Standard MR DFT scheme

In this work, all implementations of projection methods are based on the GWT, which allows for deriving compact and numerically tractable expressions for off-diagonal matrix elements between Slater determinants. For an arbitrary Slater determinant $\vert\Psi\rangle$, by $\vert\tilde{\Psi}\rangle = \hat R(\Omega )\vert\Psi\rangle$ we denote the one that is rotated in space, gauge space, or isospace, for the angular-momentum, particle-number, or isospin restoration, respectively. Hereafter, we focus our attention on the AMP, but the presented ideas and methodology can be rather straightforwardly generalized to the particle-number [5] or isospin projections.

In order to bring forward the origin of singularities in energy kernels [1,2,3], it is instructive to recall principal properties of the standard GWT approach. Let us start with a one-body density-independent operator $\hat F = \sum_{ij} F_{ij} \hata _i^\dagger \hata _j$. Its off-diagonal kernel (the matrix element divided by the overlap), can be calculated with the aid of GWT, and reads [4]:

$$\frac{ \langle\Psi \vert {\hat{F}}\vert\tilde{\Psi}\rangle}{\lang... ...+_i \hata _j}\hfil$\crcr}}}\limits \equiv \sum_{ij}\; F_{ij} \tilde{\rho}_{ji},$$ (1)

where

$$\tilde{\rho}_{ji} \equiv \mathop{\vbox{\ialign{ ...$$ (2)

denotes transition density matrix. Therefore, its matrix element between the unprojected state $\vert\Psi\rangle$ and AMP state $\vert IMK\rangle = \hat{P}^I_{MK} \vert \Psi\rangle$ can be calculated from

$$F_{IMK} \equiv \langle\Psi \vert {\hat{F}} \hat{P}^I_{MK} \vert \Psi\rangle$$

$$%\frac{ \langle\Psi \vert {\hat{F}} \hat{P}^I_{MK} \vert \Psi\rangle}{\langle\Psi \vert \hat{P}^I_{MK} \vert \Psi \rangle} \nonumber \\ = \frac{2I+1}{8\pi^2} \int d\Omega\, D^{I\, ^\star}_{MK} (\Omega ) \langle\Psi \vert {\hat{F}}\vert\tilde{\Psi}\rangle,$$ (3)

where

$$\hat{P}^I_{MK} = \frac{2I+1}{8\pi^2 } \int D^{I\, *}_{M K}(\Omega ) \hat{R}(\Omega ) \, d\Omega$$ (4)

is the AMP operator, $D^{I}_{M K}(\Omega )$ is the Wigner function, and $\hat R (\Omega ) = e^{-i\alpha \hat I_z} e^{-i\beta \hat I_y} e^{-i\gamma \hat I_z}$ stands for the active rotation operator in space, parametrized in terms of Euler angles $\Omega = (\alpha, \beta, \gamma )$, and $M$ and $K$ denote the angular-momentum components along the laboratory and intrinsic $z$-axis, respectively [6,7].

The immediate conclusion stemming from Eqs. (1)-(2) is that the overlaps, which appear in the denominators of the matrix element and transition density matrix, cancel out, and the matrix element $\langle\Psi \vert {\hat{F}}\vert\tilde{\Psi}\rangle$ of an arbitrary one-body density-independent operator $\hat F$ is free from singularities and can be safely integrated, as in Eq. (3).

Let us now turn our attention to two-body operators. The most popular two-body effective interactions used in nuclear structure calculations are the zero-range Skyrme [8,9] and finite-range Gogny [10] effective forces. Because of their explicit density dependence, they should be regarded, for consistency reasons, as generators of two-body part of the nuclear EDF. The transition matrix element of the two-body generator reads:

$$\langle\Psi\vert\hat V_{{\rm 2B}}\vert\tilde{\Psi}\rangle = \frac... ...langle\Psi\vert\hata ^+_i \hata ^+_j \hata _l \hata _k\vert\tilde{\Psi}\rangle,$$ (5)

where $\bar{V}_{ijkl}\left[\tilde{\rho}\right]$ denotes the antisymmetrized transition-density-dependent matrix element. Gogny and Skyrme effective interactions both contain local terms proportional to $\rho^{\,\eta}$ which, in the MR DFT formulation, are usually replaced with the transition (mixed) density $\rho^{\,\eta} \rightarrow \tilde \rho^{\,\eta}$ [11]. Such a procedure, although somewhat arbitrary, is very common, because it fulfills a set of internal consistency criteria formulated in Refs. [12,13]. These include hermiticity, independence of scalar observables on the orientation of the intrinsic system, and consistency with the underlying mean field. The alternative way of proceeding is to substitute density-dependent terms with projected density [14] or average density [15]. These scenarios do not fulfill all the consistency criteria and will not be discussed here.

