We re-examine the charge transport induced by a weak electric field in two-dimensional quantum Hall systems in a finite, periodic box at very low temperatures. Our model covers random vector and electrostatic potentials and electron–electron interactions. The resulting linear response coefficients consist of the time-independent term σxy corresponding to the Hall conductance and the linearly time-dependent term γsy · t in the transverse and longitudinal directions s=x,y in a slow switching limit for adiabatically applying the initial electric field. The latter terms γsy · t are due to the acceleration of the electrons by the uniform electric field in the finite and isolated system, and so the time-independent term σyy corresponding to the diagonal conductance which generates dissipation of heat always vanishes. The well-known topological argument yields the integral and fractional quantization of the averaged Hall conductance [Formula: see text] over gauge parameters under the assumption that there exists a spectral gap above the ground state. In addition to this fact, we show that the averaged acceleration coefficients [Formula: see text] vanish under the same assumption. In the non-interacting case, the spectral gap between the neighboring Landau levels persists if the vector and the electrostatic potentials together satisfy a certain condition, and then the Hall conductance σxy without averaging exhibits the exact integral quantization with the vanishing acceleration coefficients in the infinite volume limit. We also estimate their finite size corrections. In the interacting case, the averaged Hall conductance [Formula: see text] for a non-integer filling of the electrons is quantized to a fraction not equal to an integer under the assumption that the potentials satisfy certain conditions in addition to the gap assumption. We also discuss the relation between the fractional quantum Hall effect and the Atiyah–Singer index theorem for non-Abelian gauge fields.
Scaling in Plateau-to-Plateau Transition: A Direct Connection of Quantum Hall Systems with the Anderson Localization Model
