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Shell Model Monte Carlo studies of neutron-rich nuclei in the 1s-0d-1p-0f shells

We demonstrate the feasibility of realistic Shell-Model Monte Carlo (SMMC) calculations spanning multiple major shells, using a realistic interaction whose bad saturation and shell properties have been corrected by a newly developed general prescription. Particular attention is paid to the approximate restoration of translational invariance. The model space consists of the full sd-pf shells. We include in the study some well-known T=0 nuclei and several unstable neutron-rich ones around N=20,28. The results indicate that SMMC can reproduce binding energies, B(E2) transitions, and other observables with an interaction that is practically parameter free. Some interesting insight is gained on the nature of deep correlations. The validity of previous studies is confirmed.


I. INTRODUCTION
Studies of extremely neutron-rich nuclei have revealed a number of intriguing new phenomena. Two sets of these nuclei that have received particular attention are those with neutron number N in the vicinity of the 1s0d and 0f 7/2 shell closures (N ≈ 20 and N ≈ 28). Experimental studies of neutron-rich Mg and Na isotopes indicate the onset of deformation, as well as the modification of the N = 20 shell gap for 32 Mg and nearby nuclei [1]. Inspired by the rich set of phenomena occurring near the N = 20 shell closure when N ≫ Z, attention has been directed to nuclei near the N = 28 (sub)shell closure for a number of S and Ar isotopes [2,3] where similar, but less dramatic, effects have been seen as well.
In parallel with the experimental efforts, there have been several theoretical studies seeking to understand and, in some cases, predict properties of these unstable nuclei. Both mean-field [4,5] and shell-model calculations [2,3,[6][7][8][9][10] have been proposed. The latter require a severe truncation to achieve tractable model spaces, since the successful description of these nuclei involves active nucleons in both the sd-and the pf -shells. The natural basis for the problem is therefore the full sd-pf space, which puts it out of reach of exact diagonalization on current hardware.
Shell-Model Monte Carlo (SMMC) methods [11][12][13] offer an alternative to direct diagonalization when the bases become very large. Though SMMC provides limited detailed spectroscopic information, it can predict, with good accuracy, overall nuclear properties such as masses, total strengths, strength distributions, and deformation -precisely those quantities probed by the recent experiments. It thus seems natural to apply SMMC methods to these unstable neutron-rich nuclei. Two questions will arise -center-of-mass motion and choice of the interaction -that are not exactly new, but demand special treatment in very large spaces.
The center-of-mass problem concerns momentum conservation. It was investigated for the first time by Elliott and Skyrme in 1955 [14], and a vast literature on the subject has developed, but as of now, the methods proposed have not managed to reconcile rigor and applicability. Section II will be devoted to explaining why this is so and to describe howshort of ensuring exact momentum conservation -it is possible within an SMMC context to assess the damage and control it in order to perform meaningful calculations.
There has long been a consensus that G-matrices derived from potentials consistent with NN data [15] are the natural shell-model choice. Unfortunately, such interactions give results that rapidly deteriorate as the number of particles increases. Two alternative cures have been proposed: sets of fitted matrix elements (all the shell-model work quoted above), or minimal "monopole" modifications [16]. The latter restricts the fit to far fewer quantities: some average matrix elements, which are the ones that suffer from the bad saturation and shell properties of the realistic potentials. Both approaches have the common shortcoming of needing data to determine the fitted numbers, but recently a general parametrization of the monopole field (H m ) has become available that could be used to replace the Gmatrix centroids for any model space [17]. The interaction we present in Section III is the first monopole modified G-matrix free of parameters other than the six entering the independently derived H m .
Section IV contains results for a number of unstable, neutron-rich nuclei near the N = 20 and 28 shell closures and compares them to experiment and to other truncated shell-model calculations. Section V is devoted to a discussion of what we have accomplished and surveys further applications of such calculations.

II. SMMC AND CENTER-OF-MASS MOTION
By momentum conservation, a many-body wavefunction must factorize as Ψ(r) = φ(R)Ψ(r rel ), where R is the center-of-mass coordinate and r rel the relative ones. There are formalisms in which the latter are constructed explicitly, but they lead to very hard problems of antisymmetrization. What can be done in a shell-model context is to work with a basis that ensures that the eigenstates automatically factorize as requested. This is accomplished by taking φ(R) to be a harmonic oscillator state, which implies that the basis must produce eigenstates of In order to diagonalize this one-body Hamiltonian in the SM basis, we have to rewrite it using along with a similar expression for the coordinates. Then H CM = h 1 + h 2 , where h 1 is a one-body oscillator spectrum, and h 2 an oscillator two-body force. If one considers the matrix element n 1 l 1 n 2 l 2 |h 2 |n 3 l 3 n 4 l 4 , it is quite easy to convince oneself -using a general property of the Talmi-Moshinsky transformation and the oscillator form of h 2 -that 2n 1 + l 1 + 2n 2 + l 2 = 2n 3 + l 3 + 2n 4 + l 4 . In other words, H CM conserves the number of oscillator quanta. This implies that if a basis contains all states of (or up to) nhω excitations, diagonalizing a translationally invariant Hamiltonian would ensure the factorization of the center-of-mass wavefunction. To separate the wavefunctions with 0hω center-of-mass quanta it would be sufficient to do the calculations with choosing a large β CM (not to be confused with the SMMC inverse temperature). Thus, the procedure to deal with the center-of-mass problem is conceptually straightforward. Practically, things are not so simple. In 32 Mg, for example, the sd-pf basis will contain states having between 0 and 16hω quanta; however, it is very far from containing them all, and it does not even contain all those of 1hω . Then, and this point is crucial, the restriction of H CM to the basis is no longer H CM . As a consequence, the prescription in Eq. (3) is no longer a prescription to remove unwanted center-of-mass excitations, but a prescription to remove something else. Still, whatever the restricted H CM is in the model space, it is the operator most closely connected with the true one. Hence, rather than removing unwanted excitations, which is now impossible in general, we may try to assess and control the damage by using Eq. (3) to construct a set of states |β CM and see how β CM |H|β CM behaves. Since the problem is variational, the best we can do is choose a β CM that minimizes the energy.
