  $\alpha$ cluster structure in nuclei has been long understood based on the
geometrical configuration picture. By using the spatially localized Brink
$\alpha$ cluster model in the generator coordinate method, it is shown that the
$\alpha$ cluster structure has the apparently opposing duality of crystallinity
and condensation, a property of supersolids. To study the condensation aspects
of the $\alpha$ cluster structure a field theoretical superfluid cluster model
(SCM) is introduced, in which the order parameter of condensation is
incorporated by treating rigorously the Nambu-Goldstone mode due to spontaneous
symmetry breaking of the global phase. The $\alpha$ cluster structure of
$^{40}$Ca, which has been understood in the crystallinity picture, is studied
by the SCM with ten $\alpha$ clusters. It is found that the $\alpha$ cluster
structure of $^{40}$Ca is reproduced by the SCM in addition to $^{12}$C
reported in a previous paper, which gives support to the duality of the
$\alpha$ cluster structure. The emergence of the mysterious $0^+$ state at the
lowest excitation energy near the $\alpha$ threshold energy is understood to be
a manifestation of the Nambu-Goldstone zero mode, a soft mode, due to the
condensation aspect of the duality similar to the Hoyle state in $^{12}$C. The
duality of $\alpha$ cluster structure with incompatible crystallinity and
coherent wave nature due to condensation is the consequence of the Pauli
principle, which causes clustering.
