  $\alpha$-cluster correlations in the ground states of $^{12}$C and $^{16}$O
are studied. Because of the $\alpha$ correlations, the intrinsic states of
$^{12}$C and $^{16}$O have triangle and tetrahedral shapes, respectively. The
deformations are regarded as spontaneous symmetry breaking of rotational
invariance, and the resultant oscillating surface density is associated with a
density wave (DW) state caused by the instability of Fermi surface with respect
to a kind of $1p$-$1h$ correlations. To discuss the symmetry breaking between
uniform density states and the oscillating density state, a schematic model of
a few clusters on a Fermi gas core in a one-dimensional finite box was
introduced. The model analysis suggests structure transitions from a Fermi gas
state to a DW-like state via a BCS-like state, and to a Bose Einstein
condensation (BEC)-like state depending on the cluster size relative to the box
size. It was found that the oscillating density in the DW-like state originates
in Pauli blocking effects.