Before proceeding, it is worth going quickly through the history of the subject, under the light of these very elementary considerations which are often ignored, thereby creating unnecessary confusion. The pioneers of the subject were Elliott and Skyrme [14] who treated a simple case, the 1hω JT π =1 − 0 excitations on a sp shell core, showing that one of them ( 5 6 p −1 d − 1 6 p −1 s) was simply the 1hω center-of-mass state. Other early important contributions are [18,19]. The first cross-shell calculation in a full space appeared in 1968 [20]: (p 1/2 s 1/2 d 5/2 ) n , which successfully accounted for the spectra in the region around 16 O. The (p 1 1/2 s 1/2 )J π T = 1 − 0 state contained a spurious component of 5.556% of the Elliott-Skyrme state. Nonetheless, Gloeckner and Lawson [21] decided to apply Eq. (3) with an arbitrarily large β CM to eliminate the spurious components; by not realizing that H CM restricted to that small space generated very little center-of-mass excitations and many genuine ones, they managed to eliminate the latter rather than the former. In spite of the criticism that ensued [22,23] showing that the procedure could not possibly make sense (except in complete spaces), no formally satisfactory arguments were advanced to replace it. Equation (3) remained a guide on where to begin to minimize the center-of-mass nuisance, and it is indeed the basis of our variational suggestion. New projection techniques have been developed [24], but they rely on explicit construction of the spurious states and they are not applicable in SMMC.
The recipe advocated by Whitehead et al. [23] seems quite compatible with the constrained variation sketched above, and we have adopted it, since it proves sufficient to optimize the solutions. The idea is to add β CM H CM to H , but with β CM remaining fairly small. We have found that β CM = 1 works reasonably well. This value will push spurious components up in energy byhω = 45A −1/3 − 25A −2/3 MeV ≃ 14 MeV while leaving the desired components relatively unscathed. A smaller value of β CM leaves the spurious configurations at low enough energies that they are included in the Monte Carlo sampling, while larger values of β CM (> 3) begin to remove the entire pf shell from the calculation and artificially truncate the space. Technically, H CM suffers from the sign problem, and we have to say a word about it.
SMMC methods reduce the imaginary-time, many-body evolution operator to a coherent superposition of one-body evolutions in fluctuating one-body fields. This reduction is achieved via a Hubbard-Stratonovich transformation and the resulting path integral is evaluated stochastically. SMMC methods have been applied to numerous full-basis 0hω studies. The primary difficulty in these applications arises from a sign problem due to the repulsive part of effective nucleon-nucleon interactions. A practical solution to this sign problem was obtained by considering a set of Hamiltonians close to the desired realistic Hamiltonian (H) and extrapolating to the realistic case [25]. This technique has been validated in numerous studies that show the SMMC approach to be a viable and productive avenue to study extremely large many-body problems [11][12][13].
The original sign problem for realistic interactions was solved by breaking the two-body interaction into "good" (without a sign problem) and "bad" (with a sign problem) parts: H = H good + H bad . The bad part is then multiplied by a parameter, g, with values typically lying in the range −1 ≤ g ≤ 0. The Hamiltonian H = f (g)H good +gH bad has no sign problem for g in this range. The function f (g) is used to help in extrapolations. It is constructed such that f (g = 1) = 1, and takes the form [1−(1−g)/χ], with χ = 4. The SMMC observables are evaluated for a number of different negative g-values and the true observables are obtained by extrapolation to g = 1. If we fix the sign problem in the same manner as above for H CM we are no longer dealing with a Hamiltonian that pushes all spurious components to higher energies -some components might even be lowered for g < 0. We will see shortly that this is not a real problem.
We typically choose a minimal extrapolation (linear, quadratic, etc.) in the extrapolation parameter that gives a χ 2 per datum of ≃ 1. In much of our work most quantities extrapolate either linearly or quadratically. We measure the center-of-mass contamination by calculating the expectation value of H CM . In Fig. 1a, we show the value of H CM in 32 Mg for several different values of β CM . 1 It is apparent that H CM decreases as β CM increases. We also find that near β CM = 1, H CM ≪ 2hω ≃ 28 MeV showing that the center-of-mass contamination is minimal. Note that at approximately β CM = 1.5 the average of the two different techniques of extrapolation presented in Fig. 1a give H CM ≃ 0 MeV, and the calculations could be fine tuned for each nucleus to obtain this value. Figure 1a contains two different data sets corresponding to two different methods of extrapolating H CM to the physical case (g = 1). The solid circles show the results of a simple linear extrapolation where for this observable χ 2 per datum is approximately 1. It has been established [11] that H obeys a variational principle such that the extrapolating curve must have a minimum (slope = 0) at the physical value (g = 1). As we sample values of the quantityH, it is perhaps reasonable to extrapolate H CM using this constraint as well (ifH were truly separable, this would be an exact procedure). A cubic extrapolation embodying this constraint corresponds to the open circles in Fig. 1a.
