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$α+α+t$ cluster structures and Hoyle-analogue states in $^{11}$B

The structure of $3/2^{-}$ and $1/2^{+}$ states in $^{11}$B is investigated with an $α+α+t$ orthogonality condition model (OCM) based on the Gaussian expansion method. Full levels up to the $3/2^{-}_{3}$ and $1/2^{+}_2$ states around the $α+α+t$ threshold ($E_x$=11.1 MeV) are reproduced consistently with the experimental energy levels. It is shown that the $3/2_{3}^{-}$ state located around the $^{7}$Li+$α$ threshold has an $α+α+t$ cluster structure, whereas the $3/2_{1}^{-}$ and $3/2_{2}^{-}$ states have a shell-model-like compact structure. We found that the $3/2_{3}^{-}$ state does not possess an $α$-condensate-like nature similar to the $0^{+}_{2}$ state of $^{12}$C (Hoyle state) which has a dilute $3α$-condensate structure described by a $(0S_α)^3$ configuration with about $70$\% probability, although the monopole transition strength of the former is as large as that of the latter. We discuss the reasons why the $3/2_{3}^{-}$ state does not have the condensate character. On the other hand, the $1/2^{+}_{1}$ state just below the $^{7}$Li+$α$ threshold has a cluster structure which can be interpreted as a parity-doublet partner of the $3/2^{-}_3$ state. We indicate that the $12.56$-MeV state ($J^π=1/2^{+}_{2}$) just above the $α+α+t$ threshold observed in the $^7$Li($^{7}$Li,$^{11}$B$^*$)$t$ reaction etc. is of the dilute-cluster-gas-like, and is a strong candidate for the Hoyle-analogue state which has a configuration of $(0S_α)^{2}(0S_{t})$ with about $65$\% probability from the analyses of the single-cluster motions in $^{11}$B. The structure property of the $1/2^{+}$ resonant state is analyzed with the complex scaling method.