  We discuss t-J-U model on a honeycomb monolayer that has the same low-energy
description of the kinetic term as graphene bilayer, and in particular study
coexistence of antiferromagnetism and superconducting correlations that
originate from Cooper pairs without phase coherence. We show that the model is
relevant for the description of graphene bilayer and that the presence of the d
+ i d superconducting correlations with antiferromagnetism can lead to
quadratic dependence in small magnetic fields of the gap of the effective
monolayer consistent with the transport measurements of Velasco et al. on the
graphene bilayer.
