= Chern number and integer Hall conductivity
{c}
{title2=$C=\frac1{2\pi}\sum_{n\ \mathrm{occupied}}\int\Omega_n(\mathbf k)\,d^2\mathbf k$}
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.<Quantized Hall Conductance in a Two-Dimensional Periodic Potential by D. J. Thouless and others>[pp. 405–407, Kubo formula and phase winding]{ref}
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