Linear regularization scheme

The LR scheme applies, for example, to the density-independent SV interaction in its original formulation [18], that is, without the EDF tensor terms. It is also applicable to the density-dependent SIII functional [18]. The reason is that the density-dependence of this latter force does not lead, for a given type of particles, to a higher power of density. The third example, where the LR should be sufficient, is the Coulomb exchange treated in the so called Slater approximation [25], because in this approximation the exchange Coulomb transition matrix element behaves as

$$\tilde{\rho}^{\, 4/3} \, \langle \Psi \vert \tilde{\Psi}\rangle \sim \langle \Psi \vert \tilde{\Psi}\rangle^{-1/3}.$$ (19)

In cases when the LR scheme is sufficient to cancel all poles of the integrand, the set of auxiliary quantities $V^{{\rm 2B,1}}_{IMK}$ (13) can be calculated, in principle exactly, using suitably chosen quadratures with sufficient number of integration nodes.

The overlap is always regular and, therefore, it can be expanded in terms of the Wigner $D$-functions as,

$${\rm {{\cal N}}} (\Omega ) \equiv \langle \Psi \vert \tilde{\Psi}... ... = \sum_{I''M''K''} c_{I''M''K''}^{\rm {{\cal N}}} D^{I''}_{M'' K''} (\Omega ),$$ (20)

where

$$c_{IMK}^{\rm {{\cal N}}} \equiv \langle \Psi \vert \hat{P}^I_{MK}... ...\Omega\, D^{I\, ^\star}_{MK} (\Omega ) \langle \Psi \vert \tilde{\Psi}\rangle .$$ (21)

Substitution of expansions (16) and (20) into Eq. (18), taken at $n=1$, leads to:

$$\tilde{V}^{{\rm 2B,1}}_{IMK} = \frac{2I+1}{8\pi^2} \sum_{I'M'K'} \tilde{V}_{I'M'K'}^{\rm {2B}} \sum_{I''M''K''} c_{I''M''K''}^{\rm {{\cal N}}}$$

$$\int d\Omega\, D^{I\, ^\star}_{MK} (\Omega )\, D^{I'}_{M' K'} (\Omega )\, D^{I''}_{M'' K''} (\Omega ).$$ (22)

In the case of $I+I'+I''$ being half-integer, the integration over the single volume must be replaced by integration over the double volume [7]. The integral in Eq. (22) is equal [7] to

$$\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I\, ^\star}_{MK} (\Omega )\, D^{I'}_{M' K'} (\Omega )\, D^{I''}_{M'' K''} (\Omega )$$

$$= {\rm {\bf C}}^{IM}_{I''M''I'M'} {\rm {\bf C}}^{IK}_{I''K''I'K'},$$ (23)

where symbols ${\rm {\bf C}}$ stand for the Clebsch-Gordan coefficients. The integral has the same form both for integer and half-integer angular momenta.

Inserting (23) to (22), and requesting that Eq. (17) holds at $n=1$, gives rise to a set of linear equations for regularized matrix elements $\tilde{V}^{{\rm 2B}}_{I'M'K'}$:

$$V^{{\rm 2B,1}}_{IMK} = \sum_{I'M'K'} A^{IMK}_{I'M'K'} \tilde{V}^{{\rm 2B}}_{I'M'K'} ,$$ (24)

where

$$A^{IMK}_{I'M'K'} = \sum_{I''M''K''} c_{I''M''K''}^{\rm {{\cal N}}} {\rm {\bf C}}^{IM}_{I''M''I'M'} {\rm {\bf C}}^{IK}_{I''K''I'K'} .$$ (25)

Matrix $A^{IMK}_{I'M'K'}$ is quadratic for even-even and odd-odd nuclei and rectangular for odd-$A$ nuclei. The problem of finding the regularized matrix elements within the LR scheme is thus reduced to calculating auxiliary quantities (13) for $n=1$ and then solving a set of linear equations (24). In the HFODD solver, the latter is handled by using the singular-value-decomposition (SVD) technique.

