Chern number and edge states in the integer quantum Hall effect

We consider the integer quantum Hall effect on a square lattice in a uniform rational magnetic field. The relation between two different interpretations of the Hall conductance as topological invariants is clarified. One is the Thouless--Kohmoto--Nightingale--den Nijs (TKNN) integer in the infinite system and the other is a winding number of the edge state. In the TKNN form of the Hall conductance, a phase of the Bloch wave function defines U(1) vortices on the magnetic Brillouin zone and the total vorticity gives ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$. We find that these vortices are given by the edge states when they are degenerate with the bulk states.