinants using energy minimization techniques. We use an harmonicoscillator basis. The wave-function of the nucleus is written as | ψ > = N D X S =1 g S ˆ P | U S > (1) where ˆ P is a projector to good quantum numbers (e.g. good angular momentumand parity) N D is the number of Slater determinants | U S > expressed as | U S > = c ( S ) c ( S ) ...c A ( S ) | > . S = 1 , .., N D (2) The generalized creation operators c α ( S ) for α = 1 , , .., A are a linear combina-tion of the creation operators a † i in the single-particle state labeled by ic α ( S ) = N s X i =1 U i,α ( S ) a † i α = 1 , ...A (3) Here N s is the number of the single-particle states. These generalized creationoperators depend on the Slater determinant S . The complex coefficients U i,α ( S ) represent the single-particle wave-function of the particle α = 1 , , .., A . We donot impose any symmetry on the Slater determinants (axial or other) since the U i,α ( S ) are variational parameters and good quantum numbers are restored usingthe projectors. These complex coefficients are obtained by minimizing the energyexpectation values E [ U ] = < ψ | ˆ H | ψ >< ψ | ψ > (4) where ˆ H is the total Hamiltonian, which also includes the usual center of massHamiltonian β ( ˆ H cm − / hω ) , in order to suppress excitations of the center of4ass. The coefficients g S in eq. (1) are obtained by solving the generalized eigen-value problem X S < U S ′ | ˆ P ˆ H | U S > g S = E X S < U S ′ | ˆ P | U S > g S (5) for the lowest eigenvalue E . We have two versions of the method, which wecall HMD-a and HMD-b. In the first version the two-body matrix elements ofthe Hamiltonian H where 1,2,3,4 label the single-particle states with quan-tum numbers n , l , j , j z , t z ) , etc. ( n, l, j, j z and t z denote the principalquantum number, the orbital angular momentum, the angular momentum, its z-projection, and the isospin) all satisfy the relation n + l ≤ e max . In the b-versionthe single-particle quantum numbers satisfy the relation n + l + 2 n + l ≤ N max (and similarly for the states 3 and 4). The b-version has been used by theauthor in the past only to test the variational programs (using renormalized inter-action for the Deuteron binding energy an accuracy of one part in a million caneasily be achieved). In this project we use bare interactions, that is no renormal-ization steps are performed. A renormalization of the two-body interaction is nec-essary for strong interactions. The EMN interactions, especially at the M eV cutoff seem to be soft enough so that we preferred to use bare interactions. Thishas the advantage that there are no induced many-body interactions, which aredifficult to deal with. Presumably at large cutoff and medium mass nuclei a pre-liminary renormalization either with the Suzuki-Lee-Okamoto method (ref.[12]or the Similarity Renormalization Group seems advisable (ref.[13]).In this work we use the HMD-a version for excitation energies. The HMD-bversion seems more convenient for binding energies since we can perform cal-culations with much larger single-particle states ( N max ≃ ). However, theHMD-b version seems to have a strong dependence on the strength of the cen-5er of mass Hamiltonian β and this feature has not been fully analyzed yet andit will not be discussed here. Moreover for binding energies the final EVE stepis necessary. This step is not necessary for excitation energies. The reason isthe following. Consider for example the nucleus M g and the excitation energyof the first + state. We construct a sequence of approximate wave functionsconsisting of increasing numbers of Slater determinants N D and evaluate the theenergy of the ground-state and of the first +1 state. The energies E gs ( N D ) and E +1 ( N D ) are not exact but they tend to the exact values as N D becomes largerand larger. That is, the exact energies would be E gs = E gs ( N D ) + δ gs ( N D ) and E +1 = E +1 ( N D ) + δ +1 ( N D ) . As N D goes to infinity the deltas tend to zero.The deltas are the errors in the two energies and have the same negative sign.When we take the difference in order to obtain the excitation energy these er-rors cancel out. Therefore for sufficiently large N D we should obtain excitationenergies which have only a small dependence on N D . Provided of course thatwe perform the variational calculations for both states exactly at the same levelof approximation. Schematically these calculations start with N D = 1 (Hartree-Fock). We add a trial generic Salter determinant and minimize the energy ex-pectation value with respect to the last added Slater determinant. We call thisthe ”addition phase”). We then vary anew all Slater determinants for D = 1 , in sequence (”refinement phase”) until the energy changes less than a termina-tion value (typically ÷ KeV ). We then keep adding new Slater determinants.In the addition phase we vary only the one added last. After we reach a certainnumber of Slater determinants we repeat the refinement procedure to all Slaterdeterminants until the termination criterion is met. The refinement phase is per-formed after we reach specified numbers of Slater determinants typically after we6each N D = 2 , , , , , , , , , .. (these numbers are simply a pos-sible choice). Exactly the same procedure is implemented for the ground-stateand for the excited states, since we want the approximate wave-functions for theground-state and excited states to have the same degree of accuracy. Usually weuse a partial J πz projector to construct approximate wave functions. To restorethe exact angular momentum quantum numbers we take the approximate wavefunctions with N D Slater determinants and reproject them to good J π in order toobtain better approximate excitation energies as a function of the number of Slaterdeterminants N D . We focused mostly on four nuclei, Li, C , M g and Cr . Experimental valuesfor the excitation energies are from ref.