TKNN invariant

Summary

The TKNN invariant of a filled energy band of a two-dimensional crystal is the first Chern number of the band. Here the band is viewed as a line bundle over the Brillouin zone torus. David Thouless, Mahito Kohmoto, Peter Nightingale and Marcel den Nijs studied it in 1982. They showed that the Hall conductance of a band insulator is $e^2/h$ times the sum of these integers over the filled bands. This explains why the conductance in the integer quantum Hall effect is an exact integer multiple of $e^2/h$. In geometric terms it is the integral of the first Chern class of this line bundle over the torus.