We may further evaluate our extrapolation procedures by comparing SMMC and the standard shell-model results in 22 Mg. Shown in Fig. 2a is a detailed comparison for the expectation of the energy H , and in Fig. 2b a comparison for H CM . The standard shellmodel results were obtained using the code ANTOINE [26]. The SMMC results in Fig. 2a employ a constrained fit, such that d H /dg | g=1 = 0. The slight deviation from the standard shell model at g = −0.6, −0.8, −1.0 is due to increasing interaction matrix elements (with g), while dβ, the imaginary time step, is kept fixed. This deviation is also seen in Fig. 2b. Note that in Fig. 2b neither the constrained fit nor the linear fit (both with χ 2 per datum ≃ 1) give a precise description of the standard shell-model results at g = 1. An average of the two ways of extrapolation as indicated by the solid line on Fig. 2b, apparently gives the more precise result, and we shall do this for other H CM values quoted throughout this paper. The error bar for such an averaged result is given by adding in quadrature the individual errors of both extrapolations.
In Fig. 1b we show the evolution of the total B(E2) and in Fig. 1c we show the occupation of the sd-shell and the f 7/2 -shell as a function of β CM . Note that the occupation of the f 7/2 orbit decreases as β CM increases. This is due to a combination of the removal of actual centerof-mass excitations and the "pushing up" in energy of the real states. The B(E2) decreases slowly with β CM , although the uncertainties are consistent with a constant. However, the decrease, particularly at β CM = 3, is likely to be real since we are working in an incomplete nhω model space. At extremely large values of β CM we would remove the pf shell from the calculation and return to the pure sd-shell result, which is substantially smaller than the result shown here. The slow evolution of the B(E2) with β CM does open the intriguing possibility of studying B(E2)s with an interaction that has no sign problem (e.g. Pairing + Quadrupole) and no center-of-mass correction with the hope of obtaining reasonable results.
Somewhere between β CM = 3 and 5, β CM H CM begins to change so strongly as a function of g that our extrapolations become unreliable and we can extract no useful information. By β CM = 5, the extrapolated values become completely unreasonable, and numerical noise completely swamps the calculation. We thus conclude that a safe value for a generic study is β CM = 1, although for a given nucleus this value may be fine tuned to nearly eliminate all center-of-mass contamination from the statistical observables. This may be done in future studies.

III. THE EFFECTIVE INTERACTION
Numerous shell-model studies have been carried out in truncated model spaces for neutron-rich nuclei near N = 20 [6,8,7] and N = 28 [2,3,9]. The number of sd-pf shell effective interactions used almost exceeds the number of papers, but there are similarities between them. A common feature is Wildenthal's USD interaction [27] to describe the pure sd shell part of the problem. All also use some corrected version of the original Kuo-Brown (KB) G-matrix interaction [28] to describe nucleons in the pf shell. The cross-shell interaction is handled in one of two different ways: matrix elements are generated via a G-matrix or via the Millener-Kurath potential. As is common in this type of calculation, selected two-body matrix elements and single-particle energies have been adjusted to obtain agreement with experiment. As these interactions have been produced for use in highly truncated spaces (usually with only 2p2h neutron excitations to the pf -shell), they are not suitable for use in the full space. We found that they generally scatter too many particles from the sd to the pf shell, and that the B(E2) values cannot be consistently calculated. We are not saying that the interactions are wrong, but that we did not succeed in adapting them to the full space. Perhaps it can be done, but it would be of limited interest; the years of experience these forces embody cannot be transposed to other spaces, as the pf -sdg shells for instance, where detailed fits are unthinkable. Therefore, we derived a new effective interaction for the region: a monopole corrected renormalized G-matrix, derived from a modern potential.
As noted in Section I, if G-matrices have not been widely used, it is because they were thought to be so flawed as to serve at best as input parameters to overall fits, as in the case of the famous USD interaction [27]. However, it had been pointed out twenty years ago [29] that practically all the problems of the KB interaction amounted to the failure to produce the N, Z = 28 closure, and could be corrected by changing at most four centroids of the interaction. A perturbative treatment in the beginning of the pf shell using these modifications (the KB3 interaction) gave good results [30], and when the ANTOINE code became available [26], the results became truly excellent ( [31,32] and references therein). In the meantime, it was confirmed in other regions that the only trouble with the G-matrices resided in their centroids, i.e., in the bad saturation and shell formation properties of the realistic potentials [16]. The rest of the interaction was excellent, and strongly dominated by collective terms (pairing and quadrupole mainly) [33]. We say interaction and not interactions, because all the realistic ones produce very similar good multipole matrix elements, and similar monopole failures. The outstanding problem is to replace case by case modifications by a general specification of H m , the monopole field, that yields all the centroids once and for all. In subsection B we shall describe the proposed solution [17] we have adopted.
There is great advantage in the SMMC context to adopt the schematic collective multipole Hamiltonian (H M ) of [33], because its main terms have good signs, thereby eliminating extrapolation uncertainties; however, it may be a premature step. For one thing, it has not been established yet that in the light nuclei the collective contribution is sufficient to give high-quality results, a project better left to exact diagonalizations where fine details may be better probed. Furthermore, the renormalization treatment in [33] is somewhat crude. A more complete treatment might yield significant differences. This could be true even though potentials consistent with the NN data yield very similar collective contributions and are therefore reasonably well fitted even by older potentials. Finally, even if it were true that realistic interactions are interchangeable, and that a crude treatment of renormalization was adequate, there would certainly be no objection to using the best forces and the most sophisticated renormalizations available. In practice this is what we do here.

