Quantised hall conductance in a perfect crystal

Using magnetic translation symmetry, the Hall conductance of an isolated magnetic band in units of e²/h is shown to satisfy the Diophantine equation p σ +qm=1, where p and q are relatively prime integers giving the number of flux quanta per unit cell area, φ=p/q, and m is an integer. This equation holds for a general periodic Schrodinger Hamiltonian with an arbitrary magnetic field and is a direct consequence of the q-fold degeneracy of magnetic bands. Extension to general real phi gives the equation φσ_H-ρ =integer with σ_H the Hall conductance and ρ the number of electrons per unit cell, from which sσ_H is uniquely determined once rho, phi and the gap structure are given.