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 l attice in a rational magnetic field phi = 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 surfac e) when the system size is commensurate with the flux. The genus of th e 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 mome ntum in the y direction. rhe Hall conductance sigma(xy) is given by th e 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 sigm a(xy) given by this treatment is the same as that given by the Diophan tine equation numerically Effects of a random potential are also discu ssed.