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Chern number and edge states in the integer quantum Hall effect. (English) Zbl 0972.81712
Summary: 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 $$\sigma_{xy}$$. We find that these vortices are given by the edge states when they are degenerate with the bulk states.

##### MSC:
 81V80 Quantum optics
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