A. The renormalized G-matrix
In order to obtain a microscopic effective shell-model interaction which spans both the 1s0d and the 0f 1p shells, our many-body scheme starts with a free nucleon-nucleon interaction V which is appropriate for nuclear physics at low and intermediate energies. At present there are several potentials available. The most recent versions of Machleidt and co-workers [34], the Nimjegen group [35], and the Argonne group [36] have a χ 2 per datum close to 1 with respect to the Nijmegen database [37]. The potential model of Ref. [34] is an extension of the one-boson-exchange models of the Bonn group [38], where mesons like π, ρ, η, δ, ω, and the fictitious σ meson are included. In the charge-dependent version of Ref. [34], the first five mesons have the same set of parameters for all partial waves, whereas the parameters of the σ meson are allowed to vary. The recent Argonne potential [36] is also a charge-dependent version of the Argonne V 14 [39] potential. The Argonne potential models are local potentials in coordinate space and include a π-exchange plus parametrizations of the short-range and intermediate-range part of the potential. The Nimjegen group [35] has constructed potentials based on meson exchange and models parametrized in similars ways as the Argonne potentials. Another important difference between, e.g., the Bonn potentials and the Argonne and Nimjegen potentials is the strength of the much debated tensor force [40]. Typically, the Bonn potentials have a smaller D-state admixture in the deuteron wave function than the Argonne and Nimjegen potentials, as well as other potential models. A smaller (larger) D-state admixture in the ground state of the deuteron means that the tensor force is weaker (stronger). The strength of the tensor force has important consequences in calculations of the binding energy for both finite nuclei and infinite nuclear matter (see, e.g., the discussion in Ref. [15]). A potential model with a weak tensor force tends to yield more attraction in a nuclear system than a potential with a strong tensor force; however, all these modern nucleon-nucleon interactions yield very similar excitation spectra. Moreover, in calculations of Feynman-Goldstone diagrams in perturbation theory, a potential with a weak tensor force tends to suppress certain intermediate states of long-range character, like particle-hole excitations [41]. In this paper, we choose to work with the charge-dependent version of the Bonn potential models, as found in Ref. [34].
The next step in our many-body scheme is to handle the fact that the repulsive core of the nucleon-nucleon potential V is unsuitable for perturbative approaches. This problem is overcome by introducing the reaction matrix G given by the solution of the Bethe-Goldstone equation where ω is the unperturbed energy of the interacting nucleons, and H 0 is the unperturbed Hamiltonian. The projection operator Q, commonly referred to as the Pauli operator, prevents the interacting nucleons from scattering into states occupied by other nucleons.
In this work, we solve the Bethe-Goldstone equation for several starting energies Ω, by way of the so-called double-partitioning scheme discussed in Ref. [15]. For the closed-shell core in the G-matrix calculation we choose 16 O and employ a harmonic-oscillator basis for the singleparticle wave functions, with an oscillator energyhΩ given byhΩ = 45A −1/3 −25A −2/3 = 13.9 MeV, A = 16 being the mass number. Finally, we briefly sketch how to calculate an effective two-body interaction for the chosen model space in terms of the G-matrix. Since the G-matrix represents just the summation to all orders of ladder diagrams with particle-particle diagrams, there are obviously other terms which need to be included in an effective interaction. Long-range effects represented by core-polarizations terms are also needed. The first step then is to define the so-called Q-box given by TheQ-box is made up of non-folded diagrams which are irreducible and valence linked. We can then obtain an effective interaction H eff =H 0 + V eff in terms of theQ-box with [15] where (n) and (n − 1) refer to the effective interaction after n and n − 1 iterations. The zeroth iteration is represented by just theQ-box. Observe also that the effective interaction V eff (n) is evaluated at a given model space energy ω, as is the case for the G-matrix as well. Here we choose ω = −20 MeV. The final interaction is obtained after folding results in eigenvalues which depend rather weakly on the chosen starting energy (see, e.g., Ref. [42] for a discussion). All non-folded diagrams through second-order in the interaction G are included. For further details, see Ref. [15]. Finally, the reader should note that when one defines an effective interaction for several shells, the effective interaction may be strongly non-hermitian. This non-hermiticity should arise already at the level of the G-matrix. However, since the G-matrix is calculated at a fixed starting energy for both incoming and outgoing states, it is by construction hermitian. Since we are calculating an effective interaction at a fixed starting energy, the individual diagrams entering the definition of theQ-box are thereby also made hermitian. The non-hermiticity which stems from folded diagrams is made explicitly hermitian through the approach of Suzuki et al. in Ref. [43].

B. The monopole field
As results concerning the monopole field are scattered through many papers [16,33,44,45], the most relevant of which is not yet published [17], this subsection offers a compact presentation of the main ideas.
