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The structure of the $N=Z$ nucleus $^{28}$Si is studied by resorting to anIBM-type formalism with $s$ and $d$ bosons representing isospin $T=0$ andangular momentum $J=0$ and $J=2$ quartets, respectively. $T=0$ quartets arefour-body correlated structures formed by two protons and two neutrons. Themicroscopic nature of the quartet bosons, meant as images of the fermionicquartets, is investigated by making use of a mapping procedure and is supportedby the close resemblance between the phenomenological and microscopicallyderived Hamiltonians. The ground state band and two low-lying side bands, a$\beta$ and a $\gamma$ band, together with all known $E2$ transitions andquadrupole moments associated with these states are well reproduced by themodel. An analysis of the potential energy surface places $^{28}$Si, only knowncase so far, at the critical point of the U(5)-$\overline{\rm SU(3)}$transition of the IBM structural diagram.

Quartet structure of $N=Z$ nuclei in a boson formalism: the case of $^{28}$Si