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We present a complete topological characterization of a bilayer composite oftwo Chern insulators (specifically, Haldane models) and explicitly establishthe bulk-boundary correspondences. We show that an appropriately defined Chernnumber accurately maps out all the possible phases of the system and remainswell-defined even in the presence of degeneracies in the occupied bands.Importantly, our result paves the way for realizing adiabatic preparation ofmonolayer Chern insulators. This has been a major challenge till date, giventhe impossibility of unitarily connecting inequivalent topological phases. Weshow that this difficulty can be circumvented by adiabatically varying theinterlayer coupling in such a way that the system remains gapped at all times.In particular, a complete knowledge of the phase diagram of the bilayercomposite immediately allows one to identify all such adiabatic passages whichmay connect the different Chern inequivalent phases of the individualmonolayers.

Bilayer Haldane system: Topological characterization and adiabatic passages connecting Chern phases