Abstract
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 σxy. We find that these
vortices are given by the edge states when they are degenerate with
the bulk states.
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