Multi-cluster dynamics in $^{13}_\Lambda{\rm C}$ and analogy to clustering in $^{12}{\rm C}$

We investigate structure of $^{13}_\Lambda{\rm C}$ and discuss the difference and similarity between the structures of $^{12}{\rm C}$ and $^{13}_\Lambda{\rm C}$ by answering the questions if the linear-chain and gaslike cluster states, which are proposed to appear in $^{12}{\rm C}$, survives, or new structure states appear or not. We introduce a microscopic cluster model called, Hyper-Tohsaki-Horiuchi-Schuck-R\"opke (H-THSR) wave function, which is an extended version of the THSR wave function so as to describe $\Lambda$ hypernuclei. We obtained two bound states and two resonance (quasi-bound) states for $J^\pi=0^+$ in $^{13}_\Lambda{\rm C}$, corresponding to the four $0^+$ states in $^{12}{\rm C}$. However, the inversion of level ordering between the spectra of $^{12}{\rm C}$ and $^{13}_\Lambda{\rm C}$, i.e. that the $0_3^+$ and $0_4^+$ states in $^{13}_\Lambda{\rm C}$ correspond to the $0_4^+$ and $0_3^+$ states in $^{12}{\rm C}$, respectively, is shown to occur. The additional $\Lambda$ particle reduces sizes of the $0_2^+$ and $0_3^+$ states in $^{13}_\Lambda{\rm C}$ very much, but the shrinkage of the $0_4^+$ state is only a half of the other states. In conclusion, the Hoyle state becomes quite a compact object with ${^{9}_\Lambda{\rm Be}}+\alpha$ configuration in $^{13}_\Lambda{\rm C}$ and is no more gaslike state composed of the $3\alpha$ clusters. Instead, the $0_4^+$ state in $^{13}_\Lambda{\rm C}$, coming from the $^{12}{\rm C}(0_3^+)$ state, appears as a gaslike state composed of $\alpha+\alpha+^{5}_\Lambda{\rm He}$ configuration, i.e. the Hoyle analog state. A linear-chain state in a $\Lambda$ hypernucleus is for the first time predicted to exist as the $0_3^+$ state in $^{13}_\Lambda{\rm C}$ with more shrunk arrangement of the $3\alpha$ clusters along $z$-axis than the $3\alpha$ linear-chain configuration realized in the $^{12}{\rm C}(0_4^+)$ state.