The centroids we have often mentioned are -in a neutron proton (np) representation -the average matrix elements where xx ′ stands for neutrons or protons in orbits rs respectively. Technically, the monopole field H m is that part of the interaction containing all the quadratic two-body forms in scalar products of fermion operators a + rx · a sx ′ (same parity for r and s). The clean extraction of these forms from the total H (i.e., the separation H = H m + H M , M for multipole) is not altogether trivial [44]. It is conceptually important because it makes H m closed under unitary transformations of the a + , a operators, and therefore closed under spherical Hartree-Fock variation. The expectation values we may want to vary are those of the H d m , the diagonal part of H m in a given basis. Calling m rx the x-number operator for orbit r, we obtain a standard result (it is the extraction of the non diagonal terms that is more complicated). The expectation value of H d m for any state is the average energy of the configuration to which it belongs (a configuration is a set of states with fixed m rx for each orbit). In particular, H d m reproduces the exact energy of closed shells (cs) and single-particle (or hole) states built on them ((cs) ± 1), since for this set (cs ± 1) each configuration contains a single member. Consequently, it is uncontaminated by direct configuration mixing. As an example, in 56 Ni, the two-body (no Coulomb) contribution to the binding energy in the pf shell is approximately 73 MeV, and configuration mixing (i.e., H M ) is responsible for only 5 MeV; the rest is monopole. If we compare to the total binding of 484 MeV, it is clear that the monopole part becomes overwhelming, even allowing for substantial cross shell mixing (which, incidentally is included in the present calculations).
Therefore, H In a nutshell, the idea in [17] is to fit H d m to the (cs) ± 1 set, the single-particle and single-hole spectra around doubly magic nuclei. It is assumed that the bulk and surface terms can be separated, and by cancelling the kinetic energy K =hω/2 p (p + 3/2)m p , m p is the number of particles in harmonic oscillator (HO) shell p, against the collective monopole term [33,45], the leading term in H m . Defining W =hω( p (m p / D p ) 2 /4, one obtains an expression of order O(A 1/3 ) that has strong shell effects producing the HO closures. To this one adds l · s and l · l one-body terms that produce the observed splittings around HO closures. The filling order is now established, and as the largest orbit-which comes lowest-is full, it alters significantly the splitting of its neighbors (e.g., the spectrum of 57 Ni is totally different from that of 41,49 Ca). This is taken care of by strictly two-body terms. With a total of six parameters (two for the W − 4K + l · s + l · l part, and four for the two-body contributions), the fit yields an rms deviation of 220 keV for 90 data points.
All terms have a common scaling inhω =40/ρ, obtained using a very accurate fit to the radii r 2 = 0.9ρA 1/3 , where Note that due to this scaling it is possible to use the same functional form from A = 5 to A = 209. Figure 3 shows the mechanism of shell formation for nuclei with T = 4. There is an overall unbinding drift of O(A 1/3 ), with pronounced HO closures due to W − 4K at (N, Z) = (16,8), (20,12), and (28,20). The addition of the l · s + l · l terms practically destroys the closures except for the first ( 24 O), and creates a fictitious one at 40 S. It is only through the two-body terms that closure effects reappear, but now the magic numbers are 6 ( 20 C), 14 ( 20 C, 36 Si), and 28 ( 48 Ca, 64 Ni). Note that the shell effect in 32 Mg is minuscule. The same is true for 30 Ne among the T = 5 nuclei. Among the four two-body terms in H m , there is one that is overwhelmingly responsible for the new (EI, for intruder, extruder) magic numbers. It produces an overall (T=1, mainly) repulsion between the largest (extruded) orbit of the shell and the others. The extruder becomes the intruder in the shell below. This is the term that is missing in the realistic interactions. The problems in the excitation spectra of 47 Ca, 48 Ca, and 49 Ca [29,15] disappear if the realistic centroids are replaced by those-even more realistic, apparently-of H m .
To close this subsection we give some useful formulas to relate the np and isospin (mT ) representations. We have which reproduces the average energies of configurations at fixed m r T r . Calling D r = 2j r + 1 the degeneracy of orbit r, we rewrite the relevant centroids incorporating explicitly the Pauli restrictions In the np scheme each orbit r goes into two rx and rx ′ and the centroids can be obtained through

C. The monopole terms in the calculations
The calculations used the preliminary version of H m [17], which for the purposes of this study should make little difference. All we have said above is valid for both the old and the new version except for details. Only one of them is worth mentioning here, and it concerns the single-particle energies shown in Table I. It is seen that the old and new values are quite close to those adopted in the calculations, though the old set puts the s 1/2 and f 5/2 orbits higher. This reflects the awkward behavior of the l · l part of H m that changes sign at the p = 3 shell. This problem was treated artificially in the old version through a single-particle mechanism that was discarded in the calculations, mainly because keeping it would have demanded a readjustment of the interaction for each nucleus -an unwanted complication in a feasibility study such as this one. As a consequence, we expect the s 1/2 orbit to be overbound with respect to its sd partners in the upper part of the shell. In the new version the mechanism becomes two-body and should do much better.
There has been much discussion about the choice of the cross-shell gap, i.e., the distance between the d 3/2 and f 7/2 orbits, which plays a crucial role in all truncated calculations. It could be thought from Table I that it is rather small. But this is an illusion since H m will make it evolve. In 29 Si it will increase to 4.5 MeV (≈ 500 keV above experiment), which grows up to 5.2 MeV in 40 Ca, now too small with respect to the binding energy (BE) difference 2BE( 40 Ca)-BE( 41 Ca)-BE( 39 Ca)= 7.2 MeV. The only way to decide whether these positionings are correct is through calculations such as the present ones. We return to this issue in Section IV.
