Tetrahedral shape and surface density wave of $^{16}$O caused by $\alpha$-cluster correlations

$\alpha$-cluster correlations in the $0^+_1$ and $3^-_1$ states of $^{12}$C and $^{16}$O are studied using the method of antisymmetrized molecular dynamics, with which nuclear structures are described from nucleon degrees of freedom without assuming existence of clusters. The intrinsic states of $^{12}$C and $^{16}$O have triangle and tetrahedral shapes, respectively, because of the $\alpha$-cluster correlations. These shapes can be understood as spontaneous symmetry breaking of rotational invariance, and the resultant surface density oscillation is associated with density wave (DW) caused by the instability of Fermi surface with respect to particle-hole correlations with the wave number $\lambda=3$. $^{16}$O($0^+_1$) and $^{16}$O($3^-_1$) are regarded as a set of parity partners constructed from the rigid tetrahedral intrinsic state, whereas $^{12}$C($0^+_1$) and $^{12}$C($3^-_1$) are not good parity partners as they have triangle intrinsic states of different sizes with significant shape fluctuation because of softness of the $3\alpha$ structure. $E3$ transition strengths from the $3^-_1$ to $0^+_1$ states in $^{12}$C and $^{16}$O are also discussed.