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 sigma +qm=1, where p and q are relatively prime integers giving the number of flux quanta per unit cell area, phi =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 phi sigma H- rho =integer with sigma H the Hall conductance and rho the number of electrons per unit cell, from which sigma H is uniquely determined once rho , phi and the gap structure are given.