On averaging the Kubo-Hall conductivity of magnetic Bloch bands leading to Chern numbers

The authors re-examine the topological approach to the integer quantum Hall effect in its original form where an average of the Kubo-Hall conductivity of a magnetic Bloch band has been considered. For the precise definition of this average it is crucial to make a sharp distinction between the discrete Bloch wave numbers k1, k2 and the two continuous integration parameters α1, α2. The average over the parameter domain 0 ≤ αj < 2π, j = 1, 2 of the Hall conductivity of the band is a sum of integrals, each associated with different discrete k1, k2. They show how this can be transformed into a single integral over the continuous magnetic Brillouin zone 0 ≤ αj < 2πnj, j = 1, 2, nj = number of unit cells in j-direction, keeping k1, k2 fixed. This average prescription for the Hall conductivity of a magnetic Bloch band is exactly the same as the one used for a many-body system in the presence of disorder