Evaluating the transition matrix element, Eq. (5), with the aid of GWT, one obtains,

$$\frac{\langle\Psi\vert\hat V_{{\rm 2B}}\vert\tilde{\Psi}\rangle}{\langle\Psi\vert\tilde{\Psi}\rangle} = \frac{1}{4}\sum_{ijkl} \bar{V}_{ijkl}\left[\tilde{\rho}\right]\; ... ...ineskip} $\hfil\displaystyle{\hata _l \hata _k}\hfil$\crcr}}}\limits \; \right.$$

$$+$$ (6)

Furthermore, for particle-number-conserving theory, contractions and vanish, whereas the remaining two contractions give products of two transition density matrices,

$$\frac{\langle\Psi\vert\hat V_{{\rm 2B}}\vert\tilde{\Psi}\rangle}{\langle\Psi\vert\tilde{\Psi}\rangle} = \frac{1}{4}\sum_{ijkl} \bar{V}_{ijkl}\left[\tilde{\rho}\right]\; ... ...tilde{\rho}_{ki}\tilde{\rho}_{lj} - \tilde{\rho}_{li}\tilde{\rho}_{kj} \right),$$ (7)

or

$$\frac{\langle\Psi\vert\hat V_{{\rm 2B}}\vert\tilde{\Psi}\rangle}{\langle\Psi\vert\tilde{\Psi}\rangle} = \frac{1}{4}\sum_{ijkl} \bar{V}_{ijkl}\left[\tilde{\rho}\right]\; ... ...hata _l\vert\tilde{\Psi}\rangle}{\langle\Psi\vert\tilde{\Psi}\rangle^2} \right.$$

$$- \left. \frac{\langle\Psi\vert\hata ^+_i \hata _l\vert\tilde{\Psi}... ...ata _k\vert\tilde{\Psi}\rangle}{\langle\Psi\vert\tilde{\Psi}\rangle^2} \right),$$ (8)

that is, the transition matrix element reads

$$\langle\Psi\vert\hat V_{{\rm 2B}}\vert\tilde{\Psi}\rangle = \frac{1}{2}\sum_{ijkl} \bar{V}_{ijkl}\left[\tilde{\rho}\right]\; ... ...a ^+_j \hata _l\vert\tilde{\Psi}\rangle}{\langle\Psi\vert\tilde{\Psi}\rangle} .$$ (9)

This defines the matrix element between the unprojected and AMP states,

$$V^{{\rm 2B}}_{IMK} % \frac{ \langle\Psi \vert {\hat{V}_{{\rm 2B}}... ...si\rangle}{\langle\Psi \vert \hat{P}^I_{MK} \vert \Psi \rangle} \nonumber \\ = \frac{2I+1}{8\pi^2} \int d\Omega\, D^{I\, ^\star}_{MK} (\Omega ) \langle\Psi \vert {\hat{V}_{{\rm 2B}}}\vert\tilde{\Psi}\rangle .$$ (10)

We note here that, because of the density dependence of the two-body interaction, the analogue the first member of Eq. (3), that is, $V^{{\rm 2B}}_{IMK} \equiv \langle\Psi \vert {\hat{V}_{{\rm 2B}}} \hat{P}^I_{MK} \vert \Psi\rangle$ is not valid. Nevertheless, expression (10) constitutes a consistent definition of the matrix element.

At variance with the one-body case discussed above, the integrand in Eq. (10) is inversely proportional to the overlap, thus containing potentially dangerous (singular) terms. The singularity disappears only if the sums in the numerator, evaluated at angles $\Omega$ where the overlap $\langle\Psi\vert\tilde{\Psi}\rangle$ equals zero, give a vanishing result; such a cancellation requires evaluating the numerator without any approximations or omitted terms. An additional singularity is created by the density dependence of the interaction.

If some approximation of the numerator is involved, the leading-order singularity goes as:

$$\langle\Psi \vert {\hat{V}_{{\rm 2B}}}\vert\tilde{\Psi}\rangle \sim \frac{1}{ \langle\Psi \vert \tilde{\Psi}\rangle ^{1+\eta}} ,$$ (11)

with the term $1/6\leq \eta \leq 1$ inherited from the direct density dependence of the Gogny or Skyrme effective forces which are, as already mentioned, commonly used to generate the modern non-relativistic nuclear EDFs. This singularity precludes, in general, determination of the integral in Eq. (10). Only in special cases, e.g., for signature-symmetry conserving states in even-even nuclei [16,17], the overlaps never vanish, and thus problems related to singular kernels do not appear.

