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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$\hbar$ 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$\hbar$ reasonably reproduce the observed three resonances extracted from the 7-$α$ de-excitation of $^{28}$Si. The $α$ clustering of these toroidal states is supported by the $α$-localization function.