  In the recent advancement in Graphene heterostructures, it is possible to
create a double layer tunnel decoupled Graphene system which has strong
interlayer electronic interaction. In this work, we restrict the parameters in
the Hamiltonian using simple symmetry arguments. We study the ground state of
this system in the Hartree-Fock approximation at $\nu_1=\nu_2=0$. In addition
to the phases found in monolayer Graphene, we found the existence of layer
correlated phase which breaks the layer $U(1)$ symmetry. At non-zero Zeeman
coupling strength ($E_z$) this layer correlated state has a small
magnetization, which vanishes as $E_z$ goes to zero. We discuss the bulk
gapless modes using the Goldstone theorem. We also comment on the edge
structure for the layer correlated phase.