We note here that the regularization procedure can be applied separately to all terms of the EDF, that is, terms that correspond to interactions can be treated within the standard AMP method, and only those which do not, should be treated within the regularization scheme.

The expansion of Slater determinant $\vert\Psi\rangle$ in terms of the AMP states reads [6],

$$\vert\Psi \rangle = \sum_{IK} \vert IKK\rangle = \sum_{IK} \hat{P}^I_{KK} \vert\Psi \rangle .$$ (26)

In turn, the sum rule, which connects mean-field averages and projected matrix elements, has the form

$$\langle \Psi \vert \hat V_{\rm {2B}} \vert\Psi \rangle = \sum_{IK... ...t V_{\rm {2B}} \hat{P}^I_{KK} \vert\Psi \rangle = \sum_{IK} V_{IKK}^{\rm {2B}}.$$ (27)

The sum rule expresses the HF mean-field average value in terms of the projected matrix elements, and thus constitutes a stringent test of the performed AMP. On the one hand, when $\hat V_{\rm {2B}}$ is a true interaction, the sum rule must be strictly obeyed. On the other hand, for singular energy kernels, its violation gives a numerical estimate of problems related to not using true interactions. Similarly, the sum rule calculated for regularized matrix elements,

$$\langle \Psi \vert \hat V_{\rm {2B}} \vert\Psi \rangle = \sum_{IK} \tilde{V}_{IKK}^{\rm {2B}},$$ (28)

tests the quality of the regularization procedure. Note that sum rules must be obeyed separately for all terms in the interaction, which allows for studying singularities of energy kernels of separate terms. In what follows, we show results obtained for the sum-rule residuals, that is, for differences between right- and left-hand sides of Eqs. (27) and (28). Apart from those, we also assess precision of the AMP by considering energy $E_{I=0}$ of the lowest $I=0$ state.

More precisely, we focus our attention on investigating stability of these two quantities in function of the highest angular momentum ${I_{\text{max}}}$ included in the calculations, and we present them versus ${I_{\text{max}}}$. The same value of ${I_{\text{max}}}$ is consistently used to define both summation ranges in Eqs. (24) and (25). Note that in Eq. (25), the range of summation should be higher than the natural cutoff dictated by the highest meaningful AMP components in the mean-field wave function, which are given by the values of amplitudes $c_{IMK}^{\rm {{\cal N}}}$. With increasing values of ${I_{\text{max}}}$, the residuals of sum rules (27) and (28), should converge to zero.

All calculations were performed using the unrestricted-symmetry solver HFODD [26,27]. We employed the Gauss-Chebyshev quadratures to integrate over the $\alpha$ and $\gamma$ Euler angles and the Gauss-Legendre quadrature to integrate over the $\beta$ Euler angle. To achieve a sufficient accuracy, for each Euler angle we used a large number of mesh points equal $N_\alpha = N_\beta = N_\gamma \equiv N = 50$.

The examples presented below pertain to odd-odd nucleus $^{26}$Al, and to the so called anti-aligned mean-field configuration, which is relevant in the context of the superallowed Fermi $\beta$-decay [19]. The most demanding task was to calculate the auxiliary integrals $V^{{\rm 2B,1}}_{IMK}$, Eq. (13). Since we were interested in comparing the standard and regularized calculations, we decided to use a relatively small configuration space, consisting of only $N_{\rm shell}=6$ spherical harmonic-oscillator shells. Such small space suppresses high angular-momentum components in the reference Slater determinant. Unless explicitly stated, in all calculations, in both direct and exchange channels the Coulomb interaction was treated exactly.

Figure 1: (Color online) Convergence of the lowest $I$=0 energy (top) and Skyrme-energy sum-rule residuals (bottom) in function of the highest angular momentum ${I_{\text{max}}}$ included in the calculations. Open and full circles represent results obtained using the standard AMP method and our LR method, respectively. Calculations were performed for the true Skyrme interaction SV$_{\rm T}$ (that is, with the tensor EDF terms included).

