 Chern number and integer Hall conductivity

For filled, separated, noninteracting magnetic Bloch bands, the Berry curvature integral over the magnetic Brillouin zone is an integer Chern number $C$, giving $|{\sigma }_{xy}|=|C|e^2/h$. Patches of Berry connection meet through phases with integer windings; continuous gap-preserving changes cannot change $C$. D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs developed this band argument; it does not establish the interacting fractional case.^[ref]