Equivalence of generator coordinate Brink cluster model and nonlocalized cluster model and supersolidity of $\alpha$ cluster structure in nuclei

It is found that $\alpha$ cluster structure has the apparently opposing dual property of crystallinity and condensation simultaneously. The mathematical equivalence of the spatially localized Brink $\alpha$ cluster model in the generator coordinate method (GCM) and the nonlocalized cluster model (NCM), which is also called the THSR (Tohsaki-Horiuchi-Schuck-R$\ddot{\rm o}$pke) wave function based on the condensation of $\alpha$ clusters, is shown. The latter is found to be an equivalent representation of the localized cluster model and it is a natural consequence that the many NCM (THSR) calculations reproduce the proceeding cluster model calculations using the GCM and the resonating group method (RGM). Localized cluster models, which have been successfully used for more than half a century, will continue to be very powerful. The equivalence is a manifestation of the duality of incompatible aspects: crystallinity and coherent wave nature due to condensation of $\alpha$ clusters, i.e. the dual properties of a supersolid. The Pauli principle causes the duality. The evidence for supersolidity, the emergence of a Nambu-Goldstone mode caused by the spontaneous symmetry breaking of the global phase, is discussed