$\alpha +\alpha +{}^3$He cluster structure in ${}^{11}$C

We study the $\alpha +\alpha +{}^3$He cluster structure of ${}^{11}$C within the microscopic cluster model. The calculations essentially reproduce the energy spectra for both negative and positive parity states, particularly the $3\mathrm{/}{2}_3^{-}$ state near the $\alpha +\alpha$+${}^3$He threshold. We also calculate the isoscalar monopole, electric quadrupole transition strengths, and root-mean-square radii for the low-lying states. These results suggest that the $3\mathrm{/}{2}_3^{-}$, $1\mathrm{/}{2}_2^{-}$, and $5\mathrm{/}{2}_3^{-}$ states have a well-developed $\alpha +\alpha$ + ${}^3$He cluster structure. The analysis of the generator coordinate method wave functions indicates the dilute gaslike nature for the $3\mathrm{/}{2}_3^{-}$, $1\mathrm{/}{2}_2^{-}$, and $5\mathrm{/}{2}_3^{-}$ states, suggesting that they could be candidates for the Hoyle-analog state. Furthermore, it is found that the $5\mathrm{/}{2}_2^{+}$ and $5\mathrm{/}{2}_3^{+}$ states may possess a linear chain structure.