Single particle spectra based of modern effective interactions

The self-consistent Green's function method is applied to 16O using a G-matrix and Vucom as effective interactions, both derived from the Argonne v18 potential. The present calculations are performed in a larger model space than previously possible. The experimental single particle spectra obtained with the G-matrix are essentially independent of the oscillator length of the basis. The results shows that Vucom better reproduces spin-orbit splittings but tends to overestimate the gap at the Fermi energy.

In ab initio methods the effective interaction is to be computed microscopically. Typically, two classes of approaches are possible to derive effective interactions from a standard realistic nucleon-nucleon force. Block-Horowitz theory makes use of the Feshbach projection formalism to devise an energy dependent interaction [8,9]. This gives solutions for every eigenstate with non zero projection onto the model space, however, the energy dependence severely complicates the calculations. The G-matrix interaction [10] is a first order approximation to the Block-Horowitz scheme. Alterna-tively, one can employ a proper unitary transformation to map a finite set of solutions of the initial Hamiltonian into states belonging to the model space. In this case, one has the advantage to work with an energy independent interaction. Examples of such approach are the Lee-Suzuki method [11], V low−k [12] and the unitary correlator operator method (UCOM) [13,14,15]. The UCOM formalism is such that one can apply the inverse transformation to reinsert SRC into the nuclear wave function. A discussion of the similarities and differences between Lee-Suzuki and Block-Horowitz is given in Ref. [16]. These methods, in principle, generate effective many-nucleon forces in addition to the two-nucleon (2N) interaction. In practice, however, calculating medium and large nuclei one is forced to work with only 2N Hamiltonians (or at most with weak 3N forces). It is therefore important to investigate how this truncation affects the results using the different approaches outlined above.
In Ref. [17] we proposed to employ a set of Faddeev equations within the self-consistent Green's function (SCGF) approach [7] to obtain a microscopical description of LRC. This allows to couple simultaneously quasiparticles to both particle-hole (ph) and particleparticle/hole-hole (pp/hh) collective excitations. Such formalism was later applied to 16 O to investigate mechanisms that could possibly quench the spectroscopic factors of mean field orbits [18]. These calculations were already performed in a no-core fashion. However, the model space employed was still somewhat limited and phenomenological corrections were applied to tune the values of specific single particle (sp) energies (that allows to study correlations by artificially suppressing the couplings among selected excitation modes). Note that here and in the following we use the terms sp energies and sp spectra to refer to the poles of the one-body Green's function (defined below Eq. (1) ). These represent the excitation energies of the A±1 neighbor nuclei, which are observable quantities. In this letter the calculations of Ref. [18] are repeated by avoiding any phenomenology Double lines represent the dressed one-particle Green's function g(ω), which propagates quasiparticles (rightward arrows) and quasiholes (leftward arrows). The ellipses propagate collective excitations of the nucleus (Eqs. (3) and (4) ).
and employing a large model space. We discuss the results of 2N interactions belonging the two types discussed above, namely a standard G-matrix and V UCOM .
We consider the calculation of the sp Green's function from which both the one-hole and one-particle spectral functions, for the removal and addition of a nucleon, can be extracted. In Eq. (1), are the spectroscopic amplitudes for the excited states of a system with A + 1 (A − 1) particles and the poles ε ) correspond to the excitation energies with respect to the A-body ground state. The one-body Green's function can be computed by solving the Dyson equation [19,20].
where the irreducible self-energy Σ * γδ (ω) acts as an effective, energy-dependent, potential that governs the single particle behavior of the system. The self-energy is expanded in a Faddeev series as in Fig. 1, which couples the exact propagator g αβ (ω) (which is itself a solution of Eq. (2) ) to other phonons in the system [17]. The relevant information regarding ph and pp/hh collective excitations is included in the polarization and the twoparticle propagators. Respectively, and which describe the one-body response and the propagation of two-particles/two-holes. In this work, Π(ω) and g II (ω) are obtained by solving the dressed RPA (DRPA) equations [21,22], which account for the redistribution of strength in the sp spectral function. Since this information is carried by the correlated propagator g αβ (ω), Eq. (2), the SCGF formalism requires an iterative solution. It can be proven that full self-consistency guarantees to satisfy the conservation of the number of particles and other basic quantities [23].
