Isospin- and angular-momentum-projected DFT approach

The building block of the isospin- and angular-momentum-projected DFT approach employed in this study is the self-consistent deformed MF state $\vert\varphi \rangle$ that violates both the rotational and isospin symmetries. While the rotational invariance is of fundamental nature and is broken spontaneously, the isospin symmetry is violated both spontaneously and explicitly by the Coulomb interaction between protons. The strategy is to restore the rotational invariance, remove the spurious isospin mixing caused by the isospin SSB effect, and retain only the physical isospin mixing due to the electrostatic interaction [23,24]. This is achieved by a rediagonalization of the entire Hamiltonian, consisting the isospin-invariant kinetic energy and Skyrme force and the isospin-non-invariant Coulomb force, in a basis that conserves both angular momentum and isospin.

To this end, we first find the self-consistent MF state $\vert\varphi \rangle$ and then build a normalized angular-momentum- and isospin-conserving basis $\vert\varphi ;\, IMK;\, TT_z\rangle$ by using the projection method:

$$\vert\varphi ;\, IMK;\, TT_z\rangle = \frac{1}{\sqrt{N_{\varphi;IMK;TT_z}}} \hat P^T_{T_z,T_z} \hat P^I_{M,K} \vert\varphi \rangle ,$$ (4)

where $\hat P^T_{T_z ,T_z}$ and $\hat P^I_{M,K}$ stand for the standard isospin and angular-momentum projection operators:

$$\hat P^T_{T_z, T_z} = \frac{2T+1}{2} \int_0^\pi d^{T}_{T_z T_z}(\beta_T ) \hat{R}(\beta_T ) \sin\beta_T\, d\beta_T,$$ (5)

$$\hat P^I_{M, K} = \frac{2I+1}{8\pi^2 } \int D^{I\, *}_{M K}(\Omega ) \hat{R}(\Omega ) \, d\Omega,$$ (6)

where, $\hat{R}(\beta_T )= e^{-i\beta_T \hat{T}_y}$ is the rotation operator about the $y$-axis in the isospace, $d^{T}_{T_z T_z}(\beta_T )$ is the Wigner function, and $T_z =(N-Z)/2$ is the third component of the total isospin $\bm{T}$. As usual, $\hat{R}(\Omega )= e^{-i\gamma \hat{J}_z} e^{-i\beta \hat{J}_y} e^{-i\alpha \hat{J}_z}$ is the three-dimensional rotation operator in space, $\Omega = (\alpha, \beta, \gamma )$ are the Euler angles, $D^{I}_{M K}(\Omega )$ is the Wigner function, and $M$ and $K$ denote the angular-momentum components along the laboratory and intrinsic $z$-axis, respectively [22,25]. Note that unpaired MF states $\vert\varphi \rangle$ conserve the third isospin component $T_z$; hence, the one-dimensional isospin projection suffices.

The set of states (4) is, in general, overcomplete because the $K$ quantum number is not conserved. This difficulty is overcome by selecting first the subset of linearly independent states known as collective space [22], which is spanned, for each $I$ and $T$, by the so-called natural states $\vert\varphi;\, IM;\, TT_z\rangle^{(i)}$ [26,27]. The entire Hamiltonian - including the ISB terms - is rediagonalized in the collective space, and the resulting eigenfunctions are:

$$\vert n; \,\varphi ; \, IM; \, T_z\rangle = \sum_{i,T\geq \vert T_z\vert} a^{(n;\varphi)}_{iIT} \vert\varphi;\, IM; TT_z\rangle^{(i)} ,$$ (7)

where the index $n$ labels the eigenstates in ascending order according to their energies. The amplitudes $a^{(n;\varphi)}_{iIT}$ define the degree of isospin mixing through the so-called isospin-mixing coefficients (or isospin impurities), determined for a given $n$th eigenstate as:

$$\alpha_{\rm C}^n = 1 - \sum_i \vert a^{(n;\varphi)}_{iIT}\vert^2,$$ (8)

where the sum of norms corresponds to the isospin $T$ dominating in the wave function $\vert n; \,\varphi ; \, IM; \, T_z\rangle$.

One of the advantages of the projected DFT as compared to the shell-model-based approaches [28,3] is that it allows for a rigorous quantum-mechanical evaluation of the Fermi matrix element using the bare isospin operators:

$$\hat T_{\pm} = \frac{1}{2} \sum_{k=1}^A \left( \hat \tau^{(k)}_x \pm i \hat \tau^{(k)}_y \right) \equiv \mp \frac{1}{2} \hat T_{1\,\pm 1},$$ (9)

where $\hat T_{1\,\pm 1}$ denotes the rank-one covariant one-body spherical-tensor operators in the isospace, see the discussion in Ref. [29,30]. Indeed, noting that each $m$th eigenstate (7) can be uniquely decomposed in terms of the original basis states (4),

