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behavior between insulating and metallic regions. I n t h e in sulating region, M$_{Ω}$ linearly increases with µ$_{0}$ , as is expected from Eq. (10). The slope of the Berry-phase correction term in insulating regions is proportional to the system’s Hall conductance. In the metallic region, however, this term sensitively depends on the topological property of the band in which the chemical potential is located. On the whole the comparison between Fig. 3(b) and Fig. 3(c) shows that the metallic behavior of M is dominated by its conventional term M$_{c}$ , while in the insulating regime M$_{Ω}$ plays a main role in determining the behavior of M . 

Now let us consider more complex cases, for example, with a field f =1 / 3 and with a field f =1 / 6. In these two case, there are more energy bands and gaps appearing in the energy spectrum [see Figs. 4(a) and 5(a)]. Figures 4(b) and 4(c), and Figs. 5(b) and 5(c), for f =1 / 3 and f =1 / 6, respectively, plot the magnetization M and its two components as functions of µ$_{0}$ . Clearly from these figures, one can observe a similar variation of the magnetization by changing the electron’s fillings (the Fermi energy µ$_{0}$ ). Comparing the case of f =1 / 3 [Fig. 

