Let us initiate our analysis with the two deepest energy levels of the 16O spectrum, the 0+1 and 2+1 states, whose energy at N = 10 and a = 1.20 fm is equal to -23.60 and -18.35 MeV respectively. In order to suppress the discretization effects to at least≲10−1 MeV, we first fix the lattice spacing to a ≈0.59 fm. With the aim of allowing for an extension of the analysis to larger volumes by means of the Worldline algorithm for excited states (cf. sec. 4.8.1 a)),
7.2. THE 0+1 AND21+ ENERGY LEVELS 145 we set the derivative improvement index for the Laplacian to one, thus constrasting slightly the reduction of finite lattice spacing effects. The subsequent diagonalization of the lattice Hamiltonian Hr for N ≤ 10, returns the solid lines in fig. 7.3, whose behaviour closely re-sembles the one of the 0+1 and 2+1 curves in figs. 5.2 and 6.2 in the small N regime. In this region, the lattice eigenstates are sharply unbound, and their energy, independently on the discretization scheme adopted for the kinetic term of the Hamiltonian, reaches hundreds of MeV. The effect is merely a consequence of the repulsive part of the isotropic Ali-Bodmer potential, that moulds the behaviour of the wavefunctions at low values of theα-α separation.
FIGURE 7.3 – Behaviour of the energies of the lowest 0+ (horizontal bars) and 2+ (ver-tical bars) eigenstates as a function of the box sizeNfor a ≈ 0.59 fm. The multiplet-averaged eigenvalue of the 2+1 state is denoted in red.
The same convention on the markers for the cubic group irreps adopted in the figures of chaps. 5 and 6 is under-stood. The dashed lines de-note the expected behaviour of the solid curves in the large N regime, but their predictive power is limited by discretization errors in-creasing withN.
With the aim of pursuing the analysis beyond the present-day limitations of the GPU cards, we allow the lattice spacing to vary in order to predict the eigenenergy at any value of N, in the large lattice size region. More precisely, for each point withN ≥11 in fig.7.3, we perform a Lanczos diagonalization with lattice size Neff equal to 10 and lattice spacingaeff =a · N/Neff, where ais the original value, equal to 0.59 fm. In this guise, we enable the exploration of the region of lattice volumes where the restoration of SO(3) symmetry takes place, at the price of loosing information on the exact eigenvalues in the continuum and infinite-volume limit.
As a consequence, the dashed part of the curves in fig. 7.3 is a result of an interweaving of both finite volume and discretization effects. Despite the increase of the latter towards the infiniteN limit, theE and T2 multiplets become degenerate within ≈0.42 MeV atN = 25 and the overall behaviour of the curves in the asymptotic region recalls rather accurately the one of the 2+E and 2+T2 states in figs. 5.2and 6.2. Besides, it is likely to expect that both the 0+1 and the 2+E (2+T2) multiplets approach the asymptotic eigenvalue from below (above), as observed in the 8Be and 12C case.
Such a faithful reproduction of the infinite-volume behaviour of the curves was also ensured by the increase of the derivative improvement index to 4, thus quenching the growth of dis-cretization artifacts in the reconstructed energy eigenvalues withN. However, considered the result of the infinite-volume extrapolation in fig. 7.1, it is evident that the asymptotic energy eigenvalues in fig.7.3may differ from the exact ones of about 100 to∼101 MeV.
Next, we concentrate on the behaviour of the average values of the total squared angular
146 CHAPTER 7. THE 16O NUCLEUS momentum operator with the lattice size N. Implementing the same strategy adopted for the energy eigenvalues, we obtain for the 0+1 and 2+1 states the solid and dashed curves in fig. 7.4.
As it can be observed, both the theree curves display an evident peak in the region withN ∼7, as in the12C case in fig.6.5, followed by a rather smooth settling ofL2 towards the asymptotic average value. Although the squared angular momenta of the 2+E and 2+T2 multiplets overlap within 0.17 units of ℏ2 of precision at N = 25, the interplay between residual finite volume and discretization effects results into a discrepancy of about 6 ℏ2 with respect to the exact eigenvalue ofL2. A similar conclusion holds for the ground state, where the closest approach of the squared angular momentum to zero, equal to ≈3.6ℏ2, is recorded at N = 20.
FIGURE 7.4 – Average value of the squared angular momentum for the 0+1 and 2+1 states as a function of the lattice size for a ≈ 0.59 fm. The multiplet average ofL2 for the 2+1 state is denoted by a red line. Dashed lines denote the expected behaviour of L2 in the largeN regime, but their predictive power is limited by discretization errors increasing withN.
FIGURE 7.5 – Behaviour of the averageα-αdistance for the 0+1 and 2+1 multiplets as a function of the lattice size fora ≈0.59 fm. The multiplet average of R for the 2+1 state is denoted with a red line.
