Theoretical formalism: isospin restoration and Coulomb rediagonalization scheme

The first step and the starting point of our approach is the determination of the isospin-symmetry-broken single-particle (s.p.) Slater determinant $\vert\textrm{HF} \rangle$ calculated by using the Hartree-Fock (HF) theory including the isospin-invariant Skyrme ($\hat V^S$) and the isospin-symmetry-breaking Coulomb ($\hat V^C$) interactions:

$$\hat H = \hat H^S + \hat V^C \quad \textrm{where} \quad \hat H^S =\hat T + \hat V^S.$$ (1)

The isospace-deformed state $\vert\textrm{HF} \rangle$ admixes higher isospin components $T\geq \vert T_z\vert$:

$$\vert\textrm{HF} \rangle = \sum_{T\geq \vert T_z\vert}b_{T,T_z}\vert\eta; T,T_z\rangle ,$$ (2)

where $T$ and $T_z$ are the total isospin and its third component, respectively, $\eta$ labels all other quantum numbers pertaining to the $\vert\textrm{HF} \rangle$ state, and the coefficients $b_{T,T_z}$ are such that $\sum_{T\geq \vert T_z\vert} \vert b_{T,T_z}\vert^2 = 1$.

In the second step we create the good-isospin states $\vert\eta; T,T_z\rangle$ by projecting them out from the Slater determinant $\vert\textrm{HF} \rangle$:

$$\vert\eta ; T,T_z\rangle = \frac{1}{b_{T,T_z}} \hat P^T_{T_z T_z}\vert\textrm{HF} \rangle .$$ (3)

In the following, we denote the mixing coefficients $b_{T,T_z}$ and average energies $E^{\textrm{BR}}_{T,T_z}$:

$$\vert b_{T,T_z}\vert^2 = \langle \textrm{HF} \vert \hat P^T... ...} = \langle\eta ; T,T_z\vert \hat{H}\vert \eta ; T,T_z\rangle,$$ (4)

as being obtained before rediagonalization. In the above formulae, $\hat P^T_{T_z T_z}$ denotes the conventional[11] SO(3) projection operator reduced to one dimension due to the $T_z$ quantum number conservation, that is:

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

where $\hat R(\beta ) = e^{-i \beta \hat T_y}$ denotes active-rotation operator by the Euler angle $\beta$ in the isospace and $d^{T}_{T_zT_z} (\beta)$ is the Wigner $d$-function[12].

In the third step we mix the projected states,

$$\vert\eta; n,T_z\rangle = \sum_{T\geq \vert T_z\vert}a^n_{T,T_z}\vert\eta; T,T_z\rangle ,$$ (6)

and determine the mixing coefficients $a^n_{T,T_z}$ by diagonalizing Hamiltonian (1) in the space of projected states,

$$\sum_{T'\geq \vert T_z\vert}\langle\eta; T,T_z\vert\hat{H}\v... ...T_z\rangle a^n_{T',T_z} = E^{\textrm{AR}}_{n,T_z}a^n_{T,T_z},$$ (7)

where $n$ enumerates the obtained eigenstates. In the following, we denote the mixing coefficients $a^n_{T,T_z}$ and eigenenergies $E^{\textrm{AR}}_{n,T_z}$ as being obtained after rediagonalization. The lowest-energy solution, for $n=1$, corresponds to the isospin mixing in the ground state.

The Skyrme Hamiltonian, $\hat H^S$, is an isoscalar operator; hence, it contributes only to the diagonal matrix elements of the Hamiltonian (1), $\langle \eta; T,T_z \vert \hat H^S \vert \eta; T,T_z \rangle$, which can be obtained from:

$$\langle\textrm{HF}\vert \hat H^{S} \hat P^T_{T_zT_z}\vert\... ...trm{HF}\vert \hat H^{S} \hat R(\beta)\vert\textrm{HF}\rangle .$$ (8)

Similarly, calculation of the diagonal and non-diagonal matrix elements of the Coulomb interaction, $\langle \eta; T,T_z \vert \hat V^C \vert \eta; T',T_z \rangle$, can be efficiently performed after decomposing $\hat V^{C}$ into the isoscalar, $\hat V^{C}_{00}$, isovector, $\hat V^{C}_{10}$, and isotensor, $\hat V^{C}_{20}$, components, and by making use of the SO(3) transformation rules for the spherical tensors under rotations in the isospace[12]. In the particular case of one-dimensional projection we deal with in this work, all matrix elements of axial spherical tensors reduce to one-dimensional integrals over the Euler angle $\beta$:

$$\langle \textrm{HF}\vert \hat{P}^{T}_{T_z T_z} \hat{V}^C_{\lambda 0} \hat{P}^{T^\prime}_{T_z T_z} \vert\textrm{HF} \rangle = C^{T T_z}_{ T^\prime T_z\; \lambda 0} \sum_{\mu' = -\lambda}^{\lambda} C^{T T_z}_{ T^\prime T^\prime_z \; \lambda \mu' }$$

$$\frac{2T^\prime +1}{2} \int_0^\pi d\beta \; \sin\beta\; d^{T^\prime}_{T^\prime_z, T_z} (\beta )\; \langle... ...rm{HF}\vert \hat V^C_{\lambda \mu'} \hat R (\beta) \vert\textrm{HF} \rangle ,$$ (9)

where $T^\prime_z = T_z-\mu'$ and $C^{T T_z}_{ T^\prime T^\prime_z \; \lambda \mu }$ denote standard Clebsch-Gordan coefficients. The Skyrme-Hamiltonian and Coulomb-interaction kernels, $\langle\textrm{HF}\vert \hat H^{S} \hat R(\beta)\vert\textrm{HF}\rangle$ and $\langle\textrm{HF}\vert \hat V^{C} \hat R(\beta)\vert\textrm{HF}\rangle$, respectively, can be evaluated by using expressions for the standard diagonal kernels[13] ($\beta=0$) and replacing there the isoscalar and isovector densities and currents with the so-called transition densities and currents. Exact direct and exchange kernels of the Coulomb interaction can be evaluated by using methods outlined in Refs.[14,15,16].