Toroidal states in ${}^{28}$Si with covariant density functional theory in 3D lattice space

The toroidal states in ${}^{28}$Si with spin extending to extremely high are investigated with the cranking covariant density functional theory on a 3D lattice. Thirteen toroidal states with spin $I$ ranging from 0 to 56$\mathrm{\hslash }$ are obtained, and their stabilities against particle emissions are studied by analyzing the density distributions and potentials. The excitation energies of the toroidal states at $I=28$, 36, 44$\mathrm{\hslash }$ reasonably reproduce the observed three resonances extracted from the 7-$\alpha$ de-excitation of ${}^{28}$Si. The $\alpha$ clustering of these toroidal states is supported by the $\alpha$-localization function.