Cluster structures of the $0^{+}$ states of ${}_{\mathrm{\Lambda }}^{13}\mathrm{C}$ with the 3$\alpha +\mathrm{\Lambda }$ four-body orthogonality condition model

We study structures of the $0^{+}$ states of ${}_{\mathrm{\Lambda }}^{13}$C within the framework of $\alpha +\alpha +\alpha +\mathrm{\Lambda }$ four-body orthogonality condition model with the Gaussian expansion method. We obtain five $0^{+}$ states of ${}_{\mathrm{\Lambda }}^{13}$C and compare them with the low-lying four $0^{+}$ states of ${}^{12}$C. We find that the $0_1^{+}$, $0_2^{+}$ and $0_4^{+}$ states correspond to the $0_1^{+}$, $0_2^{+}$ and $0_4^{+}$ states of ${}^{12}$C, respectively. We also find that both the $0_3^{+}$ and $0_5^{+}$ states have analogous nature to the $0_3^{+}$ state of ${}^{12}$C, where the additional $\mathrm{\Lambda }$ particle couples with ${}^8$Be cluster and one $\alpha$ cluster, in the $0_3^{+}$ state of ${}^{12}$C, respectively. The former is then shown to have an elongated ${}_{\mathrm{\Lambda }}^9$Be$(0_1^{+})+\alpha$ cluster structure with a higher nodal behaviour for their relative wave function, and the latter have a dilute three-body gas consisting of $2\alpha {+}_{\mathrm{\Lambda }}^5$He clusters.