  For more than half a century, the structure of $^{12}$C, such as the ground
band, has been understood to be well described by the three $\alpha$ cluster
model based on a geometrical crystalline picture. On the contrary, recently it
has been claimed that the ground state of $^{12}$C is also well described by a
nonlocalized cluster model without any of the geometrical configurations
originally proposed to explain the dilute gas-like Hoyle state, which is now
considered to be a Bose-Einstein condensate of $\alpha$ clusters. The
challenging unsolved problem is how we can reconcile the two exclusive $\alpha$
cluster pictures of $^{12}$C, crystalline vs nonlocalized structure. We show
that the crystalline cluster picture and the nonlocalized cluster picture can
be reconciled by noticing that they are a manifestation of supersolidity with
properties of both crystallinity and superfluidity. This is achieved through a
superfluid $\alpha$ cluster model based on effective field theory, which treats
the Nambu-Goldstone zero mode rigorously. For several decades, scientists have
been searching for a supersolid in nature.Nuclear $\alpha$ cluster structure is
considered to be the first confirmed example of a stable supersolid.

Nuclear Theory;Other Condensed Matter;Quantum Gases;Nuclear Experiment