It is instructive to begin the discussion by showing results for $E_{I=0}$ and Skyrme-energy sum-rule residuals obtained for the SV$_{\rm T}$ Skyrme force. Such a calculation tests the numerical implementation of the method, and can be regarded as a proof of principle of the LR scheme. The reason is, as already mentioned, that the SV$_{\rm T}$ is a true interaction and, therefore, both standard AMP method and LR method should give exactly the same values of both indicators. As can be seen in Fig. 1, this is indeed the case. It turns out that for ${I_{\text{max}}}\geq10$, the standard AMP values of $E_{I=0}$ are perfectly stable (up to a fraction of eV). However, the sum rule, which also tests the convergence of higher angular momenta, reaches a similar level of precision only above ${I_{\text{max}}}=20$. The LR values of $E_{I=0}$ converge only above ${I_{\text{max}}}=20$, which illustrates the fact that in Eq. (24), higher intermediate angular momenta must be taken into account. Note, however, that the sum rules calculated using both methods converge in a similar smooth way.

Figure 2: (Color online) Same as in Fig. 1 but for the original Skyrme functional SV, that is, with the tensor EDF terms neglected.

The density-independent Skyrme parametrization SV in its original formulation [18], that is, without the tensor EDF terms, no longer corresponds to an interaction. Fig. 2 clearly shows that even such a seemingly insignificant departure from the true Hamiltonian is immediately detectable through the indicators tested in our study. In the standard AMP, energy $E_{I=0}$ is again perfectly stable over the entire range of studied values of ${I_{\text{max}}}$. However, such stability can be misleading, because the LR value, which converges only at ${I_{\text{max}}}=20$, differs by as much as 2keV.

Note that the singularity of energy kernels leaves its fingerprint in the values of the standard-AMP sum-rule residuals. After an apparent convergence (at the level of a few keV), which is visible below ${I_{\text{max}}}=16$, at the level of a few eV, this indicator, in fact, does not converge to zero. On the other hand, the LR sum-rule residuals smoothly converge to zero with high precision. An important conclusion obtained here is that the stability of the ground-state energy does not necessarily warrant that its value be free from spurious effects.

Figure 3: Same as in Fig. 1 but for the original SIII Skyrme functional. Panels (b) and (c) show sum-rule residuals of the Skyrme and Coulomb energies, respectively. Slater approximation of the Coulomb exchange energy was used.

For the density-dependent SIII functional [18], problems encountered within the standard AMP are further magnified. In this example, we performed calculations using the Slater approximation [25] of the Coulomb exchange energy. Therefore, here we applied our LR method both to the Skyrme and Coulomb parts of the functional. The results are depicted in Fig. 3, showing the energy $E_{I=0}$ (upper panel), Skyrme-energy sum-rule residuals (middle panel), and Coulomb-energy sum-rule residuals (lower panel). Similarly as in Fig. 2, the standard AMP leads to misleadingly stable values of $E_{I=0}$; however, now the corresponding sum rules turn out to be completely unstable. For the Skyrme and Coulomb energies, they stagger around zero at the level of 50keV and 50eV, respectively. In contrast, the LR method perfectly stabilizes the sum rules, which smoothly converge to zero, and leads to stable values of $E_{I=0}$. However, the LR $E_{I=0}$ energy is now shifted down by almost 50keV, as compared to the standard AMP solution.

Figure 4: (Color online) Energies $E_{I=0}$ calculated using the standard AMP (open circles) and LR (full circles) methods, as functions of the number of mesh points $N$. Calculations were performed for the original SIII Skyrme functional. The maximum angular momentum of ${I_{\text{max}}}=20$ was used.

Finally, let us point out yet another shortcoming of the standard AMP approach. Fig. 4 shows results of the test of stability of $E_{I=0}$ with respect to the number of mesh points $N$ used in numerical integrations over the Euler angles. In this example, calculations were performed for the SIII functional [18] and exact Coulomb exchange energy. It is clearly visible that the standard-AMP values of $E_{I=0}$ vary strongly and quite erratically with $N$. This is owing to the fact that the results do depend on relative positions of mesh points with respect to singularities of the energy kernel. In contrast, the LR results are perfectly stable.