The analysis of binding energies is a delicate exercise because external parameters have to be introduced. The philosophy behind H m is to make all calculations coreless. Because of thehω propagation (which should be extended to H M ), nuclei readjust their sizes and energies as N and Z change. If the bulk terms are added, there is, in principle, no need to fit anything, and the calculated energies are absolute -not referred to any core. In the present calculations the interaction was kept fixed, and the way to proceed is the traditional one, by referring all energies to the core of 16 O. First, we estimate Coulomb effects using V c = 0.717Z(Z − 1)(A −1/3 − A −1 ), and then fit It is generally assumed that ε should be close to the single-particle energy of 17 O (-4.14 MeV), and that the quadratic terms are the average H mT over the space (from Eq. (10)). However, these assumptions do not apply here. The contribution to b from H mT is relatively small. The symmetry energy must be counted as one of the bulk terms, and the best we can do is to take it from fits to the binding energies, which yield consistently similar numbers. From [45] we adopt the form S = 22[4T (T + 1)](1 − 1.82/A 1/3 )/A, where the main term has been reduced by the approximately 6 MeV coming from H m . We cannot change these parameters; we can only check that the fit to H corr yields a b consistent with them. But there is subtlety: the isospin term vanishes at m = 1 because it is taken to be two-body, while S gives a substantial 1.15 MeV contribution at 17 O. Therefore, to use the form of H corr , ε must be S-corrected (in the same sense that we Coulomb-correct) to -4.14-1.15=-5.29 MeV. For b we must take some average S, which we choose to be the value at A = 40, i.e., b = 2.34.
Finally for a we must expect a small value, since it should come entirely from H m . The fit yields (in MeV) ε = −5.34, a = −0.319, b = 1.99, and a χ 2 per degree of freedom of 3.12.
While ε and b are very comfortably close to our expectations, a is much too large. But that is not a problem: the program that transforms H m into V T rs had been thoroughly checked for excitation energies but not for binding energies. It had a bug in it that accounts for nearly -250 keV in the a term. Hence, when S is taken into account and the bug is corrected, the fit becomes (in KeV) ε = −50, a = −59, b = 235. The numbers are now pleasingly small and the principal uncertainty stems from the parameters in S.
Our mass results are shown in Figs. 4a,b. While there is both underbinding and overbinding of the nuclei studied, the agreement is reasonably acceptable. It becomes remarkable if we consider that -in view of the smallness of H corr -it is practically parameter free. For completeness, in Fig. 4c we show H CM for the same nuclei. Notice that for the nuclei above mass 40 the center-of-mass contamination could be further corrected by fine tuning β CM . However, for our present purposes, we will be content with the removal of much of the center-of-mass energy.
As a final example of the soundness of the interaction, we show in Fig. 5 a number of lowlying states for 22 Mg calculated by direct diagonalization in the full sd-pf space, compared to both a sd-shell calculation using the USD interaction and to experiment [51]. Generally, our interaction agrees reasonably well with both experiment and the USD interaction. The more refined treatment of H m will no doubt further improve the agreement. We also note that we have checked the center-of-mass contamination for all of the excited states shown in the first column of Fig. 5, and it is as small as that shown for the ground state in Fig. 2.

A. Comparison with experiment and other calculations
There is limited experimental information about the highly unstable, neutron-rich nuclei under consideration. In many cases only the mass, excitation energy of the first excited state, the B(E2) to that state, and the β-decay rate is known, and not even all of this information is available in some cases. From the measured B(E2), an estimate of the nuclear deformation parameter, β 2 , has been obtained via the usual relation with R 0 = 1.2A 1/3 fm and B(E2) given in e 2 fm 4 .
Much of the interest in the region stems from the unexpectedly large values of the deduced β 2 , results which suggest the onset of deformation and have led to speculations about the vanishing of the N = 20 and N = 28 shell gaps. The lowering in energy of the 2 + 1 state supports this interpretation. The most thoroughly studied case, and the one which most convincingly demonstrates these phenomena, is 32 Mg with its extremely large B(E2) = 454 ± 78 e 2 fm 4 and corresponding β 2 = 0.513 [1]; however, a word of caution is necessary when deciding on the basis of this limited information that we are in the presence of well-deformed rotors: for 22 Mg, we would obtain β 2 = 0.67, even more spectacular, and for 12 C, β 2 = 0.8, well above the superdeformed bands.
Most of the measured observables can be calculated within the SMMC framework. It is well known that in deformed nuclei the total B(E2) strength is almost saturated by the 0 + gs → 2 + 1 transition (typically 80% to 90% of the strength lies in this transition). Thus the total strength calculated by SMMC should only slightly overestimate the strength of the measured transition. In Table II the SMMC computed values of B(E2, total) are compared both to the experimental B(E2; 0 + gs → 2 + 1 ) values and to the values found in various truncated shell-model calculations. Reasonable agreement with experimental data across the space is obtained when one chooses effective charges of e p = 1.5 and e n = 0.5. We also indicate in the right column of Table II the USD values for the B(E2, 0 + gs → 2 + 1 ) (with effective charges of e p = 1.5 and e n = 0.5) for the sd-shell nuclei. Note that the sd-shell results are much lower for 30 Ne and 32 Mg than is seen experimentally. All of the theoretical calculations require excitations to the pf -shell before reasonable values can be obtained. We note a general agreement among all calculations of the B(E2) for 46 Ar, although they are typically larger than experimental data would suggest. We also note a somewhat lower value of the B(E2) in this calculation as compared to experiment and other theoretical calculations in the case of 42 S. Shown in Table III are effective charges from other calculations.  Table IV gives selected occupation numbers for the nuclei considered. We first note a difficulty in extrapolating some of the occupations where the number of particles is nearly zero. This leads to a systematic error bar that we estimate at ±0.2 for all occupations shown, while the statistical error bar is quoted in the table. The extrapolations for occupation numbers were principally linear. Table IV shows that 22 Mg remains as an almost pure sd-shell nucleus, as expected. We also see that the protons in 30 Ne, 32 Mg, and 42 Si are almost entirely confined to the sd shell. This latter is a pleasing result in at least two regards. First, it shows that the interaction does not mix the two shells to an unrealistically large extent. Second, if spurious center-of-mass contamination were a severe problem, we would expect to see a larger proton f 7/2 population for these nuclei due to the 0d 5/2 -0f 7/2 "transition" mediated by the center-of-mass creation operator. The fact that there is little proton f 7/2 occupation for these nuclei confirms that the center-of-mass contamination is under reasonable control.
