  In this work, we perform ab initio calculations, based on the density
functional theory, of the effects on the graphene bilayer when we intercalate
carbon atoms between the layers. We use the unit cell of the bilayer to
construct larger unit cells (supercells), positioning a single carbon atom in
the hollow position between the monolayers and periodically replicating the
supercell. By increasing the size of the unit cell and consequently, the
periodicity of the inserted atoms, we are able to minimize the carbon-carbon
interaction and therefore infer the changes in the electronic, vibrational and
thermal behavior of the bilayer when the intercalated atoms do not interact
with each other. The main result, concerning the electronic properties, is the
appearance of a doubly degenerate flat band at the Fermi level. These states
are interpreted as coming from the periodic deformation of the bilayer due to
the inserted atoms. It acts as a non-Abelian flux network creating zero energy
at bands as predicted by San-Jose, Gonz\'alez and Guinea in 2012. Since the
periodic strain field associated to the defect array has such a strong
influence on the electronic properties of the bilayer, it may be useful for
practical applications. For instance, it can act as frozen-in magnetic-like
field flux tubes. All-carbon nanostructures can then be designed to have
electronic behavior at different regions tailored by the chosen defect pattern.
