GLYPH<s65> GLYPH<s66> GLYPH<s67> FIG. 1: (Color online). Schematic picture of the 2D kagom´ e lattice with bond length a . The dashed line represents the Wigner-Seitz unit cell, which contains three independent sites ( A , B , and C ). FIG. 2: (Color online). Hofstadter butterfly-like spectrum of the 2D Kagom´ e lattice. The numbers recorded in gaps represent the Hall conductance of this system in units of e $^{2}$/h when the Fermi energy lies in these gaps. is an integer multiple of Φ$_{0}$. In this case, the Hamiltonian (1) should be diagonalized in a “magnetic” unit cell, which contains q plaquette in order to guarantee k being good quantum numbers. As a result, the magnetic Brillouin zone (MBZ) is q times smaller than the usual BZ. Furthermore, because of the magnetic translation symmetry, the MBZ has exactly a q - fold degeneracy. For convenience we take the gauge potential as A= B ( − 1 $_{2}$( − √$_{3}$$_{x}$$_{+}$$_{y}$$_{)}$$_{,}$ 1 $_{2}$( x − 1 √$_{3}$ y ) , 0). Under this gauge, the Peierls phases, which reflect the information about the external magnetic field, are only related to n and have nothing with m . The Hamiltonian of the kagom´ e lattice now can be diagonalized in the momentum space as the following 3 q × 3 q 5