The projected DFT framework

The building block of the isospin-projected DFT is the Slater determinant, $\vert\Phi\rangle$, representing the self-consistent Skyrme-HF solution provided by the HF solver HFODD.[10] Self-consistency ensures that the balance between the long-range Coulomb force and short-range strong interaction, represented in our model by the Skyrme energy density functional (EDF), are properly taken into account. The unphysical isospin mixing is taken care of by the rediagonalization of the entire Hamiltonian in the good isospin basis, $\vert T,T_z\rangle$, as described in Refs.[8,9] This yields the eigenstates:

$$\vert n,T_z\rangle = \sum_{T\geq \vert T_z\vert}a^n_{T,T_z}\vert T,T_z\rangle$$ (1)

numbered by an index $n$. The so-called isospin-mixing coefficients (or, equivalently, isospin impurities) are defined for the $n-$th eigenstate as

$$\alpha_C^n = 1 - \vert a^n_{T,T_z}\vert _{\mbox{\rm\scriptsize {max}}}^2 ,$$ (2)

where $\vert a^n_{T,T_z}\vert _{\mbox{\rm\scriptsize {max}}}^2$ stands for the dominant amplitude in the wave function $\vert n,T_z\rangle$.

Within the isospin- and angular-momentum-projected DFT, we use the normalized basis of states $\vert I,M,K; T,T_z\rangle$ having both good angular momentum and good isospin.[11] Here, $M$ and $K$ denote the angular-momentum components along the laboratory and intrinsic $z$-axes, respectively. The $K$ quantum number is not conserved. In order to avoid problems with overcompleteness of the basis, the $K$-mixing is performed by rediagonalizing the Hamiltonian in the so-called collective space, spanned for each $I$ and $T$ by the natural states, $\vert IM;TT_z\rangle^{(i)}$, as described in Refs.[10,12] Such a rediagonalization yields the eigenstates:

$$\vert n; IM; T_z\rangle = \sum_{i,T\geq \vert T_z\vert} a^{(n)}_{iIT} \vert IM; TT_z\rangle^{(i)} ,$$ (3)

which are labeled by the index $n$ and by the conserved quantum numbers $I$, $M$, and $T_z=(N-Z)/2$ [compare Eq. (1)].

Figure 1: The absolute values of the norm kernels, $\vert{\cal N}(\beta_T; \alpha, \beta, \gamma )\vert = \vert\langle \Phi \vert \hat{R}(\beta_T ) \hat{R}(\alpha, \beta, \gamma ) \vert\Phi\rangle\vert$, for a state in $^{14}$N calculated with the SLy4 EDF, plotted versus the rotation angle in the isospace $\beta_T$. The solid curve, exhibiting the single singularity at $\beta_T = \pi$, corresponds to the pure isospin-projected DFT theory, which is regular for all Skyrme-type functionals.[8] The dotted lines correspond to two fixed sets of the Euler angles in space, with $\alpha =\gamma \approx 0.314$, and $\beta \approx 0.229$ (left curve) and $\beta \approx 1.414$ (right curve). The poles that appear inside the integration region, $0<\beta_T<\pi$, give rise to singularities in the projected DFT approach.

The isospin projection does not produce singularities in energy kernels; hence, it can be safely used with all commonly used EDFs.[8] Coupling the isospin and angular-momentum projections, however, leads to singularities in both the norm (see Fig. 1) and energy kernels. This fact narrows the applicability of the model to Hamiltonian-driven EDFs which, for Skyrme-type functionals, leaves only one option: the SV parametrization.[13] The alternative would be to use an appropriate regularization scheme, which is currently under development.[14,15]