Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function

We study edge states in the integral quantum Hall effect on a square lattice in a rational magnetic field φ=p/q. The system is periodic in the y direction but has two edges in the x direction. We have found that the energies of the edge states are given by the zero points of the Bloch function on some Riemann surface (RS) (complex energy surface) when the system size is commensurate with the flux. The genus of the RS, g=q-1, is the number of the energy gaps. The energies of the edge states move around the holes of the RS as a function of the momentum in the y direction. The Hall conductance ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$ is given by the winding number of the edge states around the holes, which gives the Thouless, Kohmoto, Nightingale, and den Nijs integers in the infinite system. This is a topological number on the RS. We can check that ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$ given by this treatment is the same as that given by the Diophantine equation numerically. Effects of a random potential are also discussed.