The coupled cluster studies of Ref. [6,24] found that eight major harmonic oscillator shells can be sufficient to obtain converging results for 16 O with G-matrix interactions. At the same time, the experience with the calculations of Ref. [18] suggests that high partial waves do not contribute sensibly. In this work, all the orbits of the first eight shells with orbital angular momentum l ≤ 4 were included. Inside this model space a G-matrix and the V UCOM potential were employed as effective interactions. The former was computed using the CENS library routines [10,25]. For the latter, the UCOM matrix-elements code [26] was employed with the constraint I ϑ = 0.09 fm 3 . This choice of the UCOM correlator reproduces, in perturbation theory, the binding energies of several nuclei up to 208 Pb [27]. In both cases the Argonne v 18 potential [28] was used as starting interaction. However, we chose to neglect the Coulomb and the other charge independence breaking terms (i.e., to set α = 0) in the present work. In applying the SCGF formalism, the Hartree-Fock (HF) equations (Brueckner-Hartree-Fock (BHF) for the G-matrix) where first solved for the unperturbed propagator g (B)HF αβ (ω). This was employed in the first iteration. After that, the (dressed) solution g αβ (ω) was used to solve the DRPA for Π(ω) and g II (ω) and then the Faddeev equations for an improved self-energy. In each iteration the two most important fragments close to the Fermi level were retained for each partial wave, both in the quasiparticle and the quasihole domains. The remaining strength do not affect sensibly the sp states that will be discussed below and was collected, at each iteration, according to the corresponding (B)HF orbitals. In the present work, calculations were iterated until reaching convergence (to within 200 keV) for the sp energies nearby the Fermi level. With the available computational resources, self-consistence was achievable for these states except in the case of the G-matrix with harmonic oscillator length b HO ≤ 1.8 fm (as discussed below). To test the iteration procedure we computed the total number of particles obtained with V UCOM . The first calculation, based on the HF propagator, gives A=16.4 (a 2.5% error). At self-consistency 15.99<A<16.02 (due to numerical errors), showing the adequacy of our approach. More details on the SCGF/Faddeev formalism are given in Ref. [7,17,18]. As already noted, however, no phenomenological corrections were applied in the present work. Calculations have been performed for oscillator lengths in the interval b HO =1.6-2.1 fm. The sp spectra obtained at self-consistency are shown in Figs. 2 and 3. In the case of a G-matrix, these orbitals appear independent, within numerical accuracy, of the oscillator frequency for b HO ≥ 1.9 fm. Below this the trend is similar. However, the structure of the quasiparticle spectral function becomes increasingly complex for the l=1 waves that correspond to the pf shell. These break in more than two relevant fragments and their sp energies depend strongly on b HO , suggesting a non negligible contamination of the c.o.m. motion. In this situation the above approach for obtaining self-consistency cannot be carried out reliably. We show the result of a typical iteration by dashed lines in Fig. 2 but discard these results in the following discussion. This complication does not happen for V UCOM , which is a softer interaction and generates, for each shell, at most one main fragment and a smaller satellite peak  near the Fermi energy. Fig. 3 shows that the spin orbit splittings for this interaction are approximately constant, although the sp energies are not yet independent of the oscillator length. This can be understood considering that these spin orbit partners correspond to particularly simple and similar configurations (one particle or one hole on top of the correlated ground state). Conversely, separation energies are linked to the to total binging energy of neighbor isotopes. Larger model spaces will probably be required for a full convergence with V UCOM . The splittings obtained from both interactions are reported in Tab. I. For the 0p orbits these are practically constant for all the oscillator lengths. In general the Gmatrix predicts about a half of the experimental value. Better solutions are obtained with the present choice of the UCOM correlator. For the 0d orbits the results for the two interactions are more similar to each other but not totally independent of the oscillator length.
The same type of LRC studied here (which are predominantly of 2p1h and 2h1p in character) were also considered in Ref. [4]. There, the effective interaction was derived in the unitary-model-operator approach and an explicit diagonalization was performed. The resulting sp energies and spin-orbit splitting, however, showed a stronger dependence on the oscillator length than the one found in this work. The spectrum obtained in Fig. 2 is nearly convergent, suggesting that the all order summation employed here and the proper accounting of the fragmentation of the sp strength allow to select relevant configurations beyond the bare 2p1h/2h1p level. Coupled cluster calculations are also available for 16 O with an analogous v 18 /G-matrix [6]. These authors report splittings of the 0p and 0d orbits larger than those of Tab. I by about 1.5 and 0.5 MeV, respectively. The present work includes LRC only in the form of coupling to small amplitude excitations of the core -which can be described at the DRPA level. More complex collective modes are also present [29] and should be included for a full solution of the many-body problem. We note, for example, that the phenomenological studies of Ref. [18] suggest further contributions to the p 3/2 quasihole wave function coming from couplings to the first excited 0 + state in oxygen. Testing this conjecture would first require being able to reproduce the correct excitation energy of this level - since it can couple effectively only when it is low enough in energy. To our knowledge this is still a challenge for the available ab initio methods. Figures 4 and 5 show the effects of LRC on the sp spectrum for b HO = 1.9 fm, and compare to the experimental values for the addition/removal of a neutron. For both interactions the coupling to collective phonons reduces the splitting of the 0p orbits, with respect to the HF approximation. Including the effects of fragmentation tends instead to compress the sd shell and to lower the whole spectrum. The self-consistent results for the energy gap between particle and hole states, ∆E F = ε d 5/2 −ε p 1/2 , are 13.0 MeV with the G-matrix and 15.4 MeV with V UCOM . Both of them exceed the experimental value of 11.5 MeV. The mean square radii obtained are r rms =2.63 fm (G-matrix) and r rms =2.45 fm (V UCOM ).
Both the UCOM and the renormalization group approaches have the capability of shifting the binding energies for A=3,4 systems along the Tjon line [30,31]. In the first case this can be achieved by varying the correlator in the tensor-isoscalar channel. For V low−k , one modifies the momentum cut off. Usually, tuning the binding energies to the experimental value increases the non locality of the interaction and leads improved spin orbit splittings, as seen in Tab. I. On the other hand, our V UCOM result for ∆E F -with 2N forces-overestimates the experiment, more than the G-matrix does. This behavior is seen already at the HF level for soft interactions like V UCOM and V low−k [27,32] and it is only slightly modified by the LRC considered here. In all cases, three-body forces appear necessary in order to reproduce the whole spectrum of observations. We note, however, that the UCOM method offers some advantages to reduce the contribution needed from many-body forces since it allows to treat SRC in different channels separately [15].
In conclusion, SCGF calculations have been performed for the first time in a large model space, including up to eight oscillator shells. Long-range correlations in the form of coupling sp to ph and pp/hh RPA modes were investigated for 16 O. A comparison was made between the results of a G-matrix and the V UCOM effective interactions, both derived from same realistic potential (Argonne v 18 ). The spectra of adjacent nuclei were found to be nearly convergent for the G-matrix, while they depend only weakly on the oscillator length for V UCOM . In general it was found that the LRC effects considered here, tend to compress the spectra of A±1 nuclei but do not affect sensibly the the gap between quasiparticle and quasihole energies at the Fermi level.