  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.
