ADIABATIC CHARGE-TRANSPORT AND TOPOLOGICAL INVARIANTS FOR ELECTRONS IN A QUASI-PERIODIC POTENTIAL AND A MAGNETIC-FIELD

An analysis is made of charge transport induced by adiabatic variations of the phases appearing in a quasi-periodic potential acting on non-interacting electrons. It is proved that this charge transport is quantised, when the chemical potential lies in a gap. The corresponding integers are topological invariants (first or higher Chern numbers), which label the gaps. Similar results are obtained for the Hall conductivity and charge transport of electrons in a periodic potential, under the influence of an irrational magnetic field.