  On a lattice model, we study the possibility of existence of gapped broken
inversion symmetry phase (GBISP) of electrons with long-range Coulomb
interaction in bilayer graphene using both self-consistent Hartree-Fock
approximation (SCHFA) and the renormalized-ring-diagram approximation (RRDA).
RRDA takes into account the charge-density fluctuations beyond the mean field.
While GBISP at low temperature and low carrier concentration is predicted by
SCHFA, we show here the state can be destroyed by the charge-density
fluctuations. We also present a numerical algorithm for calculating the
self-energy of electrons with the singular long-range Coulomb interaction on
the lattice model.