[14]-[17] respectively (see also ref.[18]).Binding energies are from ref.[19]. In all calculations we considered single-particle states with l < . All calculations use the N4LO-450 interaction. Inall cases the harmonic oscillator frequency is selected around the minimum ofthe Hartree-Fock energy. In fig. 1 we show the dependence of the excitation en-ergy of +1 state of Li as a function of the number of Slater determinants N D . Inthis case the calculations have been performed at an harmonic oscillator frequency ¯ hω = 24 M eV . Note that the calculation does not include any coupling to the con-tinuum. Experimentally the +1 state is above in energy to the threshold of α + d break-up. In fig. 2 we show the behavior of the excitation energy of the +1 of C as a function of the number of Slater determinants. In this case we used an har-monic oscillator frequency of ¯ hω = 20 M eV . For these two cases the excitation7 E + ( M e V ) N D e max =4e max =5e max =6exp Figure 1: Excitation energy in MeV of the +1 state of Li as a function of thenumber of Slater determinants N D for several values of e max for the N4LO-450interaction. The lines are only to guide the eye.8 E + ( M e V ) N D e max =3e max =4e max =5exp Figure 2: Excitation energy in MeV of the +1 state of C as a function of thenumber of Slater determinants N D for several values of e max for the N4LO-450interaction. 9 E + ( M e V ) N D e max =4e max =5e max =6e max =7exp Figure 3: Excitation energy in MeV of the +1 state of M g as a function of thenumber of Slater determinants N D for several values of e max for the N4LO-450interaction.energies are not too far off the experimental values. The nuclei M g and Cr turned out to be the surprise. The excitation energy of the +1 state of M g is sev-eral times higher than the experimental one as shown in fig. 3. The experimentalexcitation energy of the +1 state is . M eV . In all evaluations of the excitationenergies, within a few hundred KeV’s the convergence is reasonable, and it can beimproved using more Slater determinants. The calculations have been performedat ¯ hω = 20 M eV . A similar result has been obtained for the doubly open shellnucleus Cr as sown in fig. 4. The experimental excitation energy of the +1 state is . M eV
In this case we used ¯ hω = 22 M eV . Although we investigated10 E + ( M e V ) N D e max =5e max =6exp Figure 4: Excitation energy in MeV of the +1 state of Cr as a function ofthe number of Slater determinants N D for few values of e max for the N4LO-450interaction.very few cases it is striking that for medium mass nuclei we obtain excitation en-ergies too far off the experimental values. As mentioned in the introduction, weperformed a EVE analysis only for M g . For Cr it was deemed unnecessarysince even using only Slater determinants with major shells (both are smallnumbers) we reached the experimental binding energy. The interaction we usedlacks the saturating effect of the NNN interaction and the NN interaction stronglyoverbinds. For M g we used optimized Slater determinants with majorshells. The EVE analysis has been performed as follows. These Slater deter-minants | U S > with S = 1 , .., , were first determined without the use of an-11ular momentum (partial or full) and parity projector. The minimization has beenperformed as previously described. Out of these N D = 200 Slater determinantswe can form several approximate nuclear wave functions. We could constructwave functions using the first , , .., n S Slater determinants with n S = 1 , , .. upto n S = N D , determine anew the coefficients of the linear combination using theHill-Wheeler equations and determine the variance of energy for these approx-imate nuclear wave functions. However only for sufficiently large n S we havereasonably approximate wave functions. In practice we evaluate the energy andthe corresponding variance of energy for all n S = 1 , , .., N D and we keep only thepoints ( < ˆ H > − < ˆ H > , < ˆ H > ) evaluated with reasonably accurate wavefunctions (i.e. n S should be large enough) so that all points lie on a straight line.Only then we fit the coefficients a and b in E = a + b < ( H − E ) > . The inter-cept a is the estimate of the ground-state energy. The EVE plot is shown in fig.5.The final results for the coefficients a and b are a = ( − . ± . M eV and b = (0 . ± . × − ) M eV − . The experimental binding energy is . M eV.
In this work we considered the reasonably soft NN interaction N4LO-450 andperformed some calculations about excitation energies away from major shell clo-sure. In the cases of M g and Cr we did not obtain one of the typical featuresof collective behavior, i.e. low excitation energy. It could well be that the in-clusion of the three-body interaction is necessary, a difficult task to implement.Another possible cause could be that our method of evaluating excitation energies12 E ( M e V ) σ (MeV) EVElinearexp
Figure 5: EVE plot for the ground-state of M g using major shells. Theexperimental value is shown as an horizontal line.13ust be pushed to a much larger number of Slater determinants. Or, a possiblereason could be that the bare interaction couples too strongly low momentum andhigh momentum states. In other words, a further renormalization must be usedin order to obtain reasonable excitation energies. A renormalization procedure asdone in SRG decouples low momentum from high momentum states. This can betested with reasonable ease, and it will be the goal of future work. The author wishes to thank R.Machleidt for providing the EMN subroutines.Computational resources have been partially provided by a CINECA ISCRA-Cproject. 14 eferenceseferences