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 ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$. We find that these vortices are given by the edge states when they are degenerate with the bulk states.