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Topological phase transitions induced by the variation of exchange couplings in graphene

We consider a modified graphene model under exchange couplings. Various quantum anomalous phases are known to emerge under uniform or staggered exchange couplings. We introduce the twist between the orientations of two sublattice exchange couplings, which is useful for examining how such topologically nontrivial phases under different types of exchange couplings are connected to one another. The phase diagrams constructed by the variation of exchange coupling strengths and twist angles exhibit rich structures of successive topological transitions. We analyze the emergence of peculiar phases in terms of the evolution of the energy dispersions. Perturbation schemes applied to the energy levels turn out to reproduce well phase boundary lines up to moderate values of the twist angle. We also discover two close topological transitions under uniform exchange couplings, which is attributed to the interplay of the trigonal-warping deformation due to Rashba spin-orbit coupling and the staggered sublattice potential. Finally the implications of Berry curvature structure and topological excitations in real and pseudo spin textures are discussed.