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 $ u_1= u_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.
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