Monopole excitation of the Hoyle state and linear-chain state in 12C

Background: Two new $J^\pi=0^+$ states are recently observed around a few MeV above the Hoyle state (the second $0^+$ state in 12C). The characters of them are only poorly discussed in theory and are still mysterious. Purpose: I give for the first time a comprehensive understanding of the structures of the $0^+$ states by analyzing their wave functions, and discuss relationship with the Hoyle state, similarities, and differences between the states. Method: I extend a microscopic $\alpha$-cluster model called Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave function so as to incorporate $2\alpha+\alpha$ asymmetric configuration explicitly. The so-called $r^2$-constraint method to effectively eliminate spurious continuum components is also used. Results: The $0_3^+$ state is shown to have a very large squared overlap with a single configuration of the extended THSR wave function in an orthogonal space to the Hoyle state as well as to the ground state. The $0_4^+$ state has a maximal squared overlap with a single extended THSR wave function with an extremely prolately-deformed shape. Conclusions: The $0_3^+$ state appears as a family of the Hoyle state to have a higher nodal structure in the internal motions of the $3\alpha$ clusters, due to the orthogonalization to the Hoyle state. The $0_4^+$ state dominantly has a linear-chain structure, where the $3\alpha$ clusters move freely in a non-localized way, like a one-dimensional gas of the $3\alpha$ clusters.