content_image/366327.jpg
<img src='content_image/366327.jpg'>

occupied and M begins to decrease from 0. When µ$_{0}$ increases to − 1.74, the lower band is fully occupied and the magnetization decreases to − 0.7 e/h . When continue increasing µ$_{0}$ , the system enters the first insulating region. The magnetization then linearly increases with the Fermi energy until the middle band is occupied ( µ$_{0}$ =0 − , at this time M =1.03 e/h ). When further increasing the Fermi energy, the system immediately comes into the second insulating region and the magnetization jumps down to − 1.03 e/h and then linearly increases up to 0 . 7 e/h until the upper band begins to be occupied ( µ$_{0}$ =1.74). Then the magnetization decreases to 0 with the Fermi energy going through the upper band. 

As shown in Refs. [17, 18], the totally different behavior of the magnetization in the metallic and insulating regions is due to the different roles M$_{c}$ and M$_{Ω}$ play in these two regions. For further illustration, we show in Fig. 3(c) M$_{c}$ (dashed line) and M$_{Ω}$ (dotted line) as functions of the Fermi energy, their sum gives M in Fig. 3(b). One can see that overall M$_{c}$ and M$_{Ω}$ have opposite contributions to M , which implies that these two parts carry opposite-circulating currents. In each insulating regime the conventional term M$_{c}$ keeps a constant, which is due to the fact that the upper limit of the k -integral of m$_{n}$ ( k ) is invariant as the chemical potential varies in the gap. In the metallic region, however, since the occupied states varies with the Fermi energy, thus M$_{c}$ also varies with µ$_{0}$ , resulting in a decreasing slope shown in Fig. 3(c). The Berry phase term M$_{Ω}$ also displays different 

