  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.

Nuclear Theory;Nuclear Experiment