Peculiarity of the ${}^{12}$C$(0^{+})$ and ${}^{12}$C$(2^{+})$ energy spectrum in a 3$\alpha$ model

Lowest energy spectrum of the ${}^{12}$C nucleus is analyzed in the 3$\alpha$ cluster model with a deep $\alpha \alpha$-potential of Buck, Friedrich and Wheatley with Pauli forbidden states in the $S$ and $D$ waves. The direct orthogonalization method is applied for the elimination of the 3$\alpha$-Pauli forbidden states. The effects of possible first order quantum phase transition are shown in the lowest ${}^{12}$C($0_1^{+})$ and ${}^{12}$C($2_1^{+})$ states from weakly bound phase to a deep phase. The ground and lowest $2^{+}$ states of the ${}^{12}$C nucleus in the deep phase are created by the critical eigen states of the Pauli projector for the $0^{+}$ and $2^{+}$ three-alpha functional spaces, respectively.