An interesting feature of Table IV lies in the neutron occupations of the N = 20 nuclei ( 30 Ne and 32 Mg) and the N = 28 nuclei ( 42 Si, 44 S, and 46 Ar). The neutron occupations of the two N = 20 nuclei are quite similar, confirming the finding of Fukunishi et al. [8] and Poves and Retamosa [7] that the N = 20 shell gap is modified. In fact, the neutron f 7/2 orbital contains approximately two particles before the N = 20 closure, thus behaving like an intruder single-particle state. Furthermore, we see that 2p-2h excitations dominate although higher excitations also play some role. We also see that the neutrons occupying the pf -shell in N = 20 systems are principally confined to the f 7/2 sub-shell.
The conclusions that follow from looking at nuclei with N > 20, particularly those with N = 28, are that the N = 20 shell is nearly completely closed at this point, and that the N = 28 closure shell is reasonably robust, although approximately one neutron occupies the upper part of the pf shell. Coupling of the protons with the low-lying neutron excitations probably accounts for the relatively large B(E2), without the need of invoking rotational behavior.
In Table V we show the SMMC total Gamow-Teller (GT − ) strength. We compare our results to those of previous truncated calculations, where available. In all cases, our results are slightly smaller than, but in good accord with, other calculations. Since we do not calculate the strength function, we do not compute β-decay lifetimes.

B. Pairing properties
For a given angular momentum J, isospin T , and parity π, we define the pair operators as where the parity is given by (−1) la+l b . These operators are boson-like in the sense that they satisfy the expected commutation relations in the limit where the number of valence nucleons is small compared with the total number of single-particle states in the shell. In the SMMC we compute the pair matrix in the ground state as which is a hermitian and positive-definite matrix in the space of ordered orbital pairs (ab) (with a ≤ b). The total number of pairs is given by The pair matrix can be diagonalized to find the eigenbosons B † αJT Tzπ as where α = 1, 2, · · · labels the various bosons with the same angular momentum, isospin, and parity. The ψ αJT π are the eigenvectors of the diagonalization, i.e. the wavefunctions of the boson, and satisfy the relation These eigenbosons satisfy M B † αJM,T Tzπ B γJM,T Tzπ = n αJT Tzπ δ αγ , where the positive eigenvalues n αJT Tzπ are the number of JT T z π-pairs of type α. We first show the number of pairs P JT Tzπ in the J = 0, T = 1 positive-parity pairing channels. This quantity can be interpreted as the total strength for the pair transfer of particles of the given quantum numbers. Shown in Fig. 6 are our results in the protonneutron (top), proton-proton (middle), and neutron-neutron (bottom) channels as a function of the nucleus, A. Notice that only in the N=Z nucleus 36 Ar do the proton-neutron pairs play a significant role, as has been discussed in [58]. Generally, one also sees an increase in the proton-proton pairs as A is increased. Notice also that a fair amount of increase occurs in the sulfur and argonne isotope chains as one adds neutrons. This is not the case in the two Mg isotopes calculated, in which we see a significant increase in the neutron-neutron correlations, but very little change in the proton-proton sector. This holds for both the Ne and Mg chains [59]. For the heavier isotopes in the region, in general, the J = 0 neutronneutron pairs are not significantly enhanced for the nuclei that we have calculated here. Since there are many more particles and hence more pairing, one expects enhancements to occur in higher J pairs since the total number of pairs is a conserved quantity for a given number of like nucleons. We also calculated the pairing in the same channels, but with negative parity (π = (−1) la+l b = (−1) lc+l d ), and find it to be rather small in most cases.
Further insight into the pairing comes by considering diagonal elements of the pair matrix before and after diagonalization. The presence of a pair condensate in a correlated ground state will be signaled by the largest eigenvalue for a given J being much greater than any of the others. Shown in Fig. 7 are the diagonal matrix elements of the J = 0 pair matrix for 40,44 S before (left panel) and after (right panel) diagonalization. We see from the left panel that adding four neutrons to the system increases the f 5/2 -shell neutron matrix elements, while rearranging the sd-shell elements slightly. ¿From the occupation numbers we know that the neutrons are filling pf -shell orbitals, and therefore we expect little movement in the sd shell. The proton matrix elements are slightly affected by the addition of the neutrons, although there is some movement of protons out of the f 7/2 .
