## Linking the monopole part with Hartree-Fock theory, more definitions

In Eqs. (23) and (24), we have used a compact notation for the single-particle states, with the symbol $$\alpha_p$$ etc representing all possible quantum numbers except the magnetic substates $$m_{j_p}$$, that is $$\alpha_p=(n_p,l_p,j_p,t_{z_p})$$. The symbol $$\alpha_F$$ stands now for all single-particle states up to the Fermi level, excluding again the magnetic substates. In the above two-body interaction matrix elements $$\langle (\alpha_p\alpha_i)J | \hat{v}(\tilde{v}) |(\alpha_p\alpha_i)J \rangle$$ we have dropped additional quantum numbers like the isospin projection. Our interactions are diagonal in the projection of the total isospin but breaks both isospin symmetry and charge symmetry.