where Ψ$_{k}$=( c$_{k}$$_{A}$ , c$_{k}$$_{B}$ , c$_{k}$$_{C}$ ) T is the three-component electron field operator. Each component of Ψ$_{k}$ is the Fourier transform of c$_{i}$ , i.e., where we have changed notation i → ( mns ) by using ( mn ) to label the kagom´ e unit cells and s to label the three sites in a unit cell. H ( k ) is given by where p i $_{k}$=2 cos ( k · a$_{i}$ ) and a$_{1}$ =( − 1 / 2 , − √$_{3}$$_{/}$$_{2),}$ a$_{2}$ =(1 , 0), and a$_{3}$ =( − 1 / 2 $_{,}$√$_{3}$$_{/}$$_{2) represent}$ the displacements in a unit cell from the A to B site, from the B to C site, and from the C to A site, respectively. In this notation, the first BZ is a hexagon with the corners of K = ± (2 / 3) a$_{1}$ , ± (2 / 3) a$_{2}$ , and ± (2 / 3) a$_{3}$ . In the presence of a magnetic filed ( B = 0), the hopping term of the tight-binding Hamiltonian (1) is modified by phase factors from the vector potential A , where γ$_{ij}$ is the phase factor (the well known Peierls phase) between sites j and i : with Φ$_{0}$= h/e being the flux quantum. One can take different gauge potential A mathematically to satisfy the physical confine- ment ∇ × A=B . However, the flux through one unit cell is only linearly dependent on the external magnetic field B , and has no relation with the choice of the gauge potential A . So, it is convenient to measure the magnetic field in units of the flux quantum per elementary triangular plaquette of the kagom´ e lattice. After a simple algebraic calculation, one obtains $_{the flux through one triangle is Φ=}$√$_{3}$$_{B/}$$_{4. Thus we can define a parameter, called the filling}$ ratio f , as the fraction of a flux quantum throng each triangle, i.e, f =Φ / Φ$_{0}$$_{=}$√$_{3}$$_{eB/}$$_{4}$$_{h}$ [4]. The total flux through a unit cell of the kagom´ e lattice is then given as 8 f Φ$_{0}$. When f is a rational number that can be written as f = p/q , where p and q are integers with no common factors, the total flux through q triangular plaquette of the kagom´ e lattice 4 γ$_{ij}$ = 2 π Φ$_{0}$ ∫ i j A · d l (6) t → te iγ$_{ij}$ (5) H ( k ) =    0 p 1 k p 3 k p 1 k 0 p 2 k p 3 k p 2 k 0   $^{,}$ (4) c$_{k}$$_{s}$ = ∑ mn c$_{mns}$e i k · $^{r$_{mns}$}$, (3)