Impurity effect, degeneracy, and topological invariant in the quantum Hall effect

We discuss the periodic boundary condition, degeneracy of the ground state, impurity effect, and topological invariant in the quantum Hall effect. If a two-dimensional electron Hall system with a uniform background is confined on a toroidal geometry, the ground state is q-fold degenerate at filling factor ν=p/q when there is an energy gap Δ. Weak impurities make the ground state quasi- degenerate. If the driving force of the Hall current has some low but finite speed, or the applied electric field has some finite frequency ω satisfying Δi<Latin small letter h with strokeω Δ where Δi characterizes the effect of impurities, the Hall conductance is still expressed in the first Chern number on the torus. An explicit calculation shows that the weak impurities do not change this topological invariant.