The dashed lines denote the expected behaviour ofR in the large N regime, but their reliability is affected by discretization errors increasing withN.
As in sec.7.2, we now consider the average values of theα-α separation,R, as a function of the lattice size. Although at N = 25, the edge of the cubic box measures only 12.50 fm, the dashed curves of the interparticle distance for the 0+1 and 2+1 states almost reach a plateau as the ones in fig. 6.3, with R equal to ≈ 3.94 and 4.30 fm respectively. Since the equilibrium arrangement of the 4He clusters within the nucleus is tetrahedral, an estimate of the residual finite-volume and discretization errors introduced byaeff> a atN = 25 can be obtained from comparison of the ground state value of the centroid to vertex distance,
√ 3R/√
8, with the experimental charge radius of 16O in ref. [224,231], corresponding to≈2.69 fm. Despite the quite large effective lattice spacing, our lattice estimate at N = 25 of the latter is equal to
≈ 2.41 fm, in reasonably close to the observational radius and in good agrrement with the numerical result of the unitary-model-operator approach (UMOA) in ref. [232] (2.44 fm).
It follows that R, in the asymptotic region of fig.7.5, is rather insensitive to finite-volume and discretization artifacts. Concerning the 2+E and 2+T
2 states, the average values ofα-αseparation at N = 25 already overlap within an astonishing precision, equal to 0.004 fm.
We now proceed by analogy with secs. 6.2 and 7.2, displaying the behaviour of physical ob-servables with the lattice spacing a. Although the extent of residual finite-volume effects for
7.2. THE 0+1 AND21+ ENERGY LEVELS 147 the enegy eigenvalues of the 0+1 and 2+1 states at Na ∼ 20 fm is not known with precision, we assume from comparison with the data in figs. 7.2 and 8.1 that for lattices with Na ≳ 20 these artifacts lie between 100 and 10−1 MeV. Therefore, we fix the constraintNa ≥20 fm and explore the behaviour of the energy eigenvalues in the interval 2.0≤ a ≤5.0 fm for the two angular momentum multiplets (cf. fig. 7.6), denoting the curves with solid lines. In sight of the extension of the curves towards thea <2.0 fm region, we set the derivative improvement index to unity, as done in the N ≤10 region of figs. 7.3-7.5.
FIGURE 7.6 – Behaviour of the energies of the 0+1 and 2+1 eigenstates as a function of the lattice spa-cing for Na ≥20 fm. The multiplet-averaged 2+1 energy is denoted with a red line. The dashed part of the curves denote the expected behaviour ofEr
in the smalla regime, but its predictive power is limited by finite-volume effects, inversely propor-tional toa.
FIGURE 7.7 – Behaviour of the averageα-αdistance of the 0+1 and 2+1 eigenstates as a function of the lattice spacing for Na ≥ 20 fm. The multiplet-averaged values ofR for the 2+E and 2+T2 states are connected with a red line. The dashed part of the curves denote the expected behaviour of average α-α separation in the small a regime, but its pre-dictive power is limited by finite-volume effects, in-versely proportional toa.
As noticed in figs.5.5and6.6and in ref. [3], the evolution curves of the energy eigenvalue display an oscillatory behaviour and their extremal points are susceptible of a geometric inter-pretation, based on the spatial distribution of the probability density functions (PDF) associated to the lattice wavefunctions. Differently from the8Be and12C nuclei, the16O PDFs in the small-spacing limit can not be faithfully represented, due to the present limitations in the number of mesh points per dimension, N. Nevertheless, at least the region a < 2.0 fm of the dis-cretization plot in fig. 7.6(dashed lines) can be approximately reconstructed, albeit tolerating the presence of larger finite-volume errors. The points with a < 2.0, in fact, refer to lattices with only N = 10 points per dimension. Additionally, in the dashed part of the curves the derivative improvement index, K, has been restored to 4, in order to reproduce asymptotic energy eigenvalues consistent with the ones in fig. 7.3.
In the solid part of the curves for the 0+1 and 2+1 states in fig. 7.6 two well-developed minima can be observed at a ≈2.3 and 3.1 fm. In addition, the minima of the average α-α distance in fig. 7.7, detected at a ≈2.1 and 2.9 fm and the ones of the potential energyV, at a ≈2.25 and 3.05 fm, lend weight to the hypothesis that a maximum of the related PDFs is included exactly or in good approximation in the lattice configuration space. An analogous behaviour was observed for the 2+E and 2+T
2 multiplets of8Be, whose PDF in figs.5.8and5.10exhibit deep minima located on the lattice axes.
148 CHAPTER 7. THE 16O NUCLEUS Finally, it is notewothy the accidental quasi-degeneracy of the energy eigenvalues of the 0+A
1
and 2+E multiplets for a≲2.9 fm that is accompanied by a neat overlap in the average values of R (cf. fig. 7.7) and L2, a fact that may disorient in the classification of states in terms of irreps of SO(3).