  Bilayer honeycomb lattice, with inter-layer tunneling energy, has a parabolic
dispersion relation, which causes the charge imbalance between two sublattices.
Here, we investigate the metal-insulator and magnetic phase transitions on the
strongly correlated bilayer honeycomb lattice by cellular dynamical mean-field
theory combined with continuous time quantum Monte Carlo method. We find that
different kinds of magnetic spontaneous symmetry breaking on dimer and
non-dimer sites, cause a novel phase transition between normal anti-ferromagnet
and layer anti-ferromagnet. We sketch the phase diagrams as the function of
temperature, interaction and inter-layer hopping. Finally, we set up an
experimental protocol for cold atoms in optical lattice to observe these
phenomena in future experiments.