$$\vert m; \varphi;\, IM;\, T_z\rangle = \sum_{K,T} \, f_{K T }^{(\varphi ;\,m,I)} \hat P^T_{T_z, T_z} \, \hat P^I_{M, K} \vert\varphi \rangle ,$$ (10)

with microscopically determined mixing coefficients $f_{KT}^{(\varphi ; \,m,I)}$, the expression for the Fermi matrix element between the parent state $\vert m; \,\varphi ; \, IM; \, T_z\rangle$ and daughter state $\vert n; \,\psi ; \, IM; \, T_z\pm 1 \rangle$ can be written as:

$$\langle m; \,\varphi ; \, IM; \, T_z \vert \hat T_{\mp } \vert n... ..._{1\,\mp 1} \hat P^{T'}_{T_z\pm 1, T_z\pm 1} \hat P^I_{K, K'} \vert\psi \rangle$$

$$= \pm \frac{2I+1}{16\pi^2} \sum_{TT'} \sum_{KK'} f_{KT}^{(\varphi... ...\hat T_{1\,\mp 1} \hat P^{T'}_{T_z\pm 1, T_z\pm 1} \vert{\tilde \psi} \rangle ,$$ (11)

where tilde indicates the Slater determinant rotated in space: $\vert{\tilde \psi} \rangle = \vert\psi(\Omega )\rangle = \hat R(\Omega ) \vert \psi \rangle$. The matrix element appearing on the right-hand side of Eq. (11) can be expressed through the transition densities that are basic building blocks of the multi-reference DFT [31,32,33,24]. Indeed, with the aid of the identity

$$\, \hat P^{T}_{K, M} \hat T_{\lambda \, \mu} \hat P^{T'}_{M',K'} = C... ...T K}_{T' K-\nu \, \lambda \nu} \hat T_{\lambda \nu } \hat P^{T'}_{K-\nu, K'} ,$$ (12)

which results from the general transformation rule for spherical tensors under rotations or isorotations,

$$\hat R(\Omega ) \hat T_{\lambda\mu} {\hat R(\Omega )}^\dagger = \sum_{\mu'} D^\lambda_{\mu' \mu} (\Omega ) \hat T_{\lambda \mu'} ,$$ (13)

the matrix element entering Eq. (11) can be expressed as:

$$\, \langle \varphi \vert \hat P^T_{T_z, T_z} \hat T_{1\,\mp 1} \hat ... ...hat T_{1\, \nu} \hat P^{T'}_{T_z - \nu , T_z\pm 1} \vert{\tilde \psi} \rangle .$$ (14)

For unpaired Slater determinants considered here, the double integral over the isospace Euler angles in Eq. (11) can be further reduced to a one-dimensional integral over the angle $\beta_T$ using the identity

$$\hat T_{\lambda\mu} e^{i\alpha \hat T_z} = e^{-i\alpha \mu } e^{i\alpha \hat T_z} \hat T_{\lambda\mu},$$ (15)

which is the one-dimensional version of the transformation rule (13) valid for rotations around the $Oz$ axis in the isospace. The final expression for the matrix element in Eq. (14) reads:

$$\langle \varphi \vert \hat T_{1\, \nu} \hat P^{T'}_{T_z - \nu , T... ...varphi \vert \hat T_{1\, \nu} e^{-i\beta_T \hat T_y} \vert{\tilde \psi} \rangle$$

$$= (-1)^\nu \frac{2T'+1}{2} \int_0^\pi d\beta_T \sin\beta_T d^{T'}... ...nt d^3{\vec r}\, {\tilde{\tilde \rho}}_{1\, -\nu} (\Omega, \beta_T, {\vec r}) ,$$ (16)

where ${\tilde {\tilde \rho}}_{1\nu} (\Omega, \beta_T, {\vec r})$ is the isovector transition density, and the double-tilde sign indicates that the right Slater determinant used to calculate this density is rotated both in space as well as in isospace: $\vert{\tilde {\tilde \psi}} \rangle = \hat R(\beta_T )\hat R(\Omega ) \vert \psi \rangle$. The symbol ${\cal N}(\Omega, \beta_T ) = \langle \varphi \vert \hat R(\beta_T )\hat R(\Omega ) \vert \psi \rangle$ denotes the overlap kernel.

Since the natural states have good isospin, the states (7) are free from spurious isospin mixing. Moreover, since the isospin projection is applied to self-consistent MF solutions, our model accounts for a subtle balance between the long-range Coulomb polarization, which tends to make proton and neutron wave functions different, and the short-range nuclear attraction, which acts in an opposite way. The long-range polarization affects globally all s.p. wave functions. Direct inclusion of this effect in open-shell heavy nuclei is possible essentially only within the DFT, which is the only no-core microscopic framework that can be used there.

Recent experimental data on the isospin impurity deduced in $^{80}$Zr from the giant dipole resonance $\gamma$-decay studies [34] agree well with the impurities calculated using isospin-projected DFT based on modern Skyrme-force parametrizations [23,17]. This further demonstrates that the isospin-projected DFT is capable of capturing the essential piece of physics associated with the isospin mixing.