Thus, the GWT formulation of MR DFT is, in general, singular. In fact, it is well defined only for ${\hat{V}_{{\rm 2B}}}$ being a true interaction. An example of such an EDF generator is the density-independent Skyrme interaction SV$_{\rm T}$, which is the SV interaction of Ref. [18] with all the EDF tensor terms included (these were omitted in the original definition of SV). Interaction SV$_{\rm T}$ was recently used to calculate the isospin-symmetry-breaking corrections to superallowed $0^+\rightarrow 0^+$ $\beta$-decay by means of the isospin- and angular-momentum projected DFT formalism [19].

Progress in development of projection techniques and difficulties in working out reliable regularization schemes for density-dependent interactions [2] increased the demand for density-independent effective interactions and stimulated vivid activity in this field resulting in developing density-independent zero-range [20,21] as well as finite-range [22] forces. The spectroscopic quality of these new forces is, however, still far from satisfactory. In addition, the technology of performing beyond-mean-field calculations with these novel interactions is being developed only now [23,24].

The pathologies arising in the GWT description of the two-body energy kernels come from uncompensated zeros of the overlap matrix. The central idea of this work is to cure the problem by replacing the calculation of projected matrix elements with higher-order quantities, which are regularized by multiplying the integrands with an appropriately chosen power of the overlap:

$$\langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle \ri... ...{\rm {2B}} \vert \tilde{\Psi}\rangle \langle \Psi \vert \tilde{\Psi}\rangle^n .$$ (12)

The proposed regularization scheme amounts to replacing the calculation of matrix elements $V^{{\rm 2B}}_{IMK}$, given in Eq. (10), by the calculation of an auxiliary quantities defined as:

$$V^{{\rm 2B,n}}_{IMK} = \frac{2I+1}{8\pi^2} \int d\Omega\, D^{I\, ... ...{\rm {2B}} \vert \tilde{\Psi}\rangle \langle \Psi \vert \tilde{\Psi}\rangle^n .$$ (13)

The central assumption behind such a regularization method is that the two-body matrix element $\langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle$ is regularizable, meaning that there exists a regularization procedure allowing for removal of singularities and replacing the infected matrix elements by regular ones,

$$\langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle \lo... ...w \widetilde{ \langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle } ,$$ (14)

for which the projected matrix elements can be calculated as in Eq. (10),

$$\tilde{V}^{{\rm 2B}}_{IMK} = \frac{2I+1}{8\pi^2} \int d\Omega\, D^{I\, ^\star}_{MK} (\Omega ) \widetilde{\langle\Psi \vert {\hat{V}_{{\rm 2B}}}\vert\tilde{\Psi}\rangle} ,$$ (15)

and which, in turn, can be expanded on a series of the Wigner $D$-functions:

$$\widetilde{ \langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle } = \sum_{I'M'K'} \tilde{V}_{I'M'K'}^{\rm {2B}} D^{I'}_{M' K'} (\Omega ).$$ (16)

Indeed, by inserting expansion (16) into Eq. (15), and employing the orthonormality conditions of the Wigner $D-$functions [7], one straightforwardly obtains the desired result.

Finally, the regularized matrix elements $\widetilde{ \langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle }$ are determined be requesting that the auxiliary quantities (13), calculated before and after regularization are equal, that is,

$$V^{{\rm 2B,n}}_{IMK} \equiv \tilde{V}_{IMK}^{\rm {2B,n}}.$$ (17)

for

$$\tilde{V}^{{\rm 2B,n}}_{IMK} = \frac{2I+1}{8\pi^2} \int d\Omega\,... ...\rm {2B}} \vert \tilde{\Psi}\rangle} \langle \Psi \vert \tilde{\Psi}\rangle^n .$$ (18)

Let us underline that our method does not require any explicit a priori knowledge of the regularization scheme. Also note that the expansion coefficients $\tilde{V}^{{\rm 2B}}_{IMK}$, appearing in Eqs. (15) and (16), represent true regularized (two-body) matrix elements.

Two variants of such a regularization scheme, dubbed linear regularization (LR) and quadratic regularization (QR), corresponding to, respectively, $n=1$ and $n=2$, are discussed below in Secs. 2.2 and 2.3.