Edge states and the integer quantum Hall conductance in spin-chiral ferromagnetic kagome lattice

We investigate the chiral edge states in the two-dimensional ferromagntic kagom\'{e} lattice with spin anisotropies included. The system is periodic in the $x$ direction but has two edges in the $y$ direction. The Harper equation for solving the energies of edge states is derived. We find that there are two edge states in each bulk energy gap, corresponding to two zero points of the Bloch function on the complex-energy Riemann surface (RS). The edge-state energy loops parametrized by the momentum $k_{x}$ cross the holes of the RS. When the Fermi energy lies in the bulk energy gap, the quantized Hall conductance is given by the winding number of the edge states across the holes, which reads as $\sigma_{xy}^{\text{edge}}$=$-\frac{e^{2}}{h}% $sgn$(\sin\phi) $, where $\phi$ is the spin chiral parameter (see text). This result keeps consistent with that based on the topological bulk theory.