The largest eigenvalue of the neutron-neutron pair matrix, as shown in the right panel in Fig. 7, is about 1.5 times that of the next largest eigenvalue. However, the remaining eigenvalues are significant. Thus it is unlikely that there exists a pure pair condensate in the neutrons. As a further check on this conclusion, we have diagonalized the 3×3 pairing matrix resulting from only the sd-shell neutrons in these two nuclei. We find that the three eigenvalues are all of similar size and significantly smaller than the largest eigenvalue from the full sd-pf diagonalization. Thus, what neutron pair condensate does exist is a phenomenon which involves the entire model space, not just the sd shell. In the proton sector we see a similar level of pair condensation. Since the protons occupy mainly the sd-shell, only three eigenvalues are large enough to be represented in the figure.

C. Discussion
The aim of a nuclear structure calculation is to compare with, or predict, experimental results. In the present case, the comparison with other calculations is at least of equal interest. The reason comes from the problems created by truncations, in particular the (0+2)hω "catastrophe" [6], discovered long ago in a 0hω context [29]. Calling f the f 7/2 shell and r its pf partners generically, an f n calculation can produce very sensible results; f n + f n−1 r improves them considerably, but f n + f n−1 r + f n−2 r 2 is invariably disastrous, because the f n configuration is strongly pushed down by pairing with f n−2 r 2 , while f n−1 r does not benefit from a similar push from f n−3 r 2 . The remarkable thing is that when this last configuration is included, the results are not too different from the original f n + f n−1 r. If the space is expanded, there is an attenuated 4hω catastrophy. The process continues in increasingly attenuated form until the exact (f r) n space is reached. This is a general problem with truncations, and for nuclei such as 32 Mg with a (pf ) 2 (sd) n−2 ground state, where the calculations demand also the presence of (sd) n states , the adopted solution has been simply to ignore the mixing between the two configurations. It works very well. But is it true? Could it not be possible that higher excitations play an important role? To everybody's relief we can say that the present calculations confirm the basic validity of previous work. In all the cases we analyzed, the "dressing" process, whereby a dominant configuration becomes the exact ground state, does not seem to affect strongly its basic properties. Does it mean that exact calculations are unnecessary? Not exactly. For one thing, they have no parameters other than those of H m and therefore demonstrate the validity of the monopole corrected G-matrices. And then they go -for the first time in a shell-model contextto the heart of the problem of cross-shell correlations. At present, we know little about these problems, except that they hide so well that they are difficult to detect. Still, they can be seen through effects (such as quenching of Gamow-Teller strength) that tell us that they are important. The available evidence points to a much reduced discontinuity at the Fermi level with respect to the naive shell model [53][54][55]. In Table IV we find nearly normal occupancies for high T , but strong effects for T = 0 ground states, in particular for 44 Ti -a truly interesting case. A conventional (pf ) 4 calculation yields a BE2 of 514.7 e 2 f 4 (virtually identical to that of KB3). In Table II the result is at least 20% larger. This is a good example of the hiding talents of the correlations. No doubt this nucleus is a bona fide member of the pf space, and the correlation effects can be drowned in the experimental error, but it is not always the case. The region is plagued with BE2 transitions which are systematically too large for the 0hω calculations to explain, particularly for the Ca isotopes, which should be the simplest nuclei, but are the most complicated. In 44 Ti we have a first example of what a complete calculation could do.
Binding energies are no doubt one of the best ways to shed light on the matter. In Section III.C we mentioned the cross-shell gap around 40 Ca, which should be increased by about 2 MeV, which means that the correlation energy should be much larger. And since we know now that we can trust SMMC with a good interaction to within 1 MeV, probably it will not take long before we know more about this supposedly closed nucleus that is not so closed.

V. CONCLUSION
This paper was meant as a feasibility study of SMMC calculations in multi-hω spaces. Two general issues had to be tackled: translational invariance and the definition of an interaction. Concerning the first, it was shown that the trouble caused by center-of-mass excitations can be successfully mitigated by a judicious application of ideas in [23], and a possible variational approach to the problem was suggested. The interaction chosen was a G-matrix derived from a modern potential, renormalized according to state-of-the-art techniques, and monopole-corrected for the bad saturation properties of the existing NN potentials. The only parameters entering the calculations are the six "universal" constants specifying the monopole Hamiltonian, which was shown to explain quite naturally the shell formation properties of high isospin nuclei in the A=30-50 region.
The feasibility test was passed satisfactorily. Binding energies, B(E2) rates, and Gamow-Teller strengths were obtained that are in reasonable agreement with observations, and the possible origin of the remaining discrepancies has been identified.
The calculations support the validity of previous work in the region, and open the way to the study of the elusive deep correlations at the origin of Gamow-Teller quenching. In particular it provides an example, in 44 Ti, of an extremely correlated system whose behavior is quite similar to that of the uncorrelated one. The possibility to obtain orbit occupancies should help in advancing the study of the discontinuity at the Fermi surface -one of the most difficult problems in nuclear physics.
Interest was focused on neutron-rich nuclei around N =20,28 (a region of current interest), where new data has become available and many calculations have been performed. Having established the reliability of our methods, other exotic, or not so exotic, studies can be contemplated.
Most calculations presented here were performed on the 512-node Paragon at the Oak Ridge Center for Computational Science (CCS) and the T3E at the National Energy Research Scientific Computing Center (NERSC). The sd-pf model space effectively used all of the available memory on the Paragon (32 Mbytes per node) and, hence, larger spaces were not feasible there. With the advent of a new generation of massively parallel computers that are much faster and have far more memory, much more ambitious calculations are possible.