Stacking-induced Chern insulator

Graphene can be turned into a semimetal with broken time-reversal symmetry by adding valley-dependent pseudo-scalar potential that shifts the Dirac point energies in opposite directions, as modified Haldane model. We consider bilayer obtained stacking two time-reversed copies of model, where conduction and valence bands cross to give rise nodal line each valleys. In AB stacking, interlayer hopping lifts degeneracy lines induces band repulsion, leading surprisingly chiral insulator Chern number $C=\pm2$. As consequence pair edge states appears at boundaries ribbon geometry. contrast, AA does not show nontrivial topological phases. discuss possible experimental implementations our results.