  Due to effective enhancement of the Coulomb coupling strength in the
vacuum-suspended graphene, the system may turn from a semimetal into an
insulator by the formation of a gap in the fermionic spectrum. This phenomenon
is analogous to the spontaneous breaking of chiral symmetry in the
strong-coupling relativistic field theories. We study this "chiral symmetry
breaking" and associated collective excitations on graphene in the strong
coupling regime by taking U(1) lattice gauge theory as an effective model for
graphene. Both compact and non-compact formulations of the U(1) gauge action
show chiral symmetry breaking with equal magnitude of the chiral condensate
(exciton condensate) in the strong coupling limit, while they start to deviate
from the next-to-leading order in the strong coupling expansion. Phase and
amplitude fluctuations of the order parameter are also investigated: in
particular, a mass formula for the pseudo-Nambu--Goldstone mode
($\pi$-exciton), which is analogous to Gell-Mann--Oakes--Renner relation for
the pion in quantum chromodynamics (QCD), is derived from the axial
Ward-Takahashi identity. To check the applicability of the effective field
theory description, typical energy scales of fermionic and bosonic excitations
are estimated by identifying the lattice spacing of the U(1) gauge theory with
that of the original honeycomb lattice of graphene.
