GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s45>GLYPH<s50> GLYPH<s48> GLYPH<s50> GLYPH<s107> GLYPH<s120> GLYPH<s32> GLYPH<s69>GLYPH<s110>GLYPH<s101>GLYPH<s114>GLYPH<s103>GLYPH<s121> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s45>GLYPH<s52> GLYPH<s45>GLYPH<s50> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s45>GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s32> GLYPH<s32> GLYPH<s79>GLYPH<s114>GLYPH<s98>GLYPH<s105>GLYPH<s116>GLYPH<s97>GLYPH<s108>GLYPH<s32>GLYPH<s109>GLYPH<s97>GLYPH<s103>GLYPH<s110>GLYPH<s101>GLYPH<s116>GLYPH<s105>GLYPH<s122>GLYPH<s97>GLYPH<s116>GLYPH<s105>GLYPH<s111>GLYPH<s110> GLYPH<s45>GLYPH<s52> GLYPH<s45>GLYPH<s50> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s45>GLYPH<s51> GLYPH<s48> GLYPH<s51> GLYPH<s70> GLYPH<s101> GLYPH<s114>GLYPH<s109> GLYPH<s105>GLYPH<s32>GLYPH<s101> GLYPH<s110> GLYPH<s101> GLYPH<s114>GLYPH<s103> GLYPH<s121> GLYPH<s77> GLYPH<s99> GLYPH<s44>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s77> GLYPH<s70> GLYPH<s101> GLYPH<s114>GLYPH<s109> GLYPH<s105>GLYPH<s32>GLYPH<s101> GLYPH<s110> GLYPH<s101> GLYPH<s114>GLYPH<s103> GLYPH<s121> GLYPH<s40>GLYPH<s99> GLYPH<s41> GLYPH<s32> FIG. 3: (Color online). (a) Energy spectrum of the 2D ka g o m ´ e lattice with a field f =1 / 4. The shadow areas are the energy bands. The corresponding ma g n et i zation M and its two components are respectively drawn in (b) and (c) as functions of the Fermi energy µ$_{0}$ . The shaded areas in (b) and (c) are the insulating regions. The dashed and dotted lines in (c) represent M$_{c}$ and M$_{Ω}